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without a f f e c t i n g primal
However, i f X2 i s chosen w i t h deg X2 > 3, b o t h
primal f e a s i b i l i t y and t h e value o f
w i l l be a f f e c t e d . p r i m a l s o l u t i o n i n (7.2)
Thus t h e general
is x1 = 5
and
x2 ,< 3.
(7.5)
S i m i l a r l y a p p l y i n g s e n s i t i v i t y a n a l y s i s t o t h e dual s o l u t i o n i n (7.3) we f i n d t h e is
general dual s o l u t i o n i n (7.2)
y1 = 3
(7.6)
and y2 s 0.
Obviously, t h e d e t a i l s o f t h e s e n s i t i v i t y a n a l y s i s w i l l v a r y i n d i f f e r e n t numerical examples, b u t f o r any p a i r o f dual extremal L.P's such as (7.1), t h e primal and t h e dual L.P. over
both
o f t h e corresponding symmetric-dual p a i r (7.2)
Z(z) w i l l have f e a s i b l e s o l u t i o n s .
F o r t h e p r i m a l L.P. always has t h e
o r i g i n a s a f e a s i b l e p o i n t , and a f e a s i b l e s o l u t i o n t o t h e dual L.P.
i s always
o b t a i n a b l e by a s s i g n i n g values o f s u f f i c i e n t l y h i g h degree t o a l l v a r i a b l e s . Hence f o r any g i v e n choice o f t h e u n i t parameters
J,
g. these two L.P.'s
always
have optimal f e a s i b l e s o l u t i o n s , w i t h equal o p t i m a l o b j e c t i v e f u n c t i o n values Pmax ( T Y ? )
(say).
Now, f o r t h e two extremal L.P.'s
over
f o l l o w i n g weak d u a l i t y p r i n c i p l e [15]:
2,
i t i s very s t r a i g h t f o r w a r d t o prove t h e
Using fields for semiring computations
67
No f e a s i b l e v a l u e o f t h e p r i m a l o b j e c t i v e f u n c t i o n can exceed any f e a s i b l e v a l u e o f t h e dual o b j e c t i v e function. I t f o l l o w s t h a t t h e v a l u e o f deg Pmax(~,g) i s independent o f t h e c h o i c e o f values
f o r t h e u n i t parameters, e l s e we c o u l d f i n d two s e t s of u n i t parameters (C,CI),
(!',ti) such t h a t o v e r 2 ( z ) t h e dual o b j e c t i v e f u n c t i o n c o u l d a t t a i n t h e v a l u e Pmax (2,g) and t h e p r i m a l o b j e c t i v e f u n c t i o n t h e v a l u e Pmax
(y',!')
>
deg Pmax
(s,$),
(TI,!.')
w i t h deg Pmax
which would v i o l a t e t h e weak d u a l i t y p r i n c i p l e o v e r
Thus we have used c o n v e n t i o n a l L.P.
2.
t h e o r y o v e r t h e o r d e r e d f i e l d Z(z) as a con-
v e n i e n t d e m o n s t r a t i o n o f t h e s t r o n g d u a l i t y p r i n c i p l e f o r a p a i r o f d u a l extremal o v e r 2 D5]:
L.P.'s
Both extremal L.P.'s
have an o p t i m a l f e a s i b l e s o l u t i o n , w i t h e q u a l i t y o f o p t i m a l
o b j e c t i v e f u n c t i o n value.
8.
ILLUSTRATION: EXTREMAL QUADRATIC PROGRAMMING
O p t i m i s a t i o n problems o v e r (2, @ , 8 ) i n v o l v i n g q u a d r a t i c o b j e c t i v e f u n c t i o n s do n o t seem t o have f i g u r e d p r o m i n e n t l y i n t h e l i t e r a t u r e . As an example o f such a problem, c o n s i d e r m in
2 @x ( ~@ ) 2
subject t o
8x Qy
@3
@ 2 @ x @y
(8.1)
x @ 3 @ y = 1.
Over I ( z ) we c o n s i d e r m in
Z'X2
subject t o
X
t
+
Z2XY
z3Y =
+
+
z3Y2 t z2x
Y
ITZ
X>,O,Y>,O
where
'TI
i s a u n i t parameter.
N o t i n g t h a t t h e Hessian m a t r i x o f t h e minimand, i.e.
i s p o s i t i v e - d e f i n i t e , we know t h a t t h e f o l l o w i n g Kuhn-Tucker C o n d i t i o n s (8.4), (8.5),
(8.6) g i v e s u f f i c i e n t c o n d i t i o n s f o r a minimum:
2z2x
+
z 2 ~t
x +
z2Y t
22 t h
2 z 3 ~+ 1 z3Y
-
ITZ
+
-
p = 0
-
2 3 ~
q =
o
= o
(8.4)
R.A. Cuninghame-Green
68
x >,o,
Y
>,o
(8.5)
p>,o,q>,o
px = qY = 0
(8.6)
Here (8.4) are the s t a t i o n a r i t y conditions; (8.5) the non-negativity conditions on the variables X , Y and the Lagrange multipliers p,q; and (8.6) the usual complementari t y conditions.
a i s the Lagrange multiplier f o r the equality constraint. These conditions a r e s a t i s f i e d by X = O ;
-2
Y =nz
= z2tn-2nZ-2-z-3;
=
o
(8.7)
1 = -2nz-'-z-3
whence x = --;y = -2; giving an otpimal objective function value of -1. Sensitivity analysis shows t h a t x may r i s e i n value t o 1 before i t violates the equation constraint in (8.1), b u t may only r i s e to value t o -3 before i t begins t o increase the object function value. Hence the general optimal solution i s :
x
d
-3, y = -2.
In general, a minimising extremal quadratic programing problem with linear equation and/or inequality constraints, such as (8.1), may be solved via a corresponding problem such as (8.2) over Z ( z ) , provided the l a t t e r problem has p o s i t i v e definite Hessian. For then i t can be shown by standard algebraic arguments t h a t the Kuhn-Tucker conditions are s u f f i c i e n t f o r a solution, and these conditions may be solved by L.P. derived processes such a s Wolfe's method, or principal pivoting. But,if we consider e.g. mi n
x ( 2 ) 0 3 @ x @Y 0 2 QY(2)
02 @ x O y
subject t o x @ 3 @ y = 1 ,
(8.8)
then i t i s easily seen t h a t the otpimal solution i s again x \< -3, y = -2. However, the corresponding problem over 2(z) now has non-positive-definite Hessian and leads us into the realms of non-convex quadratic programming.
9
ILLUSTRATION: THEORY OF POSITIVE MATRICES
Matrices over ( I , @ .@ ) correspond t o matrices over Y ( z ) with non-negative entries. The theory of such matrices i s , of course, highly developed and we can
Using fields for semiring computations
69
f i n d i n t e r e s t i n g correspondences between t h i s theory and t h a t o f matrices over
(2, 0 , Q ), by using the isomorphism (3.10). For example a square p o s i t i v e m a t r i x always has a Perron root, i.e. p o s i t i v e eigenvalue w i t h associated p o s i t i v e eigenvector.
a greatest
Hence a square m a t r i x
over (2, @ ,@)) always has a f i n i t e l y soluble eigenvector-eigenvalue problem ( i n f a c t t h e eigenvalue i s unique [6]). Again, from the theory o f i t e r a t i v e schemes r e l a t e d t o maximal-path-finding problems over
(2, @ ,6 )
i t i s known that, i f a m a t r i x a has no p o s i t i v e cycles,
then there e x i s t s a m a t r i x y s a t i s f y i n g (9.1 1
a@v@i = Y where i i s the i d e n t i t y matrix. To deduce t h i s r e s u l t using the isomorphism (3.10) we f i r s t remark t h a t i t i s s t r a i g h t f o r w a r d t o show t h a t i f a has no
p o s i t i v e cycles then the m a t r i x I - A , where A i s t h e corresponding m a t r i x over
I ( z ) and I i s the i d e n t i t y matrix, has p o s i t i v e p r i n c i p a l minors. known theorem f o r p o s i t i v e matrices and i s p o s i t i v e .
Thus
[lo]
=
Then by a w e l l -
r (say) e x i s t s
r i s p o s i t i v e and s a t i s f i e s Ar
Hence the corresponding m a t r i x
10.
we know t h a t (I-A)-'
Y
+I
=
r.
(9.2)
e x i s t s and s a t i s f i e s (9.1).
ILLUSTRATION: POWER SERIES OVER (IR, @ ,@ )
I n n 7 ] , we studied the properties o f power-series bo @ bl 0 x 0 b2@ x(') @ over (IR,
0 ,@ )
showing t h a t they converged f o r x
...
< P
(10.1)
and diverged f o r x >
p
where p = lim ~
-'
i n f br -r
(10.2)
Via the isomorphism (3.10), P corresponds t o the usual radius o f con.vergence -1 /r l i m i n f larl (10.3) rbr Of power-series w i t h c o e f f i c i e n t s ar = 2 The theory may be extended t o power-series o f a square m a t r i x a over (R, @ bo @ bl @ a @ which [17]
converges (resp. diverges) i f h(a)
the (necessarily unique) eigenvalue o f a.
... < p
,@ ) : (10.4)
(resp. A(a)
> P)
where x(a) i s
I n the l i g h t o f the present theory,
R.A. Cuninghame-Green
70
t h i s r e s u l t corresponds t o the convergence (resp. divergence) o f a m a t r i x powers e r i e s w i t h p o s i t i v e c o e f f i c i e n t s , f o r a p o s i t i v e m a t r i x whose Perron r o o t l i e s i n s i d e (resp. outside) the c i r c l e o f convergence o f the s c a l a r power-series.
11.
THE METHOD I N GENERAL
If ( G , @ ) i s a t o t a l l y ordered group and .I i s an i n t e g r a l domain, we may by standard algebraic methods c o n s t r u c t the group-ring Io(G) = I 1 (say). Then I l is again an i n t e g r a l domain, t o t a l l y ordered i f . I
is.
(We may thus continue the
... .
sequence Io , ,I ) l
(i,@,
Arguing exactly as i n Section 3 and i n [ll],we may show t h a t @ ), i.e. ( 6 , max, @ ) i s isomorphic t o t h e algebra o f equivalence classes o f t h e p o s i t i v e
6
cone o f t h e q u o t i e n t f i e l d F1 o f 11,where
= G
UI--l.
For the case when I. and G a r e Y, the z-method i s simply a convenient and i n t u i t i v e method o f c a r r y i n g o u t t h e c o n s t r u c t i o n o f F1, which i s i n t h i s case z(Z). For general p1 there i s a mapping d:F1
+
E
analogous t o the mapping deg, i . e .
d(X) e G f o r X # 0 d(0) =
-
(11 .l)
d(ab) = d(a) @ d ( b ) d(a + b)
d(a) @ d ( b )
The mapping d i n e f f e c t defines a non-Archimedean v a l u a t i o n
on F l .
It i s a
c l a s s i c a l r e s u l t f o r f i e l d s F~ so constructed t h a t they have a topological completion i n f i e l d s of formal Laurent series, which f o r t h e z-method a r e j u s t c l a s s i c a l Laurent s e r i e s w i t h i n t e g r a l c o e f f i c i e n t s and a t most a f i n i t e number o f terms of p o s i t i v e power. t o the valuation. [l]
Such Laurent s e r i e s are always convergent r e l a t i v e
The Giffler-Wongseelashote method consists e s s e n t i a l l y o f a d i r e c t construction o f generalised Laurent s e r i e s as well-ordered sequences o f elements o f a given z r b i t r a r y t o t a l l y ordered group G, as discussed i n [lZ] when .I
,
[15].
For t h e case
and G are 2, the two methods a r e t h e r e f o r e equivalent.
F i n a l l y , i f we c a r r y o u t the c o n s t r u c t i o n described above, f o r the dual s t r u c t u r e (G, @',Q ) = (G,min, 0 ) we o b t a i n the same q u o t i e n t f i e l d , and t h e mechanics o f
Using fields for semiring computations t h e isomorphism, v a l u a t i o n and c o m p l e t i o n a r e analogous.
71 We make use o f such a
dual c o n s t r u c t i o n i n t h e n e x t s e c t i o n .
12.
ILLUSTRATION: (SEMI) GROUP MINIMISATION PROBLEM
Suppose we a r e g i v e n c e r t a i n
m.
J
E
costs (j = 1
,...,n )
(0 < ml
f o r c e r t a i n elements o f an a b e l i a n semigroup ($,
$
g. E J
(j = 1
$,
We w i s h t o f i n d t h e cheapest element o f
<
m2 < ...)
(12.1)
0)
,..., n ) .
(12.2)
c o n s t r u c t i b l e as a p r o d u c t o f t h e
g i v e n g . ’ s a t g i v e n c o s t s m . which s h a l l l i e i n a g i v e n f e a s i b l e s e t C s J J’ t h u s we w i s h t o f i n d x j E 7 t o minimise
$:
Zm.x. J J X
g1x1 Q
subject t o
... @ g n n E
x. a 0 J
(12.3)
C
( j = 1,
Using t h e z-method we may c o n s t r u c t t h e semigroup r i n g t z SkZPk ( S k E $;pk E I) k=l
...,n )
a ( $ )o f
f o r m a l expressions (12.4)
The element 1 o f 7 ( $ )where
c
n
=
c g.z
j=1
m, (12.5)
J
c o l l e c t s each g e n e r a t o r g
and i t i s easy t o see t h a t labelled with i t s cost m j’ j’ r a i s i n g Z t o a power q ( s a y ) produces a sum l i k e (12.4) which c o l l e c t s each
element o f
$ o f t o t a l power q i n { g j } , l a b e l l e d w i t h i t s t o t a l c o s t .
Hence i f we expand ( l - Z ) - ’
as a f o r m a l geometric s e r i e s ( 1 4 p = 1
+c
t
z* +
...
(12.6)
and t h e n r e a r r a n g e i n ascending power o f z, we s h a l l t h e r e b y pr;oduce a l i s t i n g o f elements of
$ generated by t h e g
j’
w i t h associated cost, l i s t e d i n increasing
order o f cost. Mechanically, t h i s i s most e a s i l y done by f o r m a l l o n g - d i v i s i o n :
R.A. Cuninghame-Green
72
1-glzl-
1+g1z1+.
....
1-g z 1 1 g121+
..... .....
". 11
(12.7)
etc. The d i v i s i o n process i s simply terminated a t t h e f i r s t - o c c u r r i n g term i n t h e q u o t i e n t whose c o e f f i c i e n t belongs t o C;
i t s exponent then g i v e s t h e minimum
cost. The q u o t i e n t i s , o f course, a forma
Laurent s e r i e s i n the completion o f t h e
belongs, and a l l formal manipulations a r e eas 1Y j u s t i f i e d i n the l i g h t o f t h e c l a s s c a l arguments discussed i n Section 1 f i e l d El,
t o which ( l - z ) - '
In [16] we described a method o f so v i n g I n t e g e r P r o g r a m i n g problems whose j u s t i f i c a t i o n i s o f e x a c t l y t h i s nature.
REFERENCES van der Waerden, B.L., Roy, B.,
Modern Algebra (Frederick Ungar Pub. Co.,
T r a n s i t i v i t e e t connexite', C.R.
1953).
Acad. Sci. P a r i s 249 (1959) 216-218
Cuninghame-Green, R.A., Process synchronisation i n a steelworks - a problem of f e a s i b i l i t y , ( i n : Proceedings of t h e 2nd I n t e r n a t i o n a l Conference on Operational Research, E n g l i s h U n i v e r s i t y Press (1960) 323-328. Bellman, R. and Karush, W., On a new f u n c t i o n a l transform i n a n a l y s i s : t h e maximum transform, B u l l . h e r . Math. SOC. 67 (1961) 501-503. Yoeli, M., A n o t e on a g e n e r a l i z a t i o n o f boolean m a t r i x theory, Amer. Math. Monthly 68 (1961) 552-557. Cuninghame-Green, R.A., Describing i n d u s t r i a l processes w i t h i n t e r f e r e n c e and approximating t h e i r steady-state behaviour, Operational Res. Quart. 13 (1962) 95-100. G i f f l e r , B., Scheduling general p r o d u c t i o n systems using schedule algebra, Naval Res. L o g i s t . Q u a r t . 10 (1963) 237-255. Hoffman, A.J., On a b s t r a c t dual l i n e a r programs, Naval Res. L o g i s t . Q u a r t . 10 (1963) 369-373. G i f f l e r , B., Schedule algebra: a progress r e p o r t , Naval Res. L o g i s t . Q u a r t . 1 5 (1968) 255-280. Nikaido, H.,
Convex S t r u c t u r e s and Economic Theory (Academic Press, 1968).
Borawitz, W.C., Asymptotische reeksontwikkelingen i n minimax algebra toegep a s t op netwerkproblemen, Bachelor's t h e s i s (T.H. Twente, Netherlands, 1975).
Using fields for semiring computations
73
[12] Wongseelashote, A., Path algebras: A multiset-theoretic approach, Ph.D. thesis (University o f Southampton, 1976). [13] Cuninghame-Green, R.A., Minimax Algebra (Lecture Notes in Economics and Mathematical Systems No. 166, Springer-Verlag, 1979).
[14] Cuninghame-Green, R.A. and Meier, P.F.J., An algebra for piecewise-linear minimax problems, Discrete Appl. Math 2 (1980)267-294. [15]
Zimnermann, U., Linear and combinatorial optimization in ordered algebraic structures (Annals o f Discrete Mathematics 10, North-Holland, 1981).
[16] Cuninghame-Green, R.A. Integer programming by l o n g division, Discrete Appl. Math. 3 (1981) 19-25.
[17] Cuninghame-Green, R.A. and Huisman, F., Convergence problems in minimax algebra, Journ. Math. Anal. & Appl. 88,l (1982) 196-203. [18] Cuninghame-Green, R.A., The characteristic maxpolynomial o f a matrix, Journ. Math. Anal. 8 Appl. 95, 1 (1983) 110-116.
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Annals of Discrete Mathematics 19 (1984) 75-82 0 Elsevier Science Publishers B.V. (North-Holland)
75
SCHEDULING BY NON-COMMUTATIVE ALGEBRA
W F
R A Cuni nghame-Green
Borawitz
de D u l f 37 8918EB Leeuwarden The Netherlands
Department o f Mathematics U n i v e r s i t y o f Birmingham Birmingham 815 2TT England
The connection i s discussed between t h e use o f t h e m a t r i x i t e r a t i o n v = U ~ D T B V over t h e semiring (zuI-), min, +) = (2,@, @), aTid tfie use o f t h e z-transform, i n r e l a t i o n t o network scheduling i n v o l v i n g a s i n g l e vehicle. By i n t r o ducing non-commuting v a r i a b l e s t h e ideas can be extended t o producing e f f i c i e n t i t i n e r a r i e s i n v o l v i n g scheduled interchanges o f t r a v e l l e r s among t h e v e h i c l e s o f a m u l t i v e h i c l e t r a n s p o r t system.
1.
THE EARLIEST ARRIVALS
Suppose 4 towns X1,X2,X3,X4
a r e served by a c i r c u l a r bus r o u t e as shown i n F i g . 1,
t h e t r a n s i t times being: from X1 t o X2,
2 u n i t s ; from X2 t o X3, 3 u n i t s ; from X3
4 u n i t s ; from X4 t o X1, 1 u n i t . The m a t r i x 0, whose ( i , j ) t h element dij (i,j = 1, 4) i s the t r a n s i t time frm Xi t o X j along t h e a r c (Xi,Xj), o r else
t o X4,
...,
is
m
i f t h e r e i s no a r c (Xi,x.),
J
is:
(For convenience, we do n o t d i s t i n g u i s h n o t a t i o n a l l y between t h e physical system and t h e graph which represents i t . ) L e t us i n t r o d u c e 4 buses i n t o the system, one each a t : X1 a t time u1 = 7; X2 a t time u2 = 9; X3 a t time u3 = 1; X4 a t time u4 = 7.
The buses then c i r c u l a t e .
What i s t h e e a r l i e s t time vi a t which a bus w i l l be a v a i l a b l e a t each Xi?
Appeal-
i n g t o the o p t i m a l i t y p r i n c i p l e , we have: vi = min(ui,
min (vk+dki)) k = l , ,4
.. .
I n the n o t a t i o n of the semiring (ZUIml,min,+),
( i = 1,...,4)
notated as (?,@,@),
) = 1 vi = ~ ~ @ ( ( v , @ d , ~ ) @ . . . @ ( v ~ @ d ~ ~ )(i
,...,4)
(1.2) i s (1.3)
R.A. Cuninghame-Green and W.F. Borawitz
76
So, i n m a t r i x - v e c t o r n o t a t i o n i t i s t h e f a m i l i a r i t e r a t i o n v = U @ -
C
T
(FJV
-
where g, a r e t h e v e c t o r s w i t h components ui, vi r e s p e c t i v e l y ( i = 1 ,. .. ,4) and T D i s t h e transposed of D. (Our v e c t o r s a r e column-vectors.)
I f we c a l l
D T @ v-
y
t h e a v a i l a b i l i t y v e c t o r , g t h e i n i t i d l a v a i l a b i l i t y v e c t o r and
t h e a r r i v a l s v e c t o r then (1.4) says t h a t t h e a v a i l a b i l i t y o f a bus a t a
town i s e i t h e r by i n i t i a l a v a i l a b i l i t y a t t h a t town o r by a r r i v a l from another town where t h e bus was p r e v i o u s l y a v a i l a b l e . I t e r a t i o n s such as ( 1 . 4 ) have been discussed by many a u t h o r s
-
the extensive reference l i s t therein developed.
-
see e.g.
[5]
and
and may be solved by t h e methods they have
I n p a r t i c u l a r we consider t h e s o l u t i o n o f (1.4) by t h e z-method which
depends upon t h e isomorphism between classes o f p o s i t i v e elements o f t h e
(2, 0 , Q )and t h e a l g e b r a o f equivalence f i e l d z(z) o f r a t i o n a l expressions i n an
indeterminate z, w i t h i n t e g e r c o e f f i c i e n t s ( c f [6]
f o r (IUI--],max,+)).
Accordingly we must now solve, over Z ( z ) , t h e r e l a t i o n y
f o r the vector
y
>/
T = y + a y
0,where z*
0
A = -
Notice t h a t
= [dij]
i s a -_ d e f i n i t e m a t r i x i n t h e sense t h a t each c i r c u i t - s u m
d..ord. t...+d. JJ JlJ2 Jp-14 m t r i x !-:has
>
2) i s positive.
Then as discussed i n [6],
the
p o s i t i v e p r i n c i p a l minors and by a well-known theorem (I-A -) - ' e x i s t s Thus y 3 i n ( 1 . 7 ) and y i s i n t e r p r e t a b l e under t h e
and i s a p o s i t i v e m a t r i x . isomorphism.
(p
0
I f o p e r a t o r z e r counts zeroes a t t h e o r i g i n :
77
Scheduling bj, non-commutative algebra
(1.8)
More g e n e r a l l y , l e t us c o n s i d e r graphs w i t h n nodes.
From now on we assume t h a t
t i m e i s measured i n i n t e g r a l u n i t s r e l a t i v e t o some datum.
When t h e w e i g h t s dij
o f a complete d i r e c t e d weighted graph have a n i n t e r p r e t a t i o n i n terms o f t r a n s i t t i m e s r a t h e r t h a n d i s t a n c e s , we s h a l l speak o f a t r a n s i t graph and we s h a l l assume t h a t f o r such a graph dij
i s either
0=
[dij]
i s a positive matrix
o r a positive integer.
-
i . e . each element
F o r t h e o f f - d i a g o n a l elements o f
r e p r e s e n t s t h e reasonable assumption t h a t d i r e c t t r a n s i t f r o m Xi
this
t o X j (i# j )
i n t h e p h y s i c a l system i s e i t h e r i m p o s s i b l e o r e l s e t a k e s a f i n i t e amount o f time. F o r t h e d i a g o n a l elements i t may seem l e s s n a t u r a l , b u t i t r e p r e s e n t s t h e f a c t t h a t we assume e v e r y element o f
0refers
t o a motion, t a k i n g time.
p o s s i b i l i t y t h a t a v e h i c l e may move f r o m Xi
t o Xi
We a l l o w t h e
( i n d e e d i n S e c t i o n 6, we s h a l l
need t h i s p o s s i b i l i t y i n m o d e l l i n g p a r t i c u l a r s i t u a t i o n s ) b u t t h a t m o t i o n always t a k e s time, so dii
i s always p o s i t i v e .
Hence f o r such n-node systems i t e r a t i o n (1.4) may be s o l v e d by
v
2.
= zer(
(L-AT ) -1 w)
(1.9)
GIFFLER’S SERIES EXPANSION
I f we expand ( 1 - z l o ) - ’ as a f o r m a l g e o m e t r i c s e r i e s , (1.7) g i v e s
The f i r s t component o f t i m e s 6,7,8
...
y
shows t h a t some bus i s a v a i l a b l e a t Xi
The t e r m 2zI7 shows t h a t
two buses
a t each o f t h e
a r e p r e s e n t a t X1 a t t i m e 17.
So we see t h a t t h e components i n (2.1) a r e j u s t t h e z-transforms o f the a v a i l a b i l i t y processes a t t h e c o r r e s p o n d i h g nodes o f t h e graph, i n t h e f o l l o w i n g sense. Suppose t h a t a t a g i v e n town Xi, The sequence ao,a,, process a t Xi,
... i s
a,
buses w i l l t u r n up a t t i m e r ( = O , l , ...).
a t i m e s e r i e s which we may say d e f i n e s t h e a v a i l a b i l i t y
and t h e power s e r i e s a.
z - t r a n s f o r m o f a g i v e n t i m e s e r i e s ao,al,
+
alz
...
+
a2z2
+ ...
i s by d e f i n i t i o n t h e
I n an analogous way, and w i t h s e l f
e v i d e n t meanings, we may d e f i n e t h e z - t r a n s f o r m s o f t h e i n i t i a l a v a i l a b i l i t y
R.A. CuninghamcGreen and W.F. Bormvitz
78 process a t Xi
( i f v e h i c l e s are f e d i n t o t h e system v i a Xi
from o u t s i d e a t v a r i o u s
times); and o f t h e a r r i v a l process a t Xi. I n b r i e f : t h e elements o f
are r a t i o n a l f u n c t i o n s o f z , which as discussed i n [6] i n f i n i t e series.
may be expanded i n t o formal
I f elements o f w a r e the z-transforms o f t h e r e s p e c t i v e i n i t i a l
a v a i l a b i l i t y processes, then t h e expanded elements o f y a r e e x a c t l y t h e z-transforms o f t h e a v a i l a b i l i t y processes a t t h e r e s p e c t i v e nodes. The technique works n o t o n l y f o r " c i r c u l a r " graphs, b u t i n general f o r f o r k - f r e e t r a n s i t graphs where a graph i s d e f i n e d t o be f o r k - f r e e i f f o r each i = 1, t h e r e i s a t most one d . . < 1J
-
( j = 1,
a l l d i r e c t e d arcs f o r which dij
=
-
... ,n).
...,n
I f we d e l e t e from a f o r k - f r e e graph
then t h e remaining graph w i l l be c a l l e d t h e
r e s i d u a l graph o f t h e g i v e n f o r k - f r e e graph.
I n such a r e s i d u a l graph t h e r e may
be a confluence o f d i r e c t e d arcs b u t never a divergence; F i g . 2 i l l u s t r a t e s some t y p i c a l examples. I n what we have o u t l i n e d so f a r , we have e s s e n t i a l l y f o l l o w e d G i f f l e r ' s p i o n e e r i n g work [l] (1968), b u t we have used t h e f i e l d Z(z) r a t h e r than G i f f l e r ' s constructed f i e l d , thus r e l a t i n g t h e technique t o t h e f a m i l i a r one o f z-transforms. another approach, see Wongseelashote [3]
3.
For
.
NON-COMMUTING VARIABLES
Now suppose various c i t i e s a r e connected among themselves by several d i f f e r e n t t r a n s p o r t a t i o n systems.
Using the foregoing techniques, we may represent t h e
p a t t e r n s of a v a i l a b i l i t y o f each vehicle-system a t t h e v a r i o u s nodes.
A traveller
may j o i n o r leave a v e h i c l e a t a p a r t i c u l a r node o n l y a t a t i m e when the v e h i c l e i s a t t h a t node.
For mathematical convenience we assume t h a t a l l v e h i c l e s a r e
c o n s t a n t l y i n motion and a r e o n l y i n s t a n t a n e o u s l y a t any node.
A t t h a t moment a
t r a v e l l e r may leave, j o i n , o r change v e h i c l e s simultaneously a t t h a t node.
In
Section 4, however, we s h a l l consider some m o d e l l i n g techniques t o account f o r t h e f a c t t h a t t r a v e l l e r s and v e h i c l e s may a c t u a l l y
wait a t
v a r i o u s p o i n t s o f the
p h y s i c a l system. Suppose, then, t h a t we have m f o r k - f r e e graphs which a l l use t h e same nodes
XI.
...,X,
b u t have separate arc-systems.
a t a given time.
A t r a v e l l e r i s present a t a given node
How should he p l a n t o reach another given node as soon as
possible, using t h e v e h i c l e s and changing as necessary?
Scheduling b.v non-commutative algebra
We i n t r o d u c e a s e p a r a t e v a r i a b l e zh f o r each graph ( h = l,...,m)
79
and i n t h e
obvious way d e f i n e t h e i n i t i a l a v a i l a b i l i t y v e c t o r s w(zh), m a t r i c e s n ( z h ) and avai 1a b i 1it y v e c t o r s y ( z h ) = (L+(Zh)
T -1
) w(zh)
(h = l,***,m)
(3.1)
We l e t t h e v a r i a b l e zo r e p r e s e n t t h e t r a v e l l e r , w i t h i n i t i a l a v a i l a b i l i t y v e c t o r w ( z o ) , and l e t t h e unknown v e c t o r
1=
l(zo,zl
,. ..z,),
g i v e the transforms o f the
r e s u l t i n g a v a i l a b i l i t i e s o f t h e t r a v e l l e r , i n t h e f o l l o w i n g sense. of
Each component
w i l l be a ( u s u a l l y i n f i n i t e ) sum o f homogeneous f u n c t i o n s o f ~ ~ , z ~ , . . . , z ~
arranged i n ascending o r d e r o f t o t a l degree.
Each such homogeneous f u n c t i o n w i l l
be a term o r sum o f terms, each t e r m b e i n g a p r o d u c t o f powers o f z ~ , z ~ , . . . , z ~ and r e p r e s e n t i n g a p a r t i c u l a r way o f b e i n g a v a i l a b l e a t a p a r t i c u l a r t i m e and place.
Thus a p r o d u c t e.g. z ~ o z 1 2 z o( o f t o t a l degree 13) o c c u r r i n g i n t h e second
component o f v would r e p r e s e n t an a v a i l a b i l i t y a t X p a t t i m e 13 as a r e s u l t o f : s t a r t i n g a t t i m e 1 ( g i v i n g z:), ( g i v i n g z);
spending two t i m e - u n i t s on v e h i c l e - s y s t e m 1
and t h e n changing and spending 10 u n i t s o f t i m e on v e h i c l e - s y s t e m 2 N o t i c e t h a t t h e o r d e r o f f a c t o r s i n such a p r o d u c t has s i g n i f i -
( g i v i n g 221').
cance: we must work w i t h non-commuting v a r i a b l e s . Our t a s k i s t o determine
1 from
t h e o t h e r d a t a o f t h e problem.
To t h i s end we
d e f i n e a new a l g e b r a i c o p e r a t i o n . m
z
m
,...,zm)
,,...,
z gr (zo,z zm) be two s e r i e s o f terms r=o r=o w i t h fr,gr homogeneous o f t o t a l degree r ( r = O , l , ...) ; fr,gr may p o s s i b l y be Let a =
fr (zo,zl
z e r o o f course.
and B =
We d e f i n e : m
a08 =
When
1 fr(l,l, r=o
...,l ) g r ( z o . z l ,..-,zm)
(3.2)
a r e sums o f power-products, we may d e s c r i b e t h e a c t i o n o f t h i s o p e r a t i o n
by s a y i n g t h a t i n aoB, t h e terms o f B o c c u r each m u l t i p l i e d by t h e number o f power-products t h e r e a r e i n a o f t h e same t o t a l degree.
In particular the effect
i s t o remove any terms w h i c h a r e n o t matched by any non-zero t e r m o f t h e same t o t a l degree i n a . Further, i f
5, b
t h e p r o d u c t do!
a r e v e c t o r s whose components ai,Bi i s d e f i n e d componentwise
So, s i n c e y ( z l ) , ...,y( z),
-
i.e.
a r e s e r i e s such as a,B, t h e n
t h e ith component o f
gok
r e p r e s e n t t h e a v a i l a b i l i t i e s o f v e h i c l e s , and
i s aio6i.
1
r e p r e s e n t s t h e a v a i l a b i l i t i e s o f t h e t r a v e l l e r , we i n f e r t h a t y(zl)ol,...,.y(zm)o~ r e p r e s e n t those a v a i l a b i l i t i e s o f t h e t r a v e l l e r which a r e c o n s i s t e n t w i t h a departure, f r o m somewhere a t some time, by t h e r e s p e c t i v e v e h i c l e s .
Hence:
80
R.A. Cuninghamc-Greenand W.F. Borawitz T T ~ ( 2 1 )( l ( ~ l ) o v ) , . . . , a ( z m ) (Y(Zm)OV)
represent h i s p o s s i b l e a r r i v a l s by t h e r e s p e c t i v e v e h i c l e s .
Since a v a i l a b i l i t y
a r i s e s by i n i t i a l a v a i l a b i l i t y o r by a r r i v a l we i n f e r t h e f o l l o w i n g i m p l i c i t relationship f o r
v:
v = w(zo) + ; ( L \ ( z h ) T i ( ( r - a ( z h ) T ) - l w ( Z h ) ) o v } ) h -
We may s o l v e such r e l a t i o n s i t e r a t i v e l y o r r e c u r s i v e l y f o r
4.
(3.3)
1as i l l u s t r a t e d next.
AN EXAMPLE
Suppose t h a t two c i t i e s C1 and C 2 a r e served by two separate t r a n s p o r t a t i o n
-
systems
a c i r c u l a r bus r o u t e and a t r a i n s h u t t l e s e r v i c e .
The bus takes 4
time u n i t s t o t r a v e l from C1 t o C2, w a i t s a t C2 f o r 2 u n i t s , t r a v e l s back t o C1 i n 3 u n i t s , w a i t s a t C1 f o r 3 u n i t s , and then repeats t h e process. The t r a i n takes 2 t i m e u n i t s t o t r a v e l from C1 t o C2.
w a i t s a t C2 f o r 2 u n i t s ,
t r a v e l s back t o C1 i n 2 u n i t s , w a i t s a t C1 f o r 2 u n i t s , and then repeats t h e process. F i g . 3 shows how we m i g h t model t h i s s i t u a t i o n .
I n o r d e r t o account f o r t h e
time which each v e h i c l e w a i t s a t each c i t y , we r e p r e s e n t each c i t y Ci nodes X i and Yi.
t o Yi
We decree t h a t each v e h i c l e which v i s i t s Ci
by
two
" t r a v e l s " from Xi
i n a time equal t o i t s w a i t i n g t i m e a t Ci.
Suppose t h e r e i s j u s t one bus i n commission, which s t a r t s from Y 1 a t time 1 and one t r a i n which s t a r t s from Y1 a t t i m e 2. wishes t o v i s i t c i t y C2,
A t t i m e 0, a t r a v e l l e r i s a t Y,.
He
r e t u r n i n g n o t l a t e r than t i m e 10, having spent as much
t i m e as p o s s i b l e i n c i t y C2.
How does he proceed?
Now, unless two v e h i c l e s a r e simultaneously present a t a g i v e n c i t y , t h e t r a v e l l w wishing t o change f r o m one t o t h e o t h e r a t t h a t c i t y must w a i t t h e r e f o r an a p p r o p r i a t e time.
We may b r i n g such w a i t i n g w i t h i n t h e scope o f o u r method by
i n t r o d u c i n g a w a i t i n g l o o p a t each node Yi
which we may t h i n k o f as r e p r e s e n t i n g
an imaginary v e h i c l e which operates on a c i r c u l a r r o u t e from Yi one u n i t o f time per c i r c u i t . v e h i c l e when he w a i t s a t Yi. city.
t o i t s e l f taking
The t r a v e l l e r i s regarded as b e i n g "on" t h a t F i g . 4 shows t h e interchange arrangements a t each
The t r a v e l l e r may change v e h i c l e s a t Xi
i f t h e two v e h i c l e s f o r t u i t o u s l y
a r r i v e simultaneously; otherwise he t r a v e l s t o Yi on t h e " w a i t i n g - t i m e v e h i c l e " and stays i n a w a i t i n g l o o p u n t i l such t i m e as t h e v e h i c l e he wishes t o j o i n reaches Yi.
Because a l l " w a i t i n g - v e h i c l e ' ' t r a n s i t s t a k e u n i t time, he misses
Scheduling by non-commutative algebra
81
no o p p o r t u n i t i e s . I n (3.3) we have f o r t h e w a i t i n g ,
bus and t r a i n v e h i c l e s r e s p e c t i v e l y
To s o l v e (3.3) r e c u r s i v e l y we assume f o r t h e jth component o f m
the form
z
ajr(zo,
...,zm)
where ejr
v ~
an expansion o f
i s an unknown t e r m o f t o t a l degree r.
r=o 117ay t h u s develop t h e r i g h t - h a n d s i d e o f (3.3) i n terms o f t h e eir
We
and equate
J
terms of equal t o t a l degree i n t h e l e f t - h a n d and r i g h t - h a n d s i d e s t o d e r i v e a sequence o f r e c u r s i o n s w i t h r e s p e c t t o r among t h e
J on t h e r i g h t w i l l be p r o v i d e d by known f u n c t i o n s i n ~ ( z , ) ,
The lowest-degree terms g i v i n g us values f o r
t h e l o w e s t degree terms on t h e l e f t , and t h e r e c u r s i o n s may be s o l v e d systematically. A l t e r n a t i v e l y , d i s c u s s i n g (3.3) i n a condensed n o t a t i o n
v = w ( z o ) + ?(Z1, we may s o l v e i t e r a t i v e l y f o r
1 by
...,Zm
;
v)
(4.3)
t a k i n g an i n i t i a l s o l u t i o n
v0
(4.4
= w(zo)
and t h e n computing
vk+' = w(zo) + I ( z l
,...,z,
k
;1 )
( k = 0,1, . . . )
(4.5
I n b o t h methods t h e s c a l a r a l g e b r a i s m u l t i p l i c a t i v e l y non-commutative and r e g a r d must be p a i d t o t h e f a c t t h a t t h e m a t r i c e s i n ( 3 . 3 ) m u l t i p l y f r o m t h e l e f t . By e i t h e r method, we f i n d t h a t t h e terms o f t o t a l degree n o t exceeding 10 i n t h e f i r s t component o f
v are:
R.A. CuninghamcCreen and W.F. Borawitz
82
(Some o f these terms occur w i t h very l a r g e c o e f f i c i e n t as a r e s u l t o f m u l t i p l e a l t e r n a t i v e s a r i s i n g through t h e w a i t i n g - v e h i c l e "reproducing i t s e l f " i n t h e w a i t i n g loop; a l t e r n a t i v e m o d e l l i n g procedures w i l l a v o i d t h i s inelegance.) I f we examine these terms t o f i n d t h a t which has t h e h i g h e s t power o f z1 as i n t e r n a l f a c t o r , we f i n d t h a t i t i s 223z13z32z12, g i v i n g t h e f o l l o w i n g i t i n e r a r y ; T r a v e l l e r a r r i v e s a t Y1 a t t i m e 0, a l l o w s t h e bus t o d e p a r t a t t i m e 1, and catches t h e t r a i n which departs a t time 2.
On a r r i v a l a t c i t y C2 he stays t h r e e
u n i t s o f time, a l l o w i n g the t r a i n t o r e t u r n w i t h o u t him. a r r i v e s and he catches i t back t o c i t y C1
Meantime t h e bus
.
REFERENCES
G i f f l e r , B. Schedule algebra: a progress r e p o r t , Naval Res. L o g i s t . Q u a r t . 15 (1968) 255-280. Borawitz, W.C., A s m p t o t i s c h e reeksontwikkelinqen i n minimax alaebra toegepast op netweikproblemen, Bachelor's t h e s i s (T.H. Twente, i e t h e r l a n d s , 1975). Wongseelashote, A, An a l g e b a f o r determining a l l path-values i n a network w i t h a p p l i c a t i o n t o k - s h o r t e s t path problems, Networks 6 (1976) 307-334. Cuninghame-Green, R.A. Minimax Algebra (Lecture Notes i n Economics and Mathematical Systems No. 166, Springer-Verlag 1979). Zimnermann, U., L i n e a r and combinatorial o p t i m i z a t i o n i n ordered a l g e b r a i c s t r u c t u r e s Annals o f D i s c r e t e Mathematics 10 (North-Holland 1981). Cuninghame-Green, R.A., Using f i e l d s f o r semiring computations, Annals o f D i s c r e t e Mathematics ( t h i s volume), Chapter 5.
Annals of Discrete Mathematics 19 (1984) 83-98
83
0 Elsevier Science Publishers B.V. (North-Holland)
PSEUDO-BOOLEAN FUNCTIONS AND STABILITY OF GRAPHS
P.L. Hamner
Ch. Ebenegger
D. d e Werra RUTCOR Ecole P o l y t e c h n i q u e E. A.U.G. Rutgers ' Center f o r F e d e r a l e de Lausanne Uni v e r s i t 6 de Gensve O p e r a t i o n s Research DCpt. de Mathgmatiques CH-1200 GenPve Rutgers U n i v e r s i t y MA Ecublens Switzerland New Brunswick CH-1015 Lausanne NJ 08540 Switzerland USA I n t h i s oaper t $ e c o n c e n t o f c o n f l i c t g r a o h a s s o c i a t e d w i t h a pseudo-Boolean f u n c t i o n i s discussed; one e x p l o i t s t h e f a c t t h a t t h e problem o f f i n d i n g a s t a b l e s e t w i t h maximum w e i g h t i n a graph can be reduced t o t h e m a x i m i s a t i o n o f a pseudo-Boolean f u n c t i o n and c o n v e r s e l y . On these Boolean f o u n d a t i o n s a graph t h e o r e t i c a l procedure i s develooed asso c i a t i n g t o any graph a n o t h e r one h a v i n g a s t r i c t l y s m a l l e r s t a b i l i t y number. Fragmentary c o m o u t a t i o n a l e x p e r i e n c e seems t o show t h a t t h i s r e d u c t i o n may be a p p l i e d e f f i c i e n t l y i n a l g o r i t h m s f o r o b t a i n i n g t h e s t a b i l i t y number o f a graph. INTRODUCTION The purpose o f t h i s n o t e i s t o show how i n some cases Boolean methods can suggest graph t h e o r e t i c a l procedures which can o f c o u r s e be d e r i v e d a p o s t e r i o r i d i r e c t l y I n t h e n e x t s e c t i o n , t h e concept of con-
by u s i n g graph t h e o r e t i c a l p r o p e r t i e s .
f l i c t g r a p h a s s o c i a t e d w i t h a pseudo-Boolean f u n c t i o n w i l l be presented; t h e r e l a t i o n between t h e m a x i m i s a t i o n o f a pseudo-Boolean f u n c t i o n and t h e determinat i o n o f a s t a b l e s e t w i t h maximum w e i g h t i n a graph w i l l be developed. I n t h e case where a s t a b l e s e t w i t h maximum c a r d i n a l i t y
C(
must b e f o u n d ( u n w e i g h t d
case), we w i 11 d e s c r i b e a t r a n s f o r m a t i o n o f t h e c o r r e s p o n d i n g pseudo-Boolean f u n c t i o n ; t h i s t r a n s f o r m a t i o n amounts t o c o n s t r u c t i n g another graph h a v i n g a-1 as i t s s t a b i l i t y number.
Section 2 w i l l contain t h i s transformation w h i l e a d i r e c t
c o n s t r u c t i o n w i l l be d e s c r i b e d i n graph t h e o r e t i c a l terms i n S e c t i o n 3. The e q u i v a l e n c e o f these procedures w i l l be proven i n S e c t i o n 4 where one w i l l show how t h e procedure can be used i n t h e weighted case.
F i n a l l y i n S e c t i o n 5 some com-
p u t a t i o n a l experiments w i l l b e r e p o r t e d . The g r a p h t h e o r e t i c a l terms w i l l be borrowed f r o m d e f i n i t i o n s , t h e reader i s r e f e r r e d t o a node a ( a i s n o t i n N ( a ) ) .
m.
w h i l e f o r pseudo-Boolean
N(a) w i l l be t h e s e t o f neighbours o f
Furthermore, a l l graphs w i l l be assumed s i m p l e (no
l o o p s and no m u l t i p l e edges a r e a l l o w e d ) , so we w i l l s i m p l y speak o f graphs.
1
.
CONFLICT GRAPHS
We s h a l l b e i n t e r e s t e d i n t h i s s e c t i o n i n t h e o p t i m i s a t i o n o f a pseudo-Boolean
Cli. Ebenegger et al.
84
f u n c t i o n and we s h a l l c o n s i d e r a graph t h e o r e t i c i n t e r p r e t a t i o n o f t h i s problem. F i r s t we need some d e f i n i t i o n s .
. . ,xn)
A pseudo-Boolean f u n c t i o n f a s s o c i a t e s w i t h e v e r y n - t u o l e (xl ,x2,.
E
IO, 1 In
a r e a l value; i t i s known t h a t a pseudo-Boolean f u n c t i o n f can always be w r i t t e n i n a polynomial fonn, i . e .
f(x1,x2,
Ifa l l wi
>
. . . ,x n )
= K
+
nr
where T . =
1 w.T i=l
0 ( i = 1 ,... ,p), we say t h a t
1
P
r
wiTi
n
x.
n
xk
kEBi
jEAi
i s a posifonn.
j=l
...,xn)
Suppose we want t o maximise a p o s i f o r m f ( x l ,
o v e r { O , 1 ) " ; we may a s s o c i a t e
w i t h f a graph G c o n s t r u c t e d as f o l l o w s : f o r each Ti we i n t r o d u c e a node ai w i t h a w e i g h t wi
and we l i n k ai and a . i f t h e c o r r e s p o n d i n g terms Ti J
fl
.e. t h e r e e x i s t s sane i n d e x k
k
Bi,
E
11 ,...,n l such t h a t k
and T 1 a r e i n conE
x.J appears i n Ti
Ai h B j o r
( o r T . ) w h i l e x . appears i n T. ( o r J J J i s c a l l e d t h e c o n f l i c t graph of t h e p o s i f o r m f. Some i l l u s t r a t i o n s o f or equivalently
Ti c o n f l i c t graphs a r e g i v e n i n
[q.
I t i s c l e a r f r o m t h e c o n s t r u c t i o n o f G t h a t t h e maximum v a l u e o f f i s equal t o t h e
maximum w e i g h t o f a s t a b l e s e t i n G.
F i g u r e 1.1 shows a c o n f l i c t g r a p h correspon-
ding t o t h e posiform f(X,Y,Z,U,V)
= 2xu
+ 3xy + 7v; + 5y
2xu
t 62
t
+
xzi +
4zu + 4xiy
3yZ -_
3x4'
7UV
F i g u r e 1 .l. C o n f l i c t g r a p h of f(x,y,z,u,v)
+
Pseudo-boolean functions and stabi1it.v of graphs
85
We observe t h a t each v a r i a b l e x . o f f i s a s s o c i a t e d i n G w i t h a complete b i p a r t i t e J ( n o t n e c e s s a r i l y induced) subgraph whose nodes correspond on one s i d e t o a l l terms containing x
and on t h e o t h e r s i d e t o a l l terms c o n t a i n i n g
j
Conversely, i f we a r e g i v e n a graph G w i t h w e i g h t s wi
xj *
a s s o c i a t e d w i t h i t s nodes,
we may c o n s t r u c t a f u n c t i o n f as f o l l o w s : i n i t i a l l y we s e t Ai
= Bi
=
fl f o r each
node i o f G and we c o n s i d e r an a r b i t r a r y c o v e r i n g o f t h e edge s e t by complete b i p a r t i t e subgraphs G1 G .; t h e n we s e t J
,.. . ,Gq ; l e t
X -,X. be t h e 2 s e t s o f nodes c o r r e s p o n d i n g t o J J
Ai = I k B 1. = { k
n Finally f =
c
wiTi i=l
n
(where Ti =
n Xk)
x
jEAi
j ..EB
i s a p o s i f o r m h a v i n g G as i t s
i
conf 1 ic t graph. 2.
REDUCTION OF THE STABILITY NUMBER
We s h a l l now deal w i t h t h e problem o f f i n d i n g t h e s t a b i l i t y number a ( G ) o f a graph G = (X, U) w i t h node s e t X = {ao,a ly...,anl.
f(xO,xl
,. .. ,xn)
We can a s s o c i a t e w i t h G a p o s i f o r m
such t h a t max f ( x o,...,xn)
= a(G) 9
and t h a t we can f i n d a p o s i f o r m g ( x o ,..., xn) such t h a t f ( x o y ...,xn) = 1 g( xo,.
. . ,xn)
f o r a1 1 values o f xO,
+,
...,xn.
Construction o f f : 1)
L e t a.
be a n a r b t r a r y node and al,a2,
term To = 2)
x1
x2
..
xP w i t h
node a.
.
Furthermore f o r each neighbour ai o f a.
n
T. = x . 1
1.
j:a.EN(a-)
-I.1
...,aP
i t s neighbours.
We a s s o c i a t e t h e
we d e f i n e a term
X
j.
3)
J<1 F o r e v e r y r e m a i n i n g node ai o f G ( i > p) we i n t r o d u c e a term Ti = xi
4)
F i n a l l y we p u t f =
n
x
jEN( i ) j '
1
Ti.
i:a.EX 1
An example o f c o n s t r u c t i o n o f f i s g i v e n i n F i g u r e 2.1.
N o t i c e t h a t xo i s n o t
used i n t h i s c o n s t r u c t i o n , so we s h a l l w r i t e s i m p l y f ( x l,...,xn).
C/I.Ebenegger el al.
86
- -
) - - - - -
f = x 1x 2x 3x 4x 5 t x
+
x 5 XIx3
x1x2 t x3 + i 3 x 4 +
ili3X5
+ X3X4X6
F i g u r e 2.1 T h i s c o n s t r u c t i o n amounts t o c o v e r i n g t h e edge s e t o f G by s t a r s centered a t al ,a2
,..., a;,
f o r i = 1 ,...,p t h e s t a r centered a t ai covers [ai, a 4 with i
edges [ai,
c
j 4 p, w h i l e f o r i
>
aJ
and a l l
p t h e s t a r centered a t ai covers a l l
edges adjacent t o ai. Fran Section 1 we have: P r o p o s i t i o n 2.1.
Max f(xl
,...,xn)
= a(G)
Remark t h a t a depends o n l y on G, w h i l e f depends a l s o on t h e choice o f a.
..., aP
t h e p a r t i c u l a r order al,
o f i t s neighbourhood.
=1 t
I n order t o o b t a i n t h e posiform g such t h a t f t h e f o l l o w i n g i d e n t i t y w i t h il
-
x
-
x. il ' 2
...
xi
= 1
9
N o t i c e t h a t o n l y t h e terms Ti our a t t e n t i o n t o
-
and o f
<
i2 <
x
-
il
(i
D)
...
g, we w i l l r e p e a t e d l y apply
< i
9
-
-
x 'il 'i2 'il 'i2 i3
-
... -
w i l l b e m o d i f i e d so t h a t we can r e s t r i c t
Pseudo-booleanfunctions and stability of graphs
and show t h a t we s h a l l g e t a p o s i f o n n go(xl,
P x
,n
When r e p l a c i n g To =
J=1
...,xo)
such t h a t f o E 1 t go.
P n i.we o b t a i n i = l 1 j< i J
by 1 - z x -
j
ajEN( ai ) D
=
1 +
Denote ui = (1
-
t h e same i d e n t i t y as b e f o r e , we g e t
k< i akkN(ai) ui =
k z< i
slk
'k
ak&N(ai)
asCN(aif
Hence
since k
<
i we have
n
j
a.EN(ai) J
x
J
n
s
Xk)
k
j
ik). Using
n
ij(' n
n
c xi i=l
-
x
S
=
's
87
Ch. Ebenegger et al.
88
q
SO go(xl
,..., xp)
it and
n
jS
7
s.q
r 5 D.
q
,..., xp)
= fo(xl
-
1 i s a p o s i f o m and
n
g(xl
,..., x n )
= go(x
f(xl
,..., x n )
= 1 t
,,..., x p ) i. r: Ti i s a l s o a p o s i f o m w i t h i=p+l g ( x l ,..., x n ) . I n p a r t i c u l a r max g ( x l ,..., xn)
max f f x l , . . . ,xn)
-
Proposition.2.2
a(")
1.
I f we denote by = a(G)
=
G, t h e c o n f l i c t g r a p h o f go, t h e n we have
- 1.
For example, s t a r t i n g f r o m t h e g r a p h i n Fig. 2.1, t a k i n g as
- - - - -
t h e node "coded" as
x1x2x3x4x5, and c h o s i n g t h e n a t u r a l o r d e r o f t h e nodes i n t h e nei9hbourhood
of, , ,a
we g e t
- -
- - - - -
+ x1 + x1x2 + x 3 + X3x4 + x1x3x5 +
f = x x x x x = l + x
- -
1
- x1
+;x +X3+'3x4+XXX 1 2 1 3 5
-
-
x1x2
-
- -
+
( 1 - X1X2)X3 + ( 1
= 1
+
(xl + Z,X2)X3
= l + x x + ip.f.3
-
x1x2x3
1
=
i .
-
= 1
- - -
x1x2x3x4
+
-
'3'4'6
- - - -
x x x x x 1 2 3 4 5
X1X2)X3X4 + (1
+ (xl + X,X2)X3X4
-
-
x1x3x4 + x1x2x3x4
- +
-X 3 X-4 X 6
t
x2x4)x1x3x5 +
x
X3X4X6
x 3x 5
+ 2x 4 )X 1
- -
t
(x*
+
x1x2x3x5 + x x x x x 1 2 3 4 5 +
'3'4'6
+ T13
- - - -
-
+
'3'4'6
- - -
- -
T23 + T 1 4 + T 2 4 + TZ5 + T45 + x3x4x6
T h i s suggests t h a t s t a r t i n g from G and i t s a s s o c i a t e d f u n c t i o n f , we c o n s t r u c t a graph G ' c o r r e s p o n d i n g t o g
5
f
-
1, such t h a t a(G') = a(G)
-
1.
Such a d i r e c t
c o n s t r u c t i o n w i l l be d e s c r i b e d i n t h e n e x t s e c t i o n .
3. THE DIRECT CONSTRUCTION We s h a l l now d e s c r i b e a g r a p h t h e o r e t i c a l t r a n s f o r m a t i o n w h i c h a s s o c i a t e s w i t h a graph G ( w i t h n(G)
> 1 ) a n o t h e r g r a p h G ' such t h a t
a(G') = a(G)
-
1.
89
Pseirdo-boolean fiinctioris and stabi1it.v of'graphs Construction o f G ' : L e t a. al,
be an a r b i t r a r y node of G and l e t t h e neighbourhood o f a.
..., a , and P
let a
P+l
,...,an
s e t o f G ' w i l l c o n s i s t of ap+,
The node
c o n s i s t o f nodes
be t h e o t h e r nodes o f G.
,...,an,
as w e l l as o f a s e t o f "new"
nodes a . . ( 1 < j 6 p) a s s o c i a t e d t o a l l t h e p a i r s i , j o f non-adjacent nodes 1J a . , a . i n t h e neighbourhood o f ao. We s h a l l r e p r e s e n t t h i s s e t o f new nodes as 1 J b e i n g p a r t i t i o n e d i n t o " l a y e r i ' Li = l a . . ,..., a . . 1 c o n s i s t i n g o f a l l new nodes 'J1 lJk a . . h a v i n g i as t h e f i r s t i n d e x . 1J The edge-set o f G ' w i l l be d e f i n e d as c o n s i s t i n g o f ( 1 ) a l l t h e edges o f t h e subgraph o f G induced by ap+l,...,an; a. . ,a. . , (il# i,) '1J1 '2J2
( 2 ) a l l t h e edges l i n k i n g new nodes
b e l o n g i n g t o two d i f f e r e n t l a y e r s ; ( 3 ) edges l i n k i n g two
nodes a . . , a , . i n t h e same l a y e r i f a . and a . were l i n k e d i n G; ( 4 ) edges J2 1J1 'J2 J1 l i n k i n g a new node a . . t o a node a, 1J i n G.
( r > p f l ) i f ar was l i n k e d t o ai o r t o a . J
An example o f t h i s c o n s t r u c t i o n i s g i v e n i n F i g u r e 3.1 f o r t h e graph G o f F i g u r e 2.1; one v e r i f i e s t h a t a(G) = 3 and a ( G ' ) = 2 .
G'
F i g u r e 3.1 Whenever no c o n f u s i o n s can a r i s e , we s h a l l w r i t e ( i , j ) f o r a . . and i f o r ai. 1J Given any s t a b l e s e t S ' i n G ' , t h e r e e x i s t s a s t a b l e s e t S i n G
Proposition 3.1. with I S /
Proof. S = S'
=
I S ' I + 1.
i f S ' c o n t a i n s no new node, t h e n
L e t S ' be a s t a b l e s e t i n G ' ;
u
{ao? i s the required stable set.
necessarily o f the form ( i , j l ) , This means t h a t t i , j,, j,
s
( i , j,)
,..., j r i
= (5' - t ( i , jl),(i,
I f S ' c o n t a i n s new nodes, t h e y a r e
,..., ( i , j r )
w i t h j,
<
j, <
...
< jr.
i s a s t a b l e s e t i n G and
,..., ( i ,
j,)
i s a s t a b l e set o f G w i t h IS1 = I S ' I
+
1.
jr)l)
U {i,j,,
End o f o r o o f .
j2,...,jr)
Ch. Ebenegger eta!.
90 P r o p o s i t i o n 3.2.
Given any nonempty s t a b l e s e t S i n G, t h e r e e x i s t s a s t a b l e s e t
S ' i n G' w i t h I S ' / = I S 1
Proof.
-
1.
n (N(ao) u { a o ] ) ;
L e t S be a s t a b l e s e t i n G and So = S
i f So = 0, t h e n
b y removing any node of S, one g e t s a s t a b l e s e t S ' o f G ' w i t h I S ' I = ( S I
I f / S o l = 1, t h e n S ' = S
-
-
IS'I = IS/
So)U'(il.
-
1.
i2),(i,,
1.
So i s t h e r e q u i r e d s t a b l e s e t .
If So = !il,i2, ..., ir!w i t h r S' = (S
-
.,
2 and il
i 3 ) ,..., ( i l ,
i2 <
<
...
c
ir ' t h e n
i r ) ) i s a stable set o f G' with
End o f p r o o f .
As a consequence o f t h e above p r o p o s i t i o n s , we can s t a t e C o r o l l a r y 3.1.
F o r t h e graDh G' c o n s t r u c t e d from G we have a(G') = z(G)
-
1.
By r e p e a t e d l y a p p l y i n g t h i s c o n s t r u c t i o n , one may compute t h e s t a b i l i t y number o f G ( i n a t most d(G) 5 n s t e p s ) ; u n f o r t u n a t e l y t h e number o f nodes i s g e n e r a l l y
i n c r e a s i n g when t h e t r a n s f o r m a t i o n i s a D p l i e d .
However t h i s p r o c e d u r e combined
w i t h sane r e d u c t i o n s may g i v e an e f f i c i e n t a l g o r i t h m ; s e c t i o n 5 w i l l m e n t i o n s m e computational experiments.
4.
EQUIVALENCE
Our purpose i s t o show t h a t t h e r e d u c t i o n procedure d e s c r i b e d i n pseudo-Boolean terns i n Section 2 i s equivalent t o t h e c o n s t r u c t i o n o f G' g i v e n i n the previous section.
Consider t h e f u n c t i o n g ( x I,...,xn)
obtained i n Section 2 n
we w i l l determine t h e c o n f l i c t graph G a s s o c i a t e d w i t h g ( x l,...,xn).
Each T qr and each Ti corresponds t o a node o f G; so we see t h a t G has one node f o r each r ) o f neighbours o f a. w h i c h a r e n o t l i n k e d . These correspond t o p a i r q,r ( q t h e new nodes i n graph G' o f S e c t i o n 3. F u r t h e r m o r e G has nodes apel,ap+2,. . ,an
.
C l e a r l y t h e c o n f l i c t graph a s s o c i a t e d w i t h generated b y a p+l,.
IfZp+, Ti i s t h e subgraph
. . ,a n * T and Ti a r e i n > D) o f G. qr i f and o n l y i f ai i s l i n k e d t o
Consider now a new node (q, r ) and a node a: ( i c o n f l i c t ( i . e . (9. r ) and ai a r e l i n k e d i n either a
q
of G
6)
o r ar i n G s i n c e t h e y have t h e f o l l o w i n g form
91
Pseudo-boolean functions and stability ojgraphs
T . = x.
II a.EN(ai) J
1
F i n a l l y c o n s i d e r 2 terms Tij
and T
T.. = 1J
qr
x
>
p.
( w i t h i < j and q < r ) .
-
i~
X.X 1
with i
j
j sci S
i < t < jX t atEN( a .) J
and
a " W ar) t h e r e w i l l b e a c o n f l i c t i f and o n l y i f
xi
or
x.J appears
in T
(or
qr
xq
or
xr
appears i n T . . ) ; t h i s can o c c u r i f and o n l y i f e i t h e r i # q ( s o t h a t Xi appears 1J i n factor n of T if i < q or appears i n T . . i f i > q ) o r i f a j E N(ar) qr 9 1J u
x
xu
x
x
corresponding f a c t o r o f T . . i f j 1J
r).
>
These a r e p r e c i s e l y t h e c o n d i t i o n s under which t h e new nodes ( i . j) and (4, r ) a r e l i n k e d i n G ' as d e s c r i b e d i n S e c t i o n 2.
We have t h u s c o n s t r u c t e d a c o n f l i c t
graph G a s s o c i a t e d t o t h e f u n c t i o n g ( x l,...,xn)
which i s isomorphic t o G ' .
This
shows t h a t t h e c o n s t r u c t i o n s a r e e q u i v a l e n t . Remarks.
1)
I t i s i n t e r e s t i n g t o observe t h a t f o r a g i v e n G i f we u s e d i f f e r e n t o r d e r i n g s graphs G '
.
t h e same G.
...,a
o f node a. we may g e t n o t n e c e s s a r i l y isomorphic P F i g u r e 4.1 a ) and b ) show two d i f f e r e n t such graphs o b t a i n e d from
o f t h e neighbours al, Taking
f = 1
+
x x 1 3
+
x1x4
+
x,x2x3
- -
+
+ x x x x 1 2 3 4 we g e t
j; x x
1 2 4
92
x1
Cii. Ebenegger et al.
x2 G’
@
x3
F i g u r e 4.1 a ) However, t a k i n g f = 1 + x x
+ x,x3
t
X3x1x4 + X1x2x3 + x
x4 x3 G’
X x x ,we f i n d 1 3 2 4
@ 2.4
X
F i g u r e 4.1 b )
2)
One should observe t h a t if G i s a c l a w - f r e e graph, t h e n t h e graph G ’ o b t a i n e d by choosing a node a.
and o r d e r i n g i t s neighbours a r b i t r a r i l y i s n o t neces-
s a r i l y c l a w - f r e e as shown b y t h e examDle o f F i g u r e 4 . 2 .
However, one sees
e a s i l y t h a t when a p p l y i n g t h e t r a n s f o r m a t i o n t o G,in t h e r e s u l t i n g graph G ’ a l l new nodes form a c l i q u e K; i f K c o n t a i n s a s i m p l i c i a 1 new node ( i . e . a new node y whose neighbourhood i s a c l i q u e ) , t h e n y from G ’ and we g e t a graph G” w i t h
ic(G”)
= a(G)
IJ N ( y ) can b e removed
- 2.
93
Pseudo-boolean functions and stabiliti, of graphs
G'
Figure 4.2
One can a l s o show t h a t i f G i s the complement of a comparability graph, then f o r an appropriate choice o f node a o , the resulting graoh G ' i s s t i l l the complement of a comparability graph. However the number of nodes i n G ' may in general be l a r g e r than in G. 3)
I n the case where G i s a s e r i e s - p a r a l l e l graph ( s e e [a]), the transformation of G i n t o G ' leads t o a good algorithm. Series-parallel graphs always have a t l e a s t one node x with d G ( x ) 6 2. If dG(x) = 1 , l e t y be t h e unique neighbour o f x , then G ' = G - x - y. d G ( x ) = 2 , l e t y and z be the neighbours o f x; i f y and z a r e linked, G' = G
- x -
y
- z
If
and i f they a r e n o t , nodes x, y and z of G a r e replaced by
the new node ( y , z ) in G ' .
Thus in a l l cases, G ' has l e s s nodes than G and one v e r i f i e s t h a t G ' i s s t i l l a s e r i e s - p a r a l l e l graph ( i . e . i t contains no homeomorphic image of K4; see
'
PI 4) The above described procedure can be extended t o the weighted case a s well. A simple way t o do t h i s i s t o consider a node a. with weight
Ch. Ebenegger et al.
94
wo = min!wilai
?I(ao)
E
u
{ a o } } where wi
w i t h G a graph G such t h a t 3w(G) = aw(G) f(xl,x2
,...,xn)
L e t aw(G)
i s t h e w e i q h t o f node ai.
The c o n s t r u c t i o n w i l l a s s o c i a t e
b e t h e maximum w e i g h t o f a s t a b l e s e t i n G.
-
wo.
S t a r t i n g from t h e f u n c t i o n
a s s o c i a t e d w i t h G, we can w r i t e
=
P
wo .l Ti i=O
e
+ 1
(wi i=l
-
n
wO)Ti
O
q,r
wiTi
=
i=p+l F
1
= w o t w
1
+
Tqr +
1=1
(wi
-
n wO)Ti
+
1
i=p+l
wiTi,
,.
where N(ao) = l a l
The a s s o c i a t e d graph G c o n t a i n s nodes al,
...,a P w i t h
weights
,...,wn
- wo,...,wp - wo, nodes a P+1 ,...,an w i t h w e i g h t s w P+ 1 new nodes ( q , r ) as i n t h e unweighted case w i t h w e i g h t wo.
w1
and t h e same
Furthermore t h e edges o f G a r e o b t a i n e d b y t a k i n g t h e subgraph generated b y {al,
...,a n
'8
=
-
X
{ao', and by l i n k i n g t h e new nodes (9, r ) i n t h e same way as
i n t h e unweighted case.
I t i s t h e n easy t o see t h a t aw(G) = aw(G)
-
wo b y
o b s e r v i n g t h a t i f new nodes ( i , j l ) ,...,( i, jr) a r e i n s t a b l e s e t S , t h e n t h e nodes i, j
5.
,,..., jr
can b e i n c l u d e d i n
s.
COMPUTATIONAL RESULTS
The t r a n s f o r m a t i o n o f G i n t o G ' d e s c r i b e d i n s e c t i o n 3 i f a m l i e d r e p e a t e d l y may b e used f o r computing a { G ) . A few experiments have been r u n , b u t no s p e c i f i c e f f o r t was made y e t t o e v a l u a t e
Before applying
t h e e f f i c i e n c y of t h e method o r compare i t w i t h o t h e r a l g o r i t h m s .
t h e t r a n s f o r m a t i o n f r o m G t o G ' we s y s t e m a t i c a l l y used t h e f o l l o w i n g two reductions : R e d u c t i o n RO.
I f x is isolated, x i s eliminated ( s i n c e u(G
-
x) = a(G)
-
1 and
we t a k e G ' = G - x ) .
I f x has degree 1, x and i t s neighbour y a r e removed ( s i n c e a ( G we t a k e G' = G
-
-
x
y).
-
x
-
y ) = a ( G ) -1,
T h i s amounts t o s a y i n g t h a t we t r y t o a p p l y i n p r i o r i t y
t h e t r a n s f o r m a t i o n t o nodes o f degree 0 o r 1 . p e d u c t i o n R1. (see [3]). N(x) - { y j g N ( y )
-
I f x and y a r e nodes of G = ( X ,
{ x i , t h e n a(G
-
E ) such t h a t [x,
y ) = a(G) and one may remove y frm G .
y]
E
E,
Pseudo-boolean functions and stability o f graphs
Random graphs.
95
F o r t h e computer experiments, random graphs have been g e n e r a t e d
i n t h e u s u a l way by a s s i g n i n g a p r o b a b i l i t y f o r a p a i r of nodes t o be a d j a c e n t .
A l l experiments were performed on a
CPC Cyber 7326; we r e p o r t i n T a b l e 1 t h e
r e s u l t s o b t a i n e d f o r graphs w i t h 40 nodes.
F o r d e n s i t i e s o f 0.1 o r 0.9, one
c o u l d d e a l w i t h graphs having 70 t o 90 nodes i n t h e same t i m e as one c o u l d h a n d l e graphs o f d e n s i t y 0.5 w i t h about 45 nodes. density
a(G)
0.1 0.2
18
0.3 0.4
no. o f t r a n s f . G G' +
t o t a l no. o f added nodes
2 10
13 10
0.6 0.7 0.8 0.9
37.6 43.8
92
161.5 85.6 40.7
70
5
52 28 12
3 2
2 2 2 3 5 3 2 2 2
0.5 9.9
76
4 3
no. o f graphs generated
0 11 32
8 7 6
0.5
CP t i m e
14.7 3.9
Graphs w i t h 40 nodes ( r e d u c t i o n RO
-
R1)
Table 1 The r e s u l t s o b t a i n e d so f a r show t h a t t h e number o f nodes which a r e i n t r o d u c e d d u r i n g t h e successive t r a n s f o r m a t i o n s (no. o f new nodes c r e a t e d e l i m i n a t e d ) may be l a r g e .
-
no. o f nodes
T h i s does n o t seem t o be c r i t i c a l from a c o m p u t a t i o n a l
p o i n t o f view; we have t r i e d t o use more r e d u c t i o n s (namely r e d u c t i o n s R2 and R3 d e s c r i b e d i n t h e remark below) b e f o r e a p p l y i n g t h e t r a n s f o r m a t i o n .
The r e s u l t was
t h a t t h e number o f nodes i n t r o d u c e d c o u l d keep much l o w e r values, b u t t h e computing times increased dramatically.
I t seems t h e r e f o r e t h a t t h e b e s t t e c h n i q u e would b e
t o u s e o n l y a few r e d u c t i o n s i n such a procedure.
I t should however be k e p t i n
mind t h a t a l l t h e s e r e s u l t s a r e s t i l l fragmentary; more experiments should b e p e r formed b e f o r e g e t t i n g t o f i r m c o n c l u s i o m r e l a t e d t o t h i s approach t o t h e s t a b i l i t y number o f graphs.
I t may a l s o b e w o r t h w h i l e c a r r y i n g o u t experiments f o r s p e c i a l
c l a s s e s o f graphs. Remark.
As mentioned, a few o t h e r r e d u c t i o n s have been t r i e d i n c o n j u n c t i o n w i t h
t h e t r a n s f o r n a t i o n ; t h e y were abandoned because t h e t i m e spent t o check i f t h e y c o u l d b e a p p l i e d was n o t compensated b y t h e i r advantages. Reduction R2: [2]. N(y)
U N(z) -
I f a node x has 2 neighbours y,
( N ( x ) U { x i ) i s a c l i q u e , then a(G
-
z such t h a t
x) =
a(G)
b, z]h
E,
and one may remove x
f r a n G. N o t i c e t h a t R2 i s a s p e c i a l case o f Lemma 7 o f [3]. Reduction R3: (Lemma 12 i n r3]).
Since i n o u r a l g o r i t h m we r e p e a t e d l y use t h e
96
Clr. Ebenegger et al.
t r a n s f o r m a t i o n o f G i n t o G ' ( w h i c h s h o u l d g i v e us a c l i q u e i n t h e l a s t s t a g e ) , i t may be h e l p f u l t o t r y t o i n t r o d u c e edges e i n G i n such a way t h a t
a(G
+ e)
= a(G)
( h e r e G + e i s t h e g r a p h o b t a i n e d by i n t r o d u c i n g an edge e between 2 nodes o f G). T h i s o p e r a t i o n s h o u l d a l s o p r e v e n t t h e number o f new nodes f r o m i n c r e a s i n g t o o fast. Since t h e transformation G
I
G' i s n o t one-to-one,
we may t r y t o c o n s t r u c t a G ' and
and go back t o a G h a v i n g more edges t h a n G, f r o m which we may o b t a i n an a l t e r n a t i v e G ' p o s s i b l y w i t h l e s s new nodes.
An example o f t h i s i s g i v e n i n F i g u r e 5.1:
f r o m G1 we o b t a i n G ' , b u t we may i n t r o d u c e edges [ Z , 4 l and [3,q
G2 which corresponds a l s o
50
we examine e v e r y node x
(N(a,)U
t
G'.
and g e t a graph
More p r e c i s e l y , when c o n s t r u c t i n g G ' f r o m G,
...
u
N ( a p ) ) - N(ao) ; i f t h e r e i s a node
a . t N(ao) - N(x) such t h a t i n 6 ' x i s l i n k e d t o e v e r y new node o f t h e f o r m ( c , j) .I o r ( j , c ) , t h e n edge -x, a.? may be i n t r o d u c e d i n t o I;. 3
Figure 5.1 ACKNOWLEDGEMENTS The a u t h o r s would l i k e t o express t h e i r g r a t i t u d e t o N. S b i h i f o r c a l l i n g t h e i r a t t e n t i o n t o t h e s p e c i a l i s a t i o n o f t h e t r a n s f o r m a t i o n t o s e r i e s - p a r a l l e l graDhs and t o A. H e r t z a t E . P . F . L .
who r a n t h e computer experiments w i t h a g r e a t
e f f i c i e n c y and a l o t of enthusiasm. s t i m u l a t i n g r e s e a r c h atmosohere.
S p e c i a l thanks a r e due t o E.P.F.L.
for the
T h i s r e s e a r c h was c a r r i e d o u t w h i l e P.L. Hammer
was v i s i t i n g t h e Ecole P o l y t e c h n i q u e F e d e r a l e de Lausanne.
Pseudo-boolean functions and stabilitj, of graphs
91
REFERENCES Berge, C., Graphs and Hypergraphs (North-Holland, Amsterdam, 1973). B i l l i o n n e t , A., Reductions e t conditions d'optima i t e dans l e Probleme de l ' e n s m b l e s t a b l e de poids maximal, R . A . I . R . O . 15 (1981) 213-231.
B u t z , L . , Hammer, P . L . , and Hausmann, D . , Reduction methods f o r t h e vertex packing problem, University of Bonn, I n s t i t u t f u r kkonometrie u n d ODerations Research, Report 7540, (1975). Chvatal, V., On c e r t a i n polytopes associated with graphs, J . o f Combinatorial Theory B 18 (1975) 138-154. Harmer, P . L . , and Rudeanu, S . , Boolean methods in Operations Research (Springer-Verlag, New York, 1968). Hamor, A. and Leorith, P., S t o r i e s o f t h e one-zero-zero-one nights: Abu Boul in Graphistan, in: Hansen, P., de Werra, D . ( e d s . ) , Regards sur l a theorie des graphes (Presses Polytechniques Romandes, Lausanne, 1980).
This Page Intentionally Left Blank
Annals of Discrete Mathematics 19 (1984) 99- 114 0 Elsevier Science Publishers B.V. (North-Holland)
99
INDEPENDENCE SYSTEMS AND PERFECT k-MATROID-INTERSECTIONS R. E u l e r
Mathematisches I n s t i t u t d e r U n i v e r s i t a t zu Koln Weyertal 86 5 Koln 41 West Germany
We g e n e r a l i z e s t a b l e - s e t independence systems o f p e r f e c t graphs, 2 - m a t r o i d - i n t e r s e c t i o n s and matching independence systems t o t h e c l a s s o f k-matroid-intersections, which a r e p e r f e c t r e l a t i v e t o t h e k-matroids g i v e n . I t i s shown t h a t a p o l y h e d r a l d e s c r i p t i o n o f such an independence system i s g i v e n by t h e i n t e r s e c t i o n o f t h e corresponding m a t r o i d p o l y h e d r a and t h a t a r e l a t e d l i n e a r system has t h e p r o p e r t y o f t o t a l d u a l i n t e g r a l i t y ( T D I ) . Conversely, i f such a l i n e a r system i s known t o have t h e T D I - p r o p e r t y , t h e n t h e r e s u l t i n g independence system i s shown t o be p e r f e c t r e l a t i v e t o t h e matroids given.
INTRODUCTION The t h e o r i e s o f p e r f e c t graphs, 2 - m a t r o i d - i n t e r s e c t i o n s
and matchings i n graphs
c o n s t i t u t e three s i g n i f i c a n t cornerstones w i t h i n combinatorial optimization. T h i s paper aims t o r e l a t e t h e u n d e r l y i n g c l a s s e s o f independence systems t o each o t h e r by g e n e r a l i z i n g t h e n o t i o n o f p e r f e c t i o n t h e r e b y g i v i n g some h i n t s on how t h e d i f f e r e n t concepts and r e s u l t s m i g h t be combined t o a s i n g l e t h e o r y . Given a f i n i t e s e t E = {el p a i r (EJ)
,... ,en} J ~
A set X
c_
and a nonempty system
3
o f subsets o f E t h e
i s c a l l e d an independence system, if
E i s independent, i f X
1
~
J2 €7. -
€ 3 , dependent o t h e r w i s e .
A base o f X, X
c
E,
i s a maximal independent subset o f X ( a base o f E i s s i m p l y c a l l e d a base) and a c i r c u i t o f (E,Yj i s a minimal dependent subset o f E (maximal and m i n i m a l w i t h respect t o s e t i n c l u s i o n ) . (EJ)
w i t h R resp. C.
We denote t h e systems o f bases resp. c i r c u i t s o f
An element e ez E i s c a l l e d
no c i r c u i t c o n t a i n i n g i t .
free ( i n
The r a n k - f u n c t i o n o f (E,3)
(EJ)),
i f there i s
i s d e f i n e d f o r a l l X G E by
r ( X ) = m a x { ( I I : 1 ~ 2 1, s x } . A matroid
(EJ) i s an independence system such t h a t f o r any X
have t h e same c a r d i n a l i t y .
I f k m a t r o i d s (E,Jl)
,...,( EJk)
E a l l bases o f X and ( E J ) a r e g i v e n Q
100
R. Eider k
n yi,
such t h a t 3 =
then (EJ)
i s called a k-matroid-intersection.
i=l
A graph G = (E,C) c o n s i s t s o f a f i n i t e s e t E, t h e v e r t i c e s , and a system C o f 2-element subsets of E, t h e edges. F o r X c_ E, G[X] denotes t h a t subgraph o f G, A c l i q u e o f G i s a subgraph o f G, i n w h i c h any two d i s -
which i s induced b y X .
t i n c t v e r t i c e s a r e j o i n e d b y an edge, i . e .
t h e c o r r e s p o n d i n g 2-element s e t i s i n
A s t a b l e s e t o f G i s a s u b s e t o f E such t h a t , i n G, no two o f i t s v e r t i c e s
C.
a r e j o i n e d by a/n edge.
If A
=
i s a f a m i l y of n o t n e c e s s a r i l y d i s t i n c t subsets o f E, t h e n
(E, ,...,Em)
T = E i s called
a transversal o f A , i f there e x i s t s a b i j e c t i o n 4 : T
-&
{ l , ...,m i
such t h a t t e E
f o r a l l t ET. 6(t) A p a r t i a l t r a n s v e r s a l o f A i s s i m p l y a t r a n s v e r s a l o f a s u b f a m i l y o f A . A part i t i o n (Xi, i = 1, . . . ,k ) o f X , X C - E , i s a f a m i l y of p o s s i b l y empty subsets o f X such t h a t t h e members o f i t a r e p a i r w i s e d i s j o i n t and t h e i r u n i o n i s t h e whole set X. B a s i c r e f e r e n c e s t o t h e s e concepts a r e t h e books o f Berge [l] and Welsh 0 7 1 .
INDEPENDENCE SYSTEMS, WHICH ARE PERFECT RELATIVE TO A SET OF MATROIDS D e f i n i t i o n 2.1. and c = (c,,..
L e t (E,Y)
. ,cn)
dence system (ECJc)
Nn.
E
be an independence system, C i t s system o f c i r c u i t s b y c i s a new indepen-
Then a m u l t i p l i c a t i o n o f (EJ)
formed as f o l l o w s :
i ) IfC I = ii: i E ' 1 , ..., n and c .1 > 1 , a n d m = I C I I , we e n l a r g e E b y m p a i r w i s e d i s j o i n t s e t s o f new elements, E i j , o f c a r d i n a l i t y c 1. . -1 f o r j = 1, ...,r n * 3 m i.e. E i s e n l a r g e d t o Ec :
=
E
u (u
E f , ) , where E '
j=l
ii)
J
: = j
B f o r i, 6 CI; J
o v e r EL we form a system C c o f c i r c u i t s as f o l l o w s : choose an element. il o f C I and add t h e system to
t h e n choose i 2 out o f C I \ { i l l
.
i 'l"2
: =
u
l(C\el
and add
)U!e'.
2
e'eE;
c;
: C
eCUCi
and ei 1
E
C i t o C U Ci
2
1
2 and so on u n t i l we o b t a i n an independence system (Ec,Jc)
having
101
Itidepetidence systems and perfect k-matroid-irztersectio~zs
as system Cc o f c i r c u i t s . We n o t e t h a t i f (E,1)
,...,
i s g i v e n as t h e i n t e r s e c k i o n o f t h e k m a t r o i d s (E,C1)
t h e n ( E .f ) i s t h e i n t e r s e c t i o n o f ( n ci)k=:kc many m a t r o i d s o v e r Ec, c c '-1 i s , up t o isomorphism, uniquATy determined. More s p e c i f i c a l l y ,
(E,Ck),
and t h a t (Ec,&)
t h e c o n s t r u c t i o n o f (Ec,)c)
may be viewed as s u b s t i t u t i n g a t r a n s v e r s a l o f t h e
family A = ({ellU Ei
f o r t h e s e t {el, C
E
...,en]
EA)
u
$ - ' ( j ) f o r t h e corresponding c i r c u i t s e .EC J Ci and i = 1, ... ,k (which we w i l l o f t e n r e f e r t o as s u b s t i t u t i n g T f o r
{ ely...
,en}),
as w e l l as
,..., { e n j u
then e n l a r g i n g t h i s new ground s e t t o E c by a d j u n c t i o n o f t h e
remaining elements as f r e e elements and f i n a l l y t a k i n g t h e i n t e r s e c t i o n o f these E q u i v a l e n t l y , one c o u l d c o n s t r u c t t h e i n t e r -
m a t r o i d s f o r a l l t r a n s v e r s a l s o f A.
s e c t i o n of a l l s u b s t i t u t i o n s f o r a f i x e d m a t r o i d and then t a k e t h e i n t e r s e c t i o n of t h e r e s u l t i n g k i n t e r s e c t i o n s .
Note t h a t f o r c = (1,
and, c l e a r l y , t h e c o n s t r u c t i o n o f (Ec,Jc)
...,1 ) ,
(Ec,&)
(€,2),
=
i s independent o f t h e o r d e r o f t h e
... ,im.
numbers il,
F o r t h i s and t h r o u g h o u t t h e L e t us now c o n s i d e r such a m u l t i p l i c a t i o n (Ec,Jc). i for i= 1 ,n and Ec = { e i ,e;(c)j, where f o l l o w i n g l e t E; = {el ,... ,eli-l}
,...
n
z(c):=
c c.,1 b e such t h a t f o r i = 1 ,... ,n i=l i e' = e. and e;-+j = ej-l i-1
where
yi
=
y.tl 1
z cm. m= 1
1
m a t r o i d s such t h a t 2 = (Ec,'Jc)
for
j = 2,...,ci,
1
Then t h e f o l l o w i n g h o l d s :
L e t (E$)
P r o p o s i t i o n 2.2
,...
be an independence system, (E,J1)
.qyi and c
1=
E
Bn,
,... ,(E,2k)
be
Then a m u l t i p l i c a t i o n o f
C'E
by c ' i s i d e n t i c a l t o a m u l t i p l i c a t i o n o f (E,Y) by a v e c t o r c "
E
INn,
which i s g i v e n by c" = (
c e;E{elWEi
P r o o f W.1.o.g.
c;
,..., e;
E I
I: en)UE;
C;).
we can r e s t r i c t o u r s e l v e s t o a v e c t o r c ' o f t h e form
c ' = ( c i ,...,c ' ,1,...,
1).
I f we now m u l t i p l y (Ec,Jc)
by ( c i , l
,...,l ) ,
rnatroids remain unchanged, where el has been s u b s t i t u t e d b y an element e
a l l those E
Ei,
102
and
R. Eider n
(.x
1=2
S i n c e we now add a s e t o f new elements EA+l
t o Ec, n can b e s u b s t i t u t e d f o r el and t h i s y i e l d s ( c i - 1 ) ( n c . ) k i=2 1
ci)k many a r e l e f t . Er;+l
a l l elements e
new m a t r o i d s .
However, we c o u l d have o b t a i n e d t h e same i n t e r s e c t i o n o f m a t r o i d s ,
i f we had mu?t i p 1 i e d (E,'J a l s o v a l i d f o r c i ,..., c;
. . ,cn).
b y ( c l + c i - l ,c2,.
, multiplying
(E,j)
d i r e c t l y by (
i s i d e n t i c a l t o m u l t i p l y j n g (E,?)
Since t h i s argumentation i s by ( c i ,..., c ' ,1,,1) c1 c; ,c 2 " . . ,cn) 9
b y c and (Ec,7,) C
e;Elel l U E i
p r o v i d e d t h e c o r r e s p o n d i n g s e t s of new elements a r e chosen a p p r o p r i a t e l y . S i m i l a r l y , we f i n d t h a t a m u l t i p l i c a t i o n o f (Ec,Jc)
by c'
E
gUZ(')
can as w e l l b e
o b t a i n e d by m u l t i p l y i n g ( E , j ) by c" as g i v e n above.
Let
D e f i n i t i o n 2.3
(E,q
m a t r o i d s such t h a t 1 = r e l a t i v e t o (E,?)
be an independence system and (E,Yl),...,(E,2k)
n 2.. Then
i s s a i d t o have t h e max-min-property
1=l 1
,... , ( E , j k ) ,
r(X) =
(EJ)
be
if
k m in 1 ri(Xi) (Xi,i=l, ..., k ) i=l
f o r a l l X C E,
i s a partition of x where r resp. ri i s t h e r a n k - f u n c t i o n o f (EJ)
D e f i n i t i o n 2.4 m a t r o i d s such
L e t (E,>)
resp. (E,Yi),
be an independence system and (E,yl)
k t h a t ;1= .n3..Then 1=k
i = 1 ,... ,k.
,...,( E,Jk)
be
we c a l l (E,^3) p e r f e c t r e l a t i v e t o (E,gl),.
(E,jk), i f f o r a l l c E N " any m u l t i p l i c a t i o n of m i n - p r o p e r t y r e l a t i v e t o (Ec13,) ,. .. ,(EC,gk 1.
(EJ)
b y c, (EcJc),
.. ,
has t h e max-
C
Obviously, f o r any X G E t h e r e s t r i c t i o n (XJX) (EJ),
given b y Y X : = { I € 3 : I
m a t r o i d s , b u t r e s t r i c t e d t o X.
C o r o l l a r y 2.5
G
Moreover, b y P r o p o s i t i o n 2.2 we i m m e d i a t e l y o b t a i n
Any m u l t i p l i c a t i o n (Ec,JC)
o f an independence system (E,J),
i s p e r f e c t r e l a t i v e t o t h e m a t r o i d s (E,jl) c o r r e s p o n d i n g m a t r o i d s (Ec,J1)
o f such an independence system
i s p e r f e c t r e l a t i v e t o t h e same s e t o f
XI,
,. . . ,(
E
,...,( E , j k ) ,
which
i s perfect relative t o the
J ). kc
We remark a t t h i s p o i n t t h a t an independence system (E,J)
need n o t b e p e r f e c t k r e l a t i v e t o any s e t of m a t r o i d s (E,Il) ,...,( E,&) such t h a t 7 = .n1., a l t h o u g h 1=1 1 i t i s p e r f e c t r e l a t i v e t o some s p e c i f i c such s e t . However, i f (EJ) i s p e r f e c t
103
Independence systems and perfect k-matroid-intersections r e l a t i v e t o (EJ1) (E,jk),
...,
,..., (E,gk),
(E,rkJ,such
gi.
that’J=.r)
t h e m a x i n i n - p r o p e r t y n o t o n l y fo:=IE,’I1) multiplications.
,...,
t p n t h i s i s t h e case f o r any s u p e r s e t (EJl)
Besides, i t i s r e a l l y necessary t o c l a i m i t s e l f , b u t a l s o f o r a l l o f i t s proper
To see t h i s c o n s i d e r f o r i n s t a n c e those m a t r o i d s (E,7A), which
c x 6 r ( A ) , i . e . whose systems o f c i r c u i t s j u s t eEA e c o n s i s t o f a l l subsets o f A h a v i n g c a r d i n a l i t y r ( A ) + l , and r e l a t i v e t o which (E,2)
a r e induced by t h e i n e q u a l i t i e s has always t h e max-min-property.
One s t a r t i n g p o i n t f o r i n t r o d u c i n g t h i s concept o f p e r f e c t i o n has been t h e c l a s s of s t a b l e set-independence systems ( E J )
o f p e r f e c t graphs, i . e .
those graphs,
f o r which minimum number o f c l i q u e s i n r ( X ) = G[X] needed t o cover t h e s e t X
for all
cE,
C l e a r l y , (E,T) can be r e p r e s e n t e d as t h e i n t e r s e c t i o n o f t h o s e m a t r o i d s , whose system o f c i r c u i t s a r e g i v e n by t h e edges o f a maximal c l i q u e . {K1,
...,K k l
has t h e max-min-property r e l a t i v e t o (E,jl) ,... ..., k. By a lemma o f Berge
n o t d i f f i c u l t t o see t h a t (EJ) (EJk), (cf.
[l])
Moreover, i f
i s t h e s e t o f a l l maximal c l i q u e s i n a p e r f e c t graoh G, t h e n i t i s
where
(E,Yi)
,
i s induced by Ki f o r i = 1,
t h e m u l t i p l i c a t i o n o f (E,’JI) by c €LNn i s a g a i n t h e s t a b l e - s e t indepen4
dence system o f a p e r f e c t graph Gc, whose maximal c l i q u e s correspond, up t o r e p e t i t i o n s , t o t h e m a t r o i d s (Ec,’Ji),
i = 1 ,... ,kc.
Hence, (Ec,Zc) has t h e max-
,. . . ,(E ,2 ) and, t h e r e f o r e , (E,2) i s p e r f e c t
m i n - p r o p e r t y r e l a t i v e t o (Ec,fl)
kc i n t h e sense o f D e f i n i t i o n 2.4.
r e l a t i v e t o (E,Y1), ...,( EJk)
A POLYHEDRAL DESCRIPTION I n t h i s s e c t i o n we w i l l prove t h a t a p o l y h e d r a l d e s c r i p t i o n o f an independence system (EJ),
which i s p e r f e c t r e l a t i v e t o (E,jl)
c
xe 6 r l ( A )
for all
,...,( E , j k ) , AS
i s g i v e n by
E,
eEA
c xe
4 rk(A)
(3.1)
f o r a l l A G E,
eEA xe
3
0
for all e
E
where ri i s t h e r a n k - f u n c t i o n o f t h e m a t r o i d (E,Ti),
E,
i = 1,
... ,k.
r e f e r t o a theorem, which has a l r e a d y been used b y Edmonds [4],
For t h i s we
Chvdtal [2]
others:
Theorem 3.2
L e t S be a f i n i t e s e t o f s o l u t i o n s x
E
W”
o f t h e system o f
and
R. Euler
104
i nequal i t i e s
xe aiexe
I
B
f o r a l l e c E,
0
for all i
\c b .
1
eF E
(3.3)
I.
E
Then the s e t of a l l s o l u t i o n s of ( 3 . 3 ) i s t h e convex hull of S i f and only i f , n f o r every vector c r Z we h a v e max =
ICX :
min
i
x
-
iLI
E
Si
W ic1,
b. : li:O i i
x
~
~
aW ~ ek.E?.~
a
c
~
icI
n Now l e t c = ( c l ,..., c n ) E 72 . W.1.o.g. we can d e l e t e a l l 5 0 - c o e f f i c i e n t s of c and go over t o t h e corresponding r e s t r i c t i o n (E;2E,). I n a d d i t i o n , we can d e l e t e the f r e e elanentsof ( E , j ) as well as t h e i r c o e f f i c i e n t s in c , s i n c e e s t a b l i s h i n g t h e max-min-equality in Theorem 3.2 f o r such a r e s t r i c t i o n (E"JE,,) of (E,Y) will immediately lead t o the r e l a t e d one f o r (E,X). I t i s now our aim t o apply Theorem 3.2 t o the system of i n e q u a l i t i e s ( 3 . 1 ) and v i a the max-min-equality given t h e r e show (3.1) t o represent a polyhedral desc r i p t i o n of (E,?I), i . e . the v e r t i c e s of t h a t polyhedron correspond exactly t o the independent s e t s . For t h i s we multiply ( E J ) by c and obtain ( E c J C ) . Let t h e r e s u l t i n g k c matroids be indexed such t h a t any of the following blocks of k of than, ( Ec,jl
1,. . . * ( E c , 3 k )
. .. ;( Ec*Ikc-k+l correspond to
(E,7,),. . . , ( E , & ) ,
3 .
; ( Ec,lk+l ) 9 .
'.*(E
..
3
(Ec,Y2k) ;
kc
i .e. ( E c J 1 ,(Ec$xk+l1 ,. . . ,(
Ec'flkc-k+,)
arise
from ( E J 1 ) by s u b s t i t u t i n g E by a transversal T i of A , where { T . ) . is a 1 1 = 1 , ... fixed sequence of a1 1 t r a n s v e r s a l s of A , ( Ec,y2),( Ec,gk+2) ,. . . ,( E c ,lkc-k+2) in exactly the same way, and so on.
a r i s e from ( E J 2 )
Claim 3.4
Proof
Let I
L
z c and
i z 1.1,. . . , n i .
Now we s t a t e the following
Then
By assumption, t h e r e e x i s t s a c i r c u i t C of (Ec,JC) i n IU : e l . B u t then, by d e f i n i t i o n o f (Ec,&), the set ( C \ i e ? ) U i e ' ! i s a l s o a c i r c u i t of (Ec,&)
105
Iirdependence systems and perfect k-matroid-in tersections for a l l e'
( I e i } U E i ) , which proves t h e c l a i m .
E
Claim 3.4 says t h e f o l l o w i n g :
(Bn E)
If B i s a base of Ec, IBI = r c ( E c ) , t h e n
E
3
and, i n p a r t i c u l a r ,
161 = cxgnE, where xBnE i s t h e i n c i d e n c e v e c t o r o f BnE.
What we s t i l l need i s a
s o l u t i o n y o f t h e system o f i n e q u a l i t i e s 1
YA k
i = 1 ,..., k,
(3.5)
i
z
c
for all A GE,
3
yA >, c j
f o r a l l j = 1,
..., n
i=l e.EAcE J such t h a t
k 161 =
C
L e t (Ei,
i = l,...,kc)
c
c
yb r i ( A ) .
i=lAGE
be a p a r t i t i o n o f E c s a t i s f y i n g rc(Ec) =
kC
c
ri(EF).
i=1
,..., (Ec,Ijtk),
To any b l o c k o f k m a t r o i d s (Ec.Ijtl)
j = O,k,Zk
,..., kc-k,
there
o f A , which has been s u b s t i t u t e d f o r E i n (j/k)+l and a l l these t r a n s v e r s a l s a r e d i s t i n c t . i n a d d i t i o n , t o any element
e x i s t s a unique transversal T (EJ),
eEEc t h e r e i s a t l e a s t one b l o c k o f k r n a t r o i d s , i n whose i n t e r s e c t i o n e i s n o t a f r e e element. Now c o n s i d e r t h e s e t s
blockwise.
Then, o b v i o u s l y , e
E
Ec i s c o n t a i n e d i n one o f t h e s e t s , say E g .
3.
'f
such t h a t e i s n o t f r e e i n fl belongs t o a b l o c k o f k s e t s EC ,..., Eilk J+1 i=l I f , however, t h i s i s n o t t h e case, i . e . e i s ., we l e a v e e i n t h a t s e t E i .
t h i s E: J+1
f r e e i n t h a t i n t e r s e c t i o n , we can t a k e i t o u t f r o m E:
and p u t i t i n a s e t E:,
such t h a t e i s n o t f r e e i n t h e corresponding i n t e r s e c t i o n . T h i s i s p o s s i b l e f o r kC C a l l e E Ec w i t h o u t i n c r e a s i n g t h e v a l u e Now, l e t (E:, i = l,...,kc) ri(Ei). be a l r e a d y m o d i f i e d a c c o r d i n g l y .
j=l
Then i n any o f t h e s e t s E F t h e r e can be a t most
one element f r o m { e i } U E; f o r i = 1,. ..,n.
Now we r e p l a c e i n any o f t h e s e t s
Ei t h e elements e ' f r o m E; b y t h e corresponding ei, r e s u l t i n g f a m i l y (At,
t = l,...,kc)
i = 1, ..., n, and c o n s i d e r t h e
( o f n o t n e c e s s a r i l y d i s t i n c t subsets o f E ) , k -1. t = l , k t l , ...,p k t l ) , where p = To any
"4
i n p a r t i c u l a r t h e s u b f a m i l y (At, 1 A s E we a s s i g n t h e v a l u e yA, which i s equal t o t h e number o f times A i s o c c u r r i n g i n t h a t subfamily.
Clearly, rl(At)
=
rt(Ei)
for t
=
1 ,k+l,.
.. ,pk+l,
106
R. Eulcr
and i n t h i s manner we o b t a i n a number o f d u a l v a r i a b l e s f o r t h e system o f inequalities
z
f o r a l l A 6 E.
x e \c r l ( A )
eEA Next, we c o n s i d e r t h e i n d i c e s 2,k+2, ...,p k+2 and f i n d a v e c t o r yf\, and so on, ,Ek u n t i l we have y
E
WL
, which i s
f e a s i b l e f o r (3.5).
Since
k
we have, t o g e t h e r w i t h Theorem 3.2:
Theorem 3.6
L e t (E,’3) be an independence systen,which i s p e r f e c t r e l a t i v e t o t h e
m a t r o i d s (E,S1)
,... ,(E,’jk).
Then a p o l y h e d r a l d e s c r i p t i o n o f (EJ),
convex h u l l of t h e i n c i d e n c e v e c t o r s o f a l l members o f f ,
i.e.
the
i s g i v e n by ( 3 . 1 ) .
I t f o l l o w s , t h a t a l s o t h e f o l l o w i n g l i n e a r system c o n s t i t u t e s a p o l y h e d r a l
d e s c r i p t i o n of such an (E,T):
x
e
3 0
for all e
E
E.
TOTAL DUAL INTEGRALITY D e f i n i t i o n 4.1
L e t A r e s p . b be a r a t i o n a l - v a l u e d mxn-matrix r e s p . m-vector.
Then we say t h a t t h e l i n e a r system Axgb has t h e p r o p e r t y o f t o t a l d u a l i n t e g r a l i t y (TDI), i f , f o r any i n t e g e r o b j e c t i v e f u n c t i o n c such t h a t max{cx: Axsbl e x i s t s , t h e r e i s an i n t e g e r optimum dual s o l u t i o n . T h i s p r o p e r t y has been i n v e s t i g a t e d w i t h i n t h e c o n t e x t o f 2 - m a t r o i d - i n t e r s e c t i o n s by G i l e s
[8l
( s e e a l s o [16]),
o f submodular f u n c t i o n s on g r a p h s b y Edmonds and
G i l e s [7l and o f i n t e g e r p o l y h e d r a by G i l e s and P u l l e y b l a n k [ll]. I n p a r t i c u l a r , S c h r i j v e r [15]
has shown t h a t any r a t i o n a l p o l y h e d r o n i s t h e s o l u t i o n s e t o f a
unique m i n i m a l i n t e g e r l i n e a r system h a v i n g t h e T D I - p r o p e r t y . The p r o o f o f Theorem 3.6 shows t h a t , i f (E,’2) i s p e r f e c t r e l a t i v e t o m a t r o i d s
( E J , ) , . .. , ( E , & ) , t h e n t h e l i n e a r system ( 3 . 1 ) has t h e T D I - p r o p e r t y . Moreover, as i n t h e case o f 2 - m a t r o i d - i n t e r s e c t i o n s (see 0 6 ] ) , a l s o t h e l i n e a r system (3.7) has t h e T D I - p r o p e r t y .
107
Independence systems and perfect k-ma troid-in tersections
L e t us now deduce a converse r e l t i t i o n :
Theorem 4 . 2
I f t h e l i n e a r system ( 3 . 1 ) has t h e T D I - p r o p e r t y , t h e n (E,Y)
proof BY T D I , we have f o r e v e r y v e c t o r c
is
,... , ( E , l k ) .
p e r f e c t r e l a t i v e t o t h e g i v e n m a t r o i d s (E,j,)
LNn
E
max {cx : x i s t h e i n c i d e n c e v e c t o r o f an independent s e t } k = min
I c
c yAi ri(A)
k :
z
z
i >, c j yA
f o r j = 1,
i yA 3 0
f o r a l l A&,
...,n,
i=l e .EASE
i=l AGE
J
and t h e r e i s always an i n t e g e r optimum s o l u t i o n y.
i = 1,
...,k l
F o r t h i s s o l u t i o n y we can
always achieve k
i c y = c . i=l e.EAC-E A J J
c
f o r j = l ,
i f we have ,, > ' I i n ( 4 . 3 ) f o r some j A o f E c o n t a i n i n g e . as w e l l as an i n d e x i J by t h e t r a n s f o r m a t i o n since,
E
E
...,n,
(4.3)
11 ,...,nl, we can c o n s i d e r a s u b s e t such t h a t y; > 0. Then
{l,...,k}
we can decrease t h e sum i n (4.3) by 1 w i t h o u t i n c r e a s i n g t h e optimum v a l u e k c z yAi ri(A), s i n c e r a n k - f u n c t i o n s a r e monotone. By r e p e t i t i o n o f such a i=lAGE t r a n s f o r m a t i o n ( 4 . 3 ) can b e achieved. Now we m u l t i p l y (E,J)
by c t o o b t a i n (Ec,&)
and f r o m o u r optimum d u a l s o l u t i o n y
we w i l l c o n s t r u c t a p a r t i t i o n (ET, i = l , . . . , k c )
o f Ec such t h a t
k-
C
z ri(E:).
rc(Ec) =
i=1 L e t AsE, i
{ l ,...,k } be such t h a t yb >, 1.
To any e . E A we choose a d i s t i n c t J i element e l f r o m { e . ) U E l and r e p l a c e A by t h e s e t { e ' : e . E A}. I f yA 3 2, J J J J J we r e p e a t t h i s replacement b y a s e t { e " : e . E A, e " # e l } and so on, u n t i l we i j~ J J g e t yA p a i r w i s e d i s j o i n t subsets o f Ec. Now we choose t h e n e x t A: i ' such t h a t i' y A , b l and proceed s i m i l a r l y t o o b t a i n ybyb, p a i r w i s e d i s j o i n t subsets o f Ec and E
.
I
so on, u n t i l we g e t a p a r t i t i o n (ET, i = 1,. ..,kc)
o f Ec.
Note t h a t a l l t h e s e
s e t s ET c o n s t i t u t e p a r t i a l t r a n s v e r s a l s Ti o f t h e f a m i l y A ( s e e s e c t i o n 2) and,
R. Eiilrr
108
therefore, we can enlarge these Tls t o t r a n s v e r s a l s of A and assign a unique 1
T h i s matroid
matroid (Ec,2j.) t o Ei.
a r i s e s from one of
(€,Il) ,...,( E,&)
by
s u b s t i t u t i n g t i e enlarged transversal f o r E and, t h e r e f o r e , a f t e r a l l these
c
Hence, r '(E ) = C
C
1 r . (E.). Jj 1
C
We can show the corresponding r e l a t i o n f o r X 5 E by choosing a vector c '
E
Zn
E X , c! 0 f o r e . i X, as well a s f o r an a r b i t r a r y J J J 3 Ec, c already given, by choosing a n appropriate vector c ' and then constructing an optimal p a r t i t i o n of X . This completes the proof.
such t h a t c! = 1 f o r e . X
C_
FURTHER EXAMPLES
Example 5 . 1 Matroids (E,g) have t h e max-min-property r e l a t i v e t o themselves. I t remains t o show t h a t they keep t h i s property a f t e r every proper m u l t i p l i c a t i o n by a vector c
g:
CN'.
Let us consider t h e family A = (ie,-U
E; ,..., {e,iL) E;).
Since multiplying ( E J ) by c does n o t depend on the order of t h e c i , we may assume t h a t c1 ;c 2 p . . . ,c holds. Now we p a r t i t i o n E ( s e e Figure 5 . 2 ) in a s e t of n C ( p a r t i a l ) t r a n s v e r s a l s T i , i = 1 , . - . , c l , o f A , such t h a t T1 = E, T 2 = { e1l ,..., e nl ) ,
.... .
I f , f o r instance, c1 a c 2
5
1 , the l a s t transversal Tc
{ e1c , - l : .
just c o n s i s t s of 1
Figure 5.2
hideperidcwce
SIs t e m
109
arid perfect k-matroid-iritersections
To any o f these ( p a r t i a l ) t r a n s v e r s a l s Ti we a s s i g n t h a t u n i q u e m a t r o i d ( E c , r . ) , Ji
which we o b t a i n from s u b s t i t u t i n g TiU T i f o r E i n t h e g i v e n m a t r o i d (E,J),
,. . . , e l T . l + l i .
where T i c o n s i s t s o f t h e s e t Ien,en-l m u l t i p l i c a t i o n o f (E,Y) f o r a1 1 i = 1 ,.
. . ,cl,
where Ei = E\T!.
__ Proof
C l e a r l y , r . (T.) = r ( E . ) 1 Ji 1
Moreover, we have t h e f o l l o w i n g
1
P r o p o s i t i o n 5.3 where r
By t h e d e f i n i t i o n o f a
by c t h i s i s always p d s s i b l e .
rc(Ec) =
I: r . (Ti),
i=l Ji
denotes t h e r a n k - f u n c t i o n o f (Ec,Tc).
C
I t i s s u f f i c i e n t t o f i n d a member
I
of
2c such
t h a t (11 =
t h i s we a p p l y a s l i g h t v a r i a n t o f t h e Greedy-Algorithm (see [5]) Step 1 )
1:=0=:J,
i:=1
c1 I: r . (Ti). i=l Ji
For
t o (Ec,IC):
Step 2)
-+
I := IU( [ei }UE; )
1
Yes :
Step 2)
J
I f i = n, Stop.
One e a s i l y v e r i f i e s t h a t I 1 f ) Ti\ C. -1
]I! = c
Step 3 )
U ieil ~ 2 ? No :
Step 3 )
+
J :=Jut ei }
+
Step 3 )
Otherwise, i:=i+l and =
f
Step 2).
r . ( T . ) f o r a l l i = 1, Ji 1
... ,cl
and t h u s
r . (T.) J .
i=1
1
1
By an analogous c o n s t r u c t i o n t h e max-min-property
can be shown f o r a l l X relative t o itself.
s Ec,
r e l a t i v e t o (Ec,Il)
so t h a t t h e m a t r o i d (EJ)
i s shown t o be p e r f e c t
We p o i n t t o t h e f a c t , t h a t t h i s r e s u l t i s c l o s e l y r e l a t e d t o
Edmonds' work on m a t r o i d s and t h e Greedy-Algorithm ( c f . constitutes
,. . . ,(Ec,gkc)
PI),which,
obviously,
a n e f f i c i e n t procedure f o r computing t h e rank o f Ec3(Ec,2,)
a m u l t i p l i c a t i o n o f t h e m a t r o i d (E,Y).
being
Such a procedure even e x i s t s f o r t h e
problem of d e t e r m i n i n g a maximum w e i g h t independent s e t i n (Ec,Tc),
given a weight-
f u n c t i o n c ' over Ec.
Example 5 . 4
L e t two m a t r o i d s (E.2,).
corresponding 2 - m a t r o i d - i n t e r s e c t i o n
(EJ2) (EJ).
be g i v e n and l e t u s c o n s i d e r t h e We do n o t know o f a d i r e c t proof
( i n t h e sense o f D e f i n i t i o n 2.4) f o r showing t h a t (E,J) (E,Jl)
and ( E , j 2 ) .
However, i t i s w e l l known ( s e e [6],
i s perfect relative to [8]) that the l i n e a r
110
R. Euler
system
z xe <
rl(A)
f o r a l l A 5 E,
z xe
6
r2(A)
f o r a l l A c; E ,
3
0
for all e
eEA
(5.5)
eEA
xe has t h e TDI-property.
E
E
Consequently, by Theorem 4 . 2 , (E,>)
i s perfect relative
(EsT2).
t o (E,?),
Example 5.6
Let G = (V,E)
be a f i n i t e , undirected, loopless graph having vertex
s e t V and edge s e t E . For S c V l e t scs)denote the s e t of edges of G having exactly one end i n S and u(s)t h a t s e t of edges having both ends i n S; moreover, l e t Q:= {S c V : I S \ >, 3, IS/ odd1 and qs:= 1/2( IS\-1) f o r a l l S E Q. Edmonds [4] showed t h a t a polyhedral d e s c r i p t i o n o f t h e matching independence system (EJ) of G , i . e . t h e convex hull o f incidence vectors of a l l those subsets of E, no two elements of which a r e incident t o a common vertex, i s given by for all e
E
E,
Xe"l
for all v
E
V,
xe
for all S
E
Q.
xe
z
b
0
(5.7)
eE6(v)
z eEY(
\c
qs
S)
proved t h a t the system (5.7) a l s o has the TDI-property so t h a t again by Theorem 4.2 (E,T) i s shown t o be perfect r e l a t i v e t o t h e matroids ( E , & ) , v E V , and ( E , I S ) , S E Q , a s induced by the i n e q u a l i t i e s z x e < 1 resp. z xe c qs.
Cunningham and Marsh [3]
ess(v)
ecy( S )
Exmple 5.8 We would l i k e t o present now an independence system (E,C), which i s not perfect r e l a t i v e t o any s e t of matroids. Consider t h e s t a b l e - s e t independence system of the graph G = (E,C), E = t l , ...,6 1 , a s shown i n Figure 5.9.
I t i s well known ( s e e f o r instance [14])
5
z x . + 2x6
4 2 i=l defines a f a c e t of t h e convex hull of incidence vectors o f s t a b l e s e t s in G .
that the inequality
'
Clearly, by Theorem 3 . 6 , (E,C) cannot be p e r f e c t r e l a t i v e t o a s e t of matroids. k To show t h i s d i r e c t l y , suppose i t i s . Then '=I = n 3. f o r a s e t of k matroids i=l (E.2,) ,..., ( E , I k ) . I n p a r t i c u l a r , r ( E ) = ,I r i ( E i ) f o r sane p a r t i t i o n 1=1
(Ei, i
=
1 , ...,k ) of E.
Since r(E) = 2 , r i ( E )
3
2 f o r i = 1 ,..-,k .
Suppose t h e r e
Independence systems and perfect k-matroid-intersections
111
1
Figure 5.9 i s an i n d e x j such t h a t r . ( E . ) = 1.
J
J
Then GEEj]
can o n l y be a s i n g l e v e r t e x , a
s i n g l e edge o r a t r i a n g l e i n G, s i n c e r ( X ) = 2 f o r any 4-element s u b s e t X of E . However, r(E\E .) = 2 f o r any such s e t E and so r ( E ) = 2 cannot be achieved. J j Hence, r . ( E - ) = 0 o r 2 f o r a l l i = 1, k and, t h e r e f o r e , r ( E ) = ri(E) = 2 f o r 1 1
...,
some i n d e x i
E
...
{l, , k l .
Now we m u l t i p l y (E,I)
by c = (l,l,l,l,l,Z)
and
observe t h a t t h e r e i s no c l i q u e o f s i z e 4 i n t h e corresponding graph Gc = (EC$,) (see F i g u r e 5.10). 1
F i g u r e 5.10 So a g a i n 2 = r ( E c ) = ri(Ec)
m u l t i p l i c a t i o n o f (E,q)
f o r some index i
ri(Ec)
2,
{l,...,k,
...,Z k } .
But a f t e r t h e
by c t h e rank o f any o f t h e m a t r o i d s (E,;Ji),
i n c r e a s e s b y 1 and s i n c e (Ec,Ii), (Ec,jk),
E
3 for a l l i
E
i = 1, ...,k,
...,2k a r e " c o p i e s " o f (Ec,I,) ,..., 11, ..., k ,..., 2k1, a c o n t r a d i c t i o n t o t h e p e r f e c t i = k+l,
ness assumed f o r ( E , j ) .
CONCLUSIONS AND OPEN PROBLEMS I n t h i s paper we have i n t r o d u c e d t h e concept o f independence systems, which a r e
R. Euler
112
p e r f e c t r e l a t i v e t o a s e t o f matroids.
I n p a r t i c u l a r , a polyhedral d e s c r i p t i o n
has been presented, an i n t e r r e l a t i o n t o t o t a l d u a l i n t e g r a l i t y has been establ i s h e d and i t c o u l d be shown t h a t t h i s c l a s s i s c l o s e d under m u l t i p l i c a t i o n . I n c o n n e c t i o n w i t h independence systems (EJ)
one i s o f t e n i n t e r e s t e d i n s o l v i n g
a problem o f t h e f o r m Maximize g i v e n a weight f u n c t i o n c
I ce s u b j e c t t o I e: I
(6.1)
EJ,
..IR E ' . The q u e s t i o n a r i s e s , i f , f o r t h e case o f an ~
i n d e p e n d e m system, w h i c h i s p e r f e c t r e l a t i v e t o ( E J l ) a l g o r i t h m f o r t h e s o l u t i o n o f (6.1) e x i s t s .
,. . . ,(E,Xk),
a polynomial
More s p e c i f i c a l l y , c a n e l l i p s o i d
methods b e used, as i n t h e case o f p e r f e c t graphs ( s e e [12]),
t o s o l v e (6.1)
polynmially? As a l r e a d y p o i n t e d o u t f o r m u l t i p l i c a t i o n s o f a m a t r o i d , p o l y n o m i a l a l g o r i t h m s f o r s o l v i n g ( 6 . 1 ) o v e r s p e c i f i c c l a s s e s o f independence systems ( E , j ) may be used t o s o l v e t h e r e l a t e d problems o v e r m u l t i p l i c a t i o n s o f (E,Y).
Moreover, i f
I C ! i s p o l y n o m i a l i n / E l , i t seems t o be p o s s i b l e t o i n v e r t t h e o D e r a t i o n o f (E,2) by an a p p r o p r i a t e c h e c k i n g o f t h e c i r c u i t s and, t h e r e b y , t o
multiplying
r e d u c e ( 6 . 1 ) o v e r (E,?)
t o a s i m i l a r problem o v e r ( E ' , Y ) ,
which i s n o t a proper
m u l t i p l i c a t i o n o f any o t h e r independence system, i n a p o l y n o m i a l number o f steps. We conclude w i t h a l i s t o f o t h e r open q u e s t i o n s w i t h i n t h i s framework:
-
Are t h e r e p o l y n o m i a l a l g o r i t h m s o f p u r e l y c o m b i n a t o r i a l a n a t u r e t o d e t e r m i n e a maximum ( w e i g h t ) independent s e t r e s p . a minimum p a r t i t i o n of E i n t o independ e n t subsets f o r a g i v e n p e r f e c t ( r e l a t i v e t o (E,gl),.
. . ,(E,Yk))
independence
system;
-
a r e t h e f a c e t s of t h e p o l y t o p e d e s c r i b e d by (3.1) g i v e n b y t h o s e subsets o f E, which a r e c l o s e d and i n s e p a r a b l e r e l a t i v e t o t h e r a n k - f u n c t i o n r o f
-
(E,2);
how do concepts and r e s u l t s on p e r f e c t graphs such as odd c y c l e s and a n t i c y c l e s , g e n e r a l i z a t i o n s of which c o u l d g i v e more i n s i g h t i n t o t h e f a c e t t i a l s t r u c t u r e o f independence system p o l y h e d r a , t h e c h a r a c t e r i z a t i o n o f p e r f e c t graphs as g i v e n b y Lov6sz L13:
e t c . c a r r y o v e r t o such p e r f e c t independence
systems; and l a s t , b u t n o t l e a s t
- how do o t h e r w e l l known c l a s s e s o f independence systems such as those, which a r i s e from d e g r e e - c o n s t r a i n e d subgraphs (see [4]) and
clq) f i t
i n t o t h i s framework,
o r m a t c h i n g - f o r e s t s ( s e e [9]
and a r e t h e r e i n t e r e s t i n g examples beyond
those p r e s e n t e d h e r e and those, which a r e known from t h e t h e o r y o f p e r f e c t graphs?
Independence systems and perfect k-ma froid-intersections
113
ACKNOWLEDGEMENT
I am most g r a t e f u l t o P r o f e s s o r Claude Benzaken f o r v e r y v a l u a b l e s u g g e s t i o n s .
REFERENCES
[l] Berge, C . ,
Graphes e t Hypergraphes (Dunod, P a r i s , 1973).
[2]
Chvdtal, V., On c e r t a i n p o l y t o p e s a s s o c i a t e d w i t h graphs, J . C o m b i n a t o r i a l Theory B 18 (1975) 138-154.
[3]
Cunningham, W.H. and Marsh, A.B., A p r i m a l a l g o r i t h m f o r optimum matching, Math. P r o g r a m i n g Study 8 (1978) 50-72.
[4]
Edmonds, J . , Maximum matching and a p o l y h e d r o n w i t h 0 , l - v e r t i c e s , Nat. Bur. Stand. Sect. B 69 (1965) 125-130.
[5]
Edmonds, J . , M a t r o i d s and t h e greedy a l g o r i t h m , Math. Programming 1 (1971) 127-136.
[6]
Edmonds, J . , M a t r o i d i n t e r s e c t i o n , Annals o f D i s c r e t e Math. 4 (1979) 39-49.
[7]
Edmonds, J . , and G i l e s , R., A min-max r e l a t i o n f o r submodular f u n c t i o n s on graphs, Annals o f D i s c r e t e Math. 1 (1977) 185-204.
[8]
G i l e s , R., Submodular f u n c t i o n s , graphs and i n t e g e r polyhedra, Thesis, Univ. o f Waterloo, 1975.
[9]
G i l e s , R.,
Optimum matching f o r e s t s I, Math. Programming 22 (1982) 1-11.
PO]
G i l e s , R.,
Optimum matching f o r e s t s 11, Math. Programming 22 (1982) 12-38.
el]
G i l e s , R., and P u l l e y b l a n k , W . , T o t a l dual i n t e g r a l i t y and i n t e g e r polyhedra, L i n e a r Algebra and i t s A p p l i c a t i o n s 25 (1979) 191-196.
e2]
G r o t s c h e l , M., Lovasz, L., and S c h r i j v e r , A., Polynomial A l g o r i t h m s f o r P e r f e c t Graphs, Report No. 81178-0R, I n s t i t u t f u r Okonometrie und Operations Research d e r U n i v e r s i t a t z u Bonn (1981).
e3]
Lovdsz, L., A c h a r a c t e r i z a t i o n o f p e r f e c t graphs, J . C o m b i n a t o r i a l Theory B 13 (1972) 95-98.
e4]
S b i h i , N., Etude des s t a b l e s dans l e s graphes sans @ t o i l e s , Thesis, Univ. S c i e n t i f i q u e e t Medicale de Grenoble ( 1 9 7 8 ) .
c5]
S c h r i j v e r , A., On t o t a l d u a l i n t e g r a l i t y , L i n e a r Algebra and i t s A p p l i c a t i o n s 38 (1981) 27-32. <
n6]
S c h r i j v e r , A., (1982).
Submodular f u n c t i o n s , Note AE N5/82, U n i v e r s i t y o f Amsterdam
n73
Welsh, D.J.A.,
M a t r o i d Theory (Academic Press, London, 1976).
J . Res.
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Annals of Discrete Mathematics 19 (1984) 115-128 0 Elsevier Science Publishers B.V. (North-Holland)
115
MATROIDS ON ORDERED SETS AND THE GREEDY ALGORITHM
U. F a i g l e I n s t i t u t f u r Okonometrie und O.R. U n i v e r s i t a t Bonn Nassestr. 2 0-5300 Bonn 1, W-Germany
G e n e r a l i z e d independence systems and c l a s s e s o f o b j e c t i v e f u n c t i o n s a r e i n v e s t i g a t e d f o r which t h e greedy a l g o r i t h m works w e l l . Those systems may b e viewed as m a t r o i d s on o r d e r e d ground s e t s and i n c l u d e , i n p a r t i c u l a r , systems o f i n t e g r a l v e c t o r s o f i n t e g r a l p o l y m a t r o i d s . The greedy a l g o r i t h m can b e understood as b e i n g performed i n an associ a t e d m a t r o i d on an unordered s e t , t h e ' D i l w o r t h c o m p l e t i o n ' . T h i s a l l o w s t o d e r i v e w o r s t case bounds f o r t h e greedy heuri s t i c f o r c e r t a i n ordered systems o f i n t e g r a l v e c t o r s .
1 INTRODIJCTION The greedy a l g o r i t h m i s a c o m b i n a t o r i a l procedure t o s e l e c t an o o t i m a l member o f a f a m i l y o f subsets o f some f i n i t e s e t E w i t h r e s p e c t t o a g i v e n w e i g h t i n g o f t h e elements o f E i n t h e "most s t r a i g h t - f o r w a r d " manner.
I t has been known f o r a l o n g
t i m e t h a t t h e greedy a l g o r i t h m works w e l l i f and o n l y i f t h e f a m i l y o f subsets i s t h e c o l l e c t i o n o f independent s e t s o f some m a t r o i d on E ( s e e Boruvka [2]
o r Gale
1121 1 . Edmonds [8]
has shown t h a t t h e greedy a l g o r i t h m f o r m a t r o i d s may b e viewed as t h e
s o l u t i o n o f a c e r t a i n l i n e a r orogram w i t h r e s p e c t t o ' p o l y m a t r o i d s ' , i . e . , p o l y E determined by submodular f u n c t i o n s on t h e power s e t o f E ( s e e a l s o
topes i n B
[9])
and t h u s has been a b l e t o f o r m u l a t e a more g e n e r a l greedy a l g o r i t h m f o r l i n -
ear o b j e c t i v e f u n c t i o n s o v e r p o l y m a t r o i d s . In
[lo],
we c o n s i d e r e d a g r e e d y - t y p e a l g o r i t h m i n t h e case where t h e s e l e c t i o n
r u l e has t o r e s p e c t a precedence c o n s t r a i n t g i v e n by a ( p a r t i a l l y ) o r d e r e d s e t P , and we gave a c h a r a c t e r i z a t i o n o f t h o s e systems f o r which t h e greedy a l g o r i t h m works w e l l w i t h r e s p e c t t o ' a d m i s s i b l e ' w e i g h t f u n c t i o n s . a l g o r i t h m f o r o r d e r e d s e t s i n S e c t i o n 2.
We r e v i e w t h e greedy
As an example, we o b t a i n t h e greedy
a l g o r i t h m f o r ( d i s t r i b u t i v e ) supermatroids.
Since t h e i n t e g r a l vectors o f an
i n t e g r a l p o l y m a t r o i d f o r m a supermatroid, t h e greedy a l g o r i t h m f o r o o l y m a t r o i d s ( S e c t i o n 3 ) can be d e r i v e d w i t h o u t t h e d u a l i t y t h e o r y o f l i n e a r proqramming. Systems f o r which t h e ordered greedy a l g o r i t h m works w e l l g i v e r i s e t o ' r a n k
116
11. 1;aigle
f u n c t i o n s ' on t h e l a t t i c e F o f ( o r d e r ) i d e a l s o f P. S e c t i o n 4.
We s t u d y r a n k f u n c t i o n s i n
Every r a n k f u n c t i o n d e f i n e s an ' o r d e r e d m a t r o i d ' on P .
Furthermore,
t h e s u b m o d u l a r i t y o f a r a n k f u n c t i o n a l l o w s t h e ' D i l w o r t h c o m p l e t i o n ' o f an unordered m a t r o i d on t h e same ground s e t .
I n S e c t i o n 5, we show t h a t t h e greedy a l g o r i t h m f o r o r d e r e d s e t s may b e performed i n such a way t h a t i n e f f e c t i t becomes t h e greedy a l g o r i t h m w i t h r e s p e c t t o t h e
p o l y m a t r o i d determined b y t h e a s s o c i a t e d r a n k f u n c t i o n s .
Thus t h e o p t i m i z a t i o n
problem o f a d m i s s i b l e w e i g h t f u n c t i o n s on o r d e r e d s e t s i s e q u i v a l e n t t o t h e o p t i m i z a t i o n problem o f l i n e a r f u n c t i o n s o v e r p o l y n a t r o i d s .
Moreover, t h e greedy
a l g o r i t h m f o r supermatroids can b e i n t e r p r e t e d as t h e g r e e d y a l g o r i t h m i n t h e D i l w o r t h completion.
T h i s o b s e r v a t i o n n o t o n l y shows t h a t t h e greedy a l g o r i t h m s
above work w e l l b u t a l s o t h a t c e r t a i n o p t i m i z a t i o n problems o v e r o r d e r e d systems can b e seen as problems o v e r independence systems o f s e t s .
As an a p p l i c a t i o n ,
i n S e c t i o n 6 t h e w o r s t case bound f o r t h e performance o f t h e greedy h e u r i s t i c d e r i v e d b y K o r t e and Hausmann [14i f o r i n t e r s e c t i o n s o f k m a t r o i d s i s i n e d i a t e l y o b t a i n e d f o r o r d e r e d systems o f i n t e g r a l v e c t o r s w h i c h a r e i n t e r s e c t i o n s o f t h e c o l l e c t i o n s o f i n t e g r a l vectors o f k i n t e g r a l polymatroids.
2
THE GREEDY ALGORITHM ON ORDERED SETS
I n t h i s s e c t i o n we b r i e f l y d e s c r i b e t h e greedy a l g o r i t h m on o r d e r e d s e t s as
presented i n
[lo].
We use a s l i g h t l y d i f f e r e n t t e r m i n o l o g y and we a l l o w w e i g h t
functions achieving p o s s i b l y negative values.
I t i s easy t o see t h a t t h e l a t t e r
may b e done. L e t P be a f i n i t e ( p a r t i a l l y ) o r d e r e d s e t .
We d e n o t e b y F = F ( P ) t h e ( d i s t r i b -
u t i v e ) l a t t i c e o f a l l ( o r d e r ) i d e a l s o f P, i . e . ,
.
x 6 y implies x
subsets A E. P s o t h a t y
A and
E
A.
A non-empty c o l l e c t i o n S o f sequences o v e r P i s c a l l e d a s e q u e n t i a l f a m i l y i f (S1)
For every x = xlx 2...
(s2)
For e v e r y where t k We s e t
E
S , xi \< x . i m a l i e s i 4 j . J
= xlx 2 . . . ~ n E S , "k =
lo:
xlx =
*...x k
E
S,
i s t h e i n i t i a l segment o f
i s finite.
o f l e n g t h k, 0
6
k
Q
n.
0, t h e emoty sequence.
Note t h a t because o f (S1) a l l elements i n of
3
E
S a r e d i s t i n c t and hence t h e l e n g t h
Maximal members o f S a r e b a s i c sequences.
Because o f p r o p e r t y
!S2), e v e r y s e q u e n t i a l f a m i l y i s c o m p l e t e l y determined b y i t s b a b i c bequences. As i n m a t r o i d t h e o r y , an element p
E
P i s an S-isthmus i f p occurs i n e v e r y b a s i c
Matroids on ordered sets and the greedy algorithm
117
sequence. O f t e n we w i l l n o t d i s t i n g u i s h between t h e sequence
E
S and t h e subset o f P
u n d e r l y i n g a. T h i s should cause no c o n f u s i o n . For A
E
F l e t S(A): = { a
E
S:n =A}.
Clearly, S ( A ) i s again a sequential family.
An S ( A ) - i s t h m u s w i l l s i m p l y be c a l l e d an A-isthmus. An a d m i s s i b l e w e i g h t f u n c t i o n on P i s a f u n c t i o n w: P 4 so t h a t x 6 y i m p l i e s w(x)
3
w(y) f o r a l l x,y
P.
E
w extends t o S as f o l l o w s :
O i f a = d for a
E
S, w ( a ) : =
z w(x) o t h e r w i s e . [xECY
The problem t h e n c o n s i s t s i n f i n d i n g a w-maximal member
o f the sequential
CY
family S. The greedy a l g o r i t h m i s t h e f o l l o w i n g procedure: Step I: Choose x1
E
P such t h a t w(xl)
>
0 i s maximal and x1
E
S.
I f no such
c h o i c e i s p o s s i b l e , s e t a = # and stop. Step k : If a = x ~ x ~ . . . x E~ S- ~i s c o n s t r u c t e d , choose x k t h a t axk x
E
E
P -Ixl
E
P
-
{X~,...,X~-~}
S and w(xk) i s maximal among those w(x) w i t h w(xk-,)
,..., X ~and -~ ax EIS.,
3
such
w(x) > O ,
I f no such c h o i c e i s p o s s i b l e , s t o p .
We say t h a t t h e greedy a l g o r i t h m works w e l l i f i t c o n s t r u c t s a w-maximal sequence E
S.
I f t h e greedy a l g o r i t h m works w e l l f o r e v e r y a d m i s s i b l e w, t h e s e q u e n t i a l
f a m i l y S i s c a l l e d greedy. Theorem 1 (GS,)
ay
(GS2)
[ l o ]:
For every E
The s e q u e n t i a l f a m i l y S i s greedy i f and o n l y i f a . E~
S w i t h ( a ( < 161, t h e r e e x i s t s x
E
B and y 6 x such t h a t
s.
F o r e v e r y A,B
E
F , A c _ B, i f p
E
A i s a B-isthmus, t h e n p i s a l s o an A -
isthmus . Remark 1 :
I f f o r a1 1
cx E
S , t h e elements o f
CY
a r e l i n e a r l y ordered, t h e n ( GS1 )
i m p l i e s ( G S 2 ) ( s e e C r o i t o r u [5]). The n e c e s s i t y o f t h e c o n d i t i o n s (GS1) and ( G S 2 ) i s e a s i ly seen by c o n s i d e r i n g s u i t a b l e weight functions.
We w i l l n o t r e p r o v e t h e s u f f i c i e n c y o f t h e c o n d i t i o n s
f o r t h e greedy a l g o r i t h m i n i t s general form above b u t c o n c e n t r a t e t o t h e f o l l o w i n g s p e c i a l form o f t h e greedy a l g o r i t h m .
L! Faigle
I18
Step 0: L i s t the p o s i t i v e elements o f P, x1,x2,
1 w(xi)
w(x,) Step 1: a
..., xn,
so t h a t xm
#
xi and
>I 0 f o r m < i;
0;
+
Thus, i f S i s greedy, t h e s p e c i a l greedy a l g o r i t h m w i l l produce a w-maximal element a
E
We w i l l show i n t h e n e x t s e c t i o n how t h i s
S a f t e r n t 1 steps.
181 greedy
greedy a l g o r i t h n g e n e r a l i z e s Edmonds' roids.
a l g o r i t h m f o r i n t e g r a l polymat-
On t h e o t h e r hand, we w i l l see i n S e c t i o n 5 t h a t i n t h e Dresence o f (GS1)
and ( G S i ) t h e s p e c i a l greedy a l g o r i t h m may be viewed as an execution o f Edmonds' greedy a l g o r i t h m and thus prove the s u f f i c i e n c y o f (GS1) and (GS2) f o r the special greedy a l g o r i t h m t o work w e l l . Remark 2: Step 0 of t h e s p e c i a l greedy a l g o r i t h m may be c a r r i e d o u t as f o l l o w s : Choose x1 as a minimal element of P of maximal weight, then choose x 2 as a minimal element o f P
-
o f maximal weight, e t c .
xl,
Remark 2 suggests t o e x h i b i t a canonical subfamily o f t h e sequential f a m i l y S. The minimal f a m i l y Smin
0
E
If a
Smin.
E
i f and o n l y i f wx
Smni
o f S i s c o n s t r u c t i b l e from S i n t h e f o l l o w i n g manner: i s a l r e a d y constructed and x
S and ay
E
E
P a r b i t r a r y , then ax
E
Smni
S f o r a l l y < x.
i s greedy i f S i s greedy.
Obviously, Smni
E
Moreover, t n e s p e c i a l greedy a l g o r i t h m
w i l l always s e l e c t a member o f Smin. Remark 3: A greedy f a m i l y i s n o t n e c e s s a r i l y a 'greedoid' i n t h e sense o f K o r t e and Lovasz [15].
A minimal greedy f a m i l y , however, i s a greedoid (see Section 4 ) .
We end t h i s s e c t i o n w i t h t h e examole o f a general c l a s s o f greedy f a m i l i e s . L e t L be a f i n i t e d i s t r i b u t i v e l a t t i c e .
A subset Q E L i s a s w e n n a t r o i d on L
(see Dunstan e t a1 . [ 7 ] ) i f
(SMO) 0
E
Q.
(SM1)
E
Q
x
and
y 4 x implies y
(SM?) For every x,y x
c
x'
x
y.
E
E
Q.
Q w i t h 1x1 < I y I , t h e r e e x i s t s x '
E
C! so t h a t
(Here 1x1 denotes t h e h e i g h t o f x i n t h e l a t t i c e L ) .
L e t P be t h e ordered s e t of j o i n - i r r e d u c i b l e elements o f P, i . e . , P = i p E L: p # 0, D = x w i t h the i d e a l P ( x ) = [ p
v
y implies p = x o r p = y l .
E
P: p
6
I d e n t i f y i n g every x E L XIof P, L may be viewed as t h e l a t t i c e F ( P ) o f
Matroids on ordered sets and the greedy algorithm
[l, p . 5 9 1 ) .
i d e a l s of P ( c f . B i r k h o f f
W i t h each x
E
119
Q, we a s s o c i a t e t h e c o l l e c -
t i o n S ( x ) of a l l sequences a = xlx 2 . . . ~ n such t h a t P ( x ) = {x,,x 2,...,~n} a maximal element of P ( x ) , x
~ i -s a~ maximal element o f P ( x )
-
and xn i s
xn e t c . ( i n t h e
language o f 0 5 1 , S ( x ) i s t h e ' s c h e d u l i n g g r e e d o i d ' o f t h e o r d e r e d s e t P ( x ) ) . S(Q): =
U iS(x) : x
E
Q1 t h e n i s a s e q u e n t i a l f a m i l y , and
i m p l i e s (GS,).
(Sbf2)
Moreover, we c l a i m t h a t (GS2) h o l d s . Indeed, c o n s i d e r t h e i d e a l B = P(z) f o r some z maximal element o f B.
Let
OL
E
L and A = B
be an A-basic sequence and p
(SM2), a can be augmented t o a B - b a s i c sequence 5. t h e i d e a l of P u n d e r l y i n g 6 .
By (SM1), P ( 5 )
sequence i n S(Q). Now a c P ( 5 ) P(B)
-
b.
Hence p
E
-
-
E
Assume p
-
b, where b i s some
A be a r b i t r a r y . E
By
5, and l e t P ( 5 ) be
b i s t h e u n d e r l y i n g i d e a l o f some
b c A, and t h e m a x i m a l i t y o f
a
i m p l i e s P(,)
=
a.
Since e v e r y sequence o f S(Q) forms an i d e a l i n P, i t i s c l e a r t h a t S(Q) i s a minimal family.
3 THE GREEDY ALGORITHM FOR POLYNATROIDS Making use o f t h e d u a l i t y t h e o r y o f l i n e a r programming, Edmonds [8]
has g e n e r a l -
i z e d t h e greedy a l g o r i t h m f o r m a t r o i d s t o a c l a s s o f p o l y h e d r a s o - c a l l e d ' p o l y matroids'.
We now o u t l i n e h i s approach and t h e n i n d i c a t e how t h e g r e e d y a l g o r i t h m
f o r p o l y m a t r o i d s can b e d e r i v e d f r o m t h e greedy a l g o r i t h m f o r ordered s e t s presented i n t h e previous section. L e t E be a f i n i t e s e t and f : 2'+R
a ground s e t r a n k f u n c t i o n , i . e . ,
a function
s a t is f y ing (GRo)
f(0) = 0
(GR1)
AE; B implies f ( A ) 6 f(B)
(GR2)
f ( A U B ) + f ( A n B) 6 f ( A ) + f ( B ) .
We d e f i n e t h e p o l y m a t r o s a s s o c i a t e d w i t h f by P ( f ) = {x
E
IRE : x
Here x A i s t h e r e s t r i c t i o n o f x o f x n o t i n A equal t o 0.
E
3 0,
lxAl c f(A) f o r a l l A c E l .
(3.1)
IRt t o t h e index s e t A by s e t t i n g a l l components
1x1 i s t h e sum o f t h e comoonents o f t h e v e c t o r x.
Given P ( f ) , t h e ground s e t r a n k f u n c t i o n f can b e r e c o v e r e d by f(A) = max{lxAl : x
E
P(f)}
f o r a l l A s E.
(3.2)
U. Faigle
120
The o p t i m i z a t i o n problem c o n s i s t s i n m a x i m i z i n g c . x , where c v e c t o r such t h a t x
E
RE i s a f i x e d
P(f).
E
The greedy a l g o r i t h m f o r p o l y m a t r o i d s proceeds as f o l l o w s : Step 0: L i s t t h e elements o f E, el,e2 t h a t c(el,)
SO
c(e2) t
2
...
,..., ek ,...,en, c(ek)
2
;
0
3
c(ek+l)
>,
...
3
c(en).
L e t A1 = [el
U
Ai = Ai-, Step 1 :
Construct xo
E
[ e 1. )
f o r i = 2,
..., k
P ( f ) by
0
x ( e i ) = f(A1) X
0
lei)
= 0
x(e,)
-
= f(Ai)
),
f(Ai-
i = 2,
...,k
f o r m >, k + 1 .
T h e o r m 2 [8J : The greedy a l g o r i t h m works w e l l f o r p o l y m a t r o i d s . The p o l y m a t r o i d P ( f ) i s i n t e g r a l i f t h e ground s e t rank f u n c t i o n f t a k e s an o n l y integer values.
I n t h i s case, we may r e s t r i c t o u r a t t e n t i o n t o t h e s e t Q ( f ) o f I n p a r t i c u l a r , t h e e q u a t i o n ( 3 . 2 ) remains v a l i d .
i n t e g r a l vectors o f tP(f).
So l e t ! P ( f ) be an i n t e g r a l p o l y m a t r o i d , and choose b
P(f), i.e.,
such t h a t x
6
b for all x
E
P(f).
ELNE as a bounding v e c t o r f o r
Then
D ( b ) = ',x € W E : x 6 b i
(3.3)
i s a d i s t r i b u t i v e l a t t i c e w i t h r e s p e c t t o componentwise o r d e r .
x
t
Note t h a t a v e c t o r
D(b) i s j o i n - i r r e d u c i b l e i n D ( b ) i f and o n l y i f x has e x a c t l y one non-vanishing
component. et al. Given c
Furthermore, Q ( f ) i s a s u p e r m a t r o i d w i t h r e s p e c t t o D(b) ( c f . Dunstan
17:). F
RE, we a s s i g n w e i g h t c ( e ) t o t h e j o i n - i r r e d u c i b l e element x
and o n l y i f t h e e - t h component o f x does n o t v a n i s h . admissible weight f u n c t i o n . Suppose now c(e,)
c(e2)
E
D(b) i f
C l e a r l y t h i s y i e l d s an
e v e_ ry x Moreover, c . x i s t h e induced w e i g h t f o r _
...
a c(ek)
>
0
c(ek+,)
E
D(b).
.
To o b t a i n a l i s t i n g as i n Step 0 of t h e s p e c i a l greedy a l g o r i t h m i n S e c t i o n 2, we l i s t a l l j o i n - i r r e d u c i b l e elements o f D(b) h a v i n g n o n - v a n i s h i n g el-th
component i n
i n c r e a s i n g o r d e r , t h e n a l l j o i n - i r r e d u c i b l e elements h a v i n g n o n - v a n i s h i n g e w - t h Component, e t c . I n view of ( 3 . 2 ) , t h e greedy a l g o r i t h m f o r i n t e g r a l p o l y m a t r o i d s t h u s reduces t o
121
Matroids on ordered sets and the greedy algorithm
t h e spec a1 greedy a l g o r i t h m w i t h r e s p e c t t o t h e supermatroid Q ( f ) . Remark 4
The p r e c e d i n g argument does n o t y e t p r o v e Theorem 2 f r o m Theorem 1 .
However, assuming t h a t t h e greedy a l g o r i t h m works w e l l f o r t h e s u p e t m a t r o i d Q ( f ) i t i s r o u t i n e t o d e r i v e t h e f u l l f o r m o f theorem 2 u s i n g McDiarmid's [17]
nique o f ' r a t i o n a l approximation'.
4
tech-
We o m i t t h e d e t a i l s .
RANK FUNCTIONS AND THE DILWORTH COMPLETION
We have seen how t h e greedy a l g o r i t h m f o r p o l y m a t r o i d s can b e d e r i v e d from t h e greedy a l g o r i t h m f o r s e q u e n t i a l f a m i l i e s .
Conversely, we now a s s o c i a t e w i t h each
s e q u e n t i a l f a m i l y a r a n k f u n c t i o n i n o r d e r t o r e l a t e greedy f a m i l i e s t o polymatroids.
T h i s l e a d s t o t h e d e f i n i t i o n o f an ' o r d e r e d m a t r o i d ' .
Employing D i l w o r t h ' s [6] c o n s t r u c t i o n , we t h e n embed e v e r y o r d e r e d m a t r o i d i n t o a T h i s w i l l a l l o w us t o e x h i b i t
c a n o n i c a l unordered m a t r o i d on t h e same ground s e t .
t h e greedy a l g o r i t h m s o f t h e p r e v i o u s s e c t i o n s as s p e c i a l cases o f t h e c l a s s i c a l greedy a l g o r i t h m f o r unordered m a t r o i d s . L e t S be a s e q u e n t i a l f a m i l y o v e r t h e o r d e r e d s e t P. define the
For every i d e a l A
E
F , we
S-rank r ( A ) = max { ( a l : a
E
S(A)).
(4.1)
I f p r o p e r t y (GS1) h o l d s f o r S, i t i s s t r a i g h t - f o r w a r d t o v e r i f y t h a t S and Smni
d e f i n e t h e same rank. The p r o o f o f
[lo,
Thm.91 may now b e c a r r i e d o v e r l i t e r a l l y t o show t h e f o l l o w i n g
Theorem 3: I f p r o p e r t i e s (GS,) r : F
-f
and (GS2) h o l d f o r S, t h e n t h e r a n k f u n c t i o n
N satisfies
(Ro)
r(0) = 0
(R1)
For
As B
E
(R2)
For
A,B
F,
E
F,
0 B r(B) r ( A U 6)
+
-
r(A) B lB
-
A(
r ( A n B) c r(A)
+
r(B).
Seemingly more g e n e r a l l y , we d e f i n e any f u n c t i o n r : F + [ N w i t h t h e p r o p e r t i e s (R,,),
(R1), and (R2) t o be a rank f u n c t i o n on F and t h e p a i r (P,r)
o r d e r e d m a t r o i d on P.
t o b e an
T h i s n o t i o n t h u s g e n e r a l i z e s m a t r o i d s on unordered s e t s P.
Note t h a t e v e r y rank f u n c t i o n r extends t o a ground s e t rank f u n c t i o n f o r a l l subsets S o f P by
r
defined
C! Faigle
122
7 ( S ) = r ( 5 ) , where
'5
i s t h e i d e a l o f P generated b y the
(4.2)
subset S. C o r o l l a r y 3.1: I f S s a t i s f i e s (GS1) and (GS,), t h e n Smni
(GSi)
F o r e v e r y u,t.
E
P r o o f : Note f i r s t t h a t
Smni
w i t h la!
?(Y)
= ( Y /f o r
<
Igl. there exists x
all Y
L e t 6 = xlx 2 . . . ~ n and choose x = xi ;(a) = ? ( a u Si-,)
I f ;(a)
= ?(c,~x),
F ( a U ei)
i s a greedoid, i . e . E
13 so t h a t ax
E
Smin.
S.
E
6 such t h a t
E
b u t ; ( a ) < ? ( a u Bi).
t h e n (GR2) i m p l i e s r< ? ( a u x ) + ? ( a U
ei-l)
-
?(a) = r ( a ) ,
c o n t r a d i c t i n g t h e c h o i c e o f x.
u
Thus ? ( J
r ( a ) and hence UY
x)
?(bi-lU
I f we can show
E
Smni
f o r some y 6 x .
? ( B ~ - ~ )i ,t w i l l f o l l o w f r o m t h e d e f i n i t i o n
y)
of S . that y = x. min
Suppose r ( 6 i - 1 U
F(,U
Bi-1
u Y)
Y ) = ?(Ei-l). 6 ?(aU
a contradiction t o Every r a n k f u n c t i o n r : F
Then, b y (GR2),
Ri-1) + F(Bi-1
r(aU si-l U y ) +
z
U Y) -
r(aU y)
?(Bi-l)
= ;(a),
?(a).
W d e f i n e s a semimodular c l o s u r e o p e r a t o r i . e . , a
c l o s u r e o p e r a t o r whose l a t t i c e o f c l o s e d s e t s i s (uDper) semimodular (see, e.g., B i r k h o f f [l]),
A
+
on F v i a
A
=
UiB
E
F : A c B, r ( A ) = r ( B ) J .
(4.3)
Conversely, e v e r y semimodular c l o s u r e o p e r a t o r on F y i e l d s a r a n k f u n c t i o n v i a For e v e r y A
E
F, r(A) =
x(R),
(4.4)
where h i s t h e r a n k f u n c t i o n o f t h e l a t t i c e o f c l o s e d s e t s . F o r more m a t r o i d a x i o m a t i c s , we r e f e r t o [lll. Another c o n s t r u c t i o n t o o b t a i n r a n k f u n c t i o n s f o r unordered m a t r o i d s i s essent i a l l y due t o D i l w o r t h [6!. case as f o l l o w s .
H i s method i s d i r e c t l y a p p l i c a b l e t o t h e o r d e r e d
Matroids on ordered sets and the greedy algorithm Let f : F
+
H
be any n o r m a l i z e d submodular f u n c t i o n ,
(Ro) and (R2), and d e f i n e , A
E
i.e.,
123
a function satisfying
F,
rf(A) = minIf(X)
+
]A
Theorem 4: r f : F +!N i s a r a n k f u n c t i o n .
-
XI : X
F}.
E
(4.5)
Moreover, i f f i s a r a n k f u n c t i o n ,
t h e n rf = f. Remark 5: If f : b
E
ZE +(N i s a ground s e t rank f u n c t i o n on t h e unordered s e t E and
W E a bounding v e c t o r f o r t h e i n t e g r a l p o l y m a t r o i d I P ( f ) , t h e n f y i e l d s a norm-
a l i z e d submodular f u n c t i o n on t h e d i s t r i b u t i v e l a t t i c e D(b) by For a
E
O(b), f ( a ) = f ( s u p p ( a ) ) ,
where supp(a) = { e
E
(4.6)
# 01.
E : a,
I n t h i s case, ( 4 . 5 ) g i v e s t h e v e c t o r rank o f t h e i n t e g r a l v e c t o r a respect t o t h e polymatroid P(f).
E
NE w i t h
It i s well-known t h a t t h e v e c t o r r a n k o f
i n t e g r a l v e c t o r s c o i n c i d e s w i t h t h e r a n k d e f i n e d by ( 4 . 1 ) . Next, we c o n s i d e r Po, t h e s e t P w i t h o u t o r d e r s t r u c t u r e . Fo = F(Po) t h e l a t t i c e o f a l l subsets o f Po. every A
E
L e t us denote by
S i m i l a r l y as above, we d e f i n e f o r
Fo, r f0( A ) = m i n { f ( X ) + I A
C o r o l l a r y 4.1: r;
: Fo +oU
-
XI : X
is a rank f u n c t i o n on Fo.
E
F}
(4.7
Moreover, f o r e v e r y A
E
0 rf(A) = rf(A).
F,
(4.8
0 I n v i e w o f C o r o l l a r y 4.1, we c a l l t h e unordered m a t r o i d (Po, r r ) t h e D i l w o r t h c o m p l e t i o n o f t h e o r d e r e d m a t r o i d (P, rf).
( D i l w o r t h completions o f submodular
f u n c t i o n s have been l o o k e d a t b e f o r e ; f o r t h e i n t e r e s t i n g c l a s s o f D i l w o r t h ccnnp l e t i o n s a r i s i n g from l o w e r t r u n c a t i o n s o f c o m b i n a t o r i a l geometries,
see Crapo
131 ) . Remark 6: The D i l w o r t h c a n o l e t i o n o f t h e i n t e g r a l v e c t o r rank f u n c t i o n o f an i n t e g r a l p o l y m a t r o i d ( c f . Remark 5) shows t h a t e v e r y i n t e g r a l p o l y m a t r o i d a r i s e s i n t h e f o l l o w i n g way: F o r a m a t r o i d (T,r)
{A1,
... ,A,}
an i n t e g r a l p o l y m a t r o i d (E,f)
on an unordered s e t T, and subsets on E = { A l,...,An}
i s given v i a the
ground s e t rank f u n c t i o n . f(X) = r ( U {A : A
E
XI),
X
C_
E.
( s e e Lovasz [16]). We end t h i s s e c t i o n by showing t h a t t h e D i l w o r t h c o m p l e t i o n i s t h e " f r e e s t " ( w i t h r e s p e c t t o dependence) embedding o f t h e m a t r o i d ( P y r f ) i n t o a m a t r o i d
(4.9)
U. Faigle
1 24
defined on Po. Theorem 5: L e t (Po,r) be a m a t r o i d on Po such t h a t r ( A ) = r f ( A ) f o r every A and IL Po an independent s e t o f (Po,r). Proof: L e t X
E
F be a r b i t r a r y . f(x!
3
-
F,
Then, by t h e d e f i n i t i o n o f rf, we have r ( X ) >I r ( 1
rf(X)
f ( X ) + 11
Hence
E
Then I i s an independent s e t o f (Pw r:).
n x)
= IIn
X I a 111, and thus r 0f ( I )
XI. 3
(11.
The D i l w o r t h completion provides a unique f r e e s t embedding b u t n o t a unique enbedding o f the ordered m a t r o i d (P,r) Example: Consider t h r e e l i n e s b,c,d a,b,c,d
i n t o an unordered m a t r o i d on Po.
o f an a f f i n e plane i n t e r s e c t i n g i n a p o i n t a.
form an ordered s e t P by s e t - t h e o r e t i c containment.
With each i d e a l A
o f P, associate as rank r ( A ) t h e rank ( = dimension p l u s one) o f t h e subspace generated by A.
Then t h i s geometric s t r u c t u r e may be embedded i n t o two non-
isomorphic m a t r o i d s on Po represented by t h e a f f i n e c o n f i g u r a t i o n s :
b,c,d
non-collinear
b,c,d
collinear
( F i g u r e 2)
(Figure 1) Figure 1 here shows t h e D i l w o r t h Completion.
Returning t o the greedy algorithm, we now show t h a t t h e s p e c i a l greedy a l g o r i t h m f o r ordered sets i s i m p l i e d by t h e greedy a l g o r i t h m f o r polymatroids.
Noting
t h a t every member o f a greedy family i s independent i n t h e D i l w o r t h completion, we then i n v e s t i g a t e t h e r o l e o f t h e D i l w o r t h completion w i t h respect t o t h e greedy algorithm.
As a r e s u l t , t h e greedy a l g o r i t h m f o r supermatroids (and
hence f o r polymatroids) w i l l be recognized as a s p e c i a l case o f t h e well-known greedy a l g o r i t h m f o r unordered m a t r o i d s . L e t S be a sequential f a m i l y s a t i s f y i n g (GS,) w i t h rank f u n c t i o n r.
and (GS2) over t h e ordered s e t P
Then S g i v e s r i s e t o t h e i n t e g r a l polymatroid P ( r ) =
P(r)
Matroids on ordered sets and the greedy algorithm
r
where
125
i s t h e induced ground s e t rank f u n c t i o n on Po,
Suppose we have l i s t e d t h e elements o f P, p1 ,p2,...,
as i n Step 0 o f t n e s p e c i a l
greedy a l g o r i t h m w i t h r e s p e c t t o t h e a d m i s s i b l e w e i g h t f u n c t i o n w : P
+
R.
Thus
'0
>I Wk+l > . wlpl* P i n t h e o r d e r of t h i s l i s t i n g . A l s o suppose t h a t we r e a d t h e v e c t o r s o f W
w1
For every
1 \< i Q k ,
Note t h a t a l l Ails
2
w2
>I Wk
. a .
l e t Ai =
Iq,,
E
P : m 6 i}.
-
a r e members o f F and t h a t 0 4 r(Ai)
r(Ai-l)
6
1, 2 4 i
6
k,
b y p r o p e r t y (R1) o f t h e r a n k f u n c t i o n r. We d e f i n e t h e v e c t o r x O
E
RP
as f o l l o w s :
( 0 otherwise. I n view o f t h e d e f i n i t i o n o f t h e rank f u n c t i o n r, i t i s apparent t h a t x o i s c o n s t r u c t e d a c c o r d i n g t o t h e s p e c i a l greedy a l g o r i t h m , and t h a t xn i s t h e 0 - 1 i n c i d e n c e v e c t o r o f some element o f Smni . greedy a l g o r i t h m works w e l l i f t h e 0
-
Hence, by Theorem 2, t h e s p e c i a l 1 i n c i d e n c e v e c t o r o f e v e r y member o f Smin
belongs t o P( r ) . Theorem 6: I f x
E
IRP i s t h e 0
-
1 incidence vector o f a
E
S,in,
then x
E
P(r).
Moreover, a i s independent i n t h e D i l w o r t h c o m p l e t i o n (Po,ro). P r o o f : L e t a = a,a 2...an. endent i n (Po,r
0
), i . e . ,
By i n d u c t i o n on la1 , we may assume t h a t an-, 0 r (an-l)
= n
-
1.
0 Supposp r !an-1)
i s indep-
0 = r (a).
0 can be embedded i n t o (Po,r ), :(a) > n 1 i m p l i e s 0 t h e e x i s t e n c e o f some a < an such t h a t r (an-l U a ) = n. By (GSl), we t h e r e f o r e Smni f o r some a \c a < an, c o n t r a d i c t i n g t h e assumption must have an-la'
-
S i n c e t h e o r d e r e d m a t r o i d (P,r)
a = a
n-lan
',in*
0 I f S i s any subset o f Po, we have r ( a n S ) 4 0 (Po,r ) . Hence 0 l x s l 6 r (s) 6 F(S) N o t e t h a t t h e v e c t o r u = (1,1,. P(r)*
.. ,1)
Ian SI
f o r every
because a i s independent i n
Sc
E Rp i s g e n e r a l l y
Po.
not a
bounding v e c t o r f o r
U.Faigle
126
Example: For P = { a
<
b ) , consider t h e rank f u n c t i o n r induced by t h e sequential
family Ib,a,ab). Then x = (xa,xb)
= (0,Z)
E
P(r).
The greedy a l g o r i t h m f o r supermatroids g e n e r a l i z e s t h e greedy a l g o r i t h m f o r unordered matroids since every unordered m a t r o i d may b e viewed as a supennatroid
on a Boolean algebra o r , e q u i v a l e n t l y , as an i n t e g r a l polymatroid bounded by u = (l,l,
..., 1).
We w i l l now t u r n our a t t e n t i o n t o t h e converse i m p l i c a t i o n .
L e t Q be a supermatroid on t h e f i n i t e d i s t r i b u t i v e l a t t i c e L. As remarked e a r l i e r every element of Q may be thought o f as an i d e a l o f t h e ordered s e t P = P(L) o f j o i n - i r r e d u c i b l e elements o f L.
I n p a r t i c u l a r , every element o f Q i s an indepen-
dent s e t of the D i l w o r t h completion
9,
on Po.
Moreover, we may i d e n t i f y Q w i t h
t h e minimal greedy f a m i l y S ( Q ) . Let w : P
+IR
be an admissible weight f u n c t i o n , and p1,p2,
i n Step 0 of t h e s p e c i a l greedy a l g o r i t h n .
Then p1,p2,
..., a
...
l i s t i n g of P as
i s a l s o a compatible
l i s t i n g o f Po i n order t o perform t h e greedy a l g o r i t h m w i t h r e s p e c t t o t h e D i l w o r t h completion .Q, Theorem 7: The greedy a l g o r i t h m w i t h r e s p e c t t o Q, and t h e greedy a l g o r i t h m w i t h respect t o Q y i e l d the sane i d e a l x 0 E Q. Hence, s i n c e Q 5 Q, t h e greedy a l g o r -
ittm works w e l l f o r supermatroids. Proof: Suppose X c Po i s a l r e a d y constructed according t o the Qo-greedy a l g o r i t h m and t h a t X i s a member o f Q.
Suppose p
E
P i s t h e n e x t element adjoined t o X by
the Qo-greedy a l g o r i t h m .
If
Xu p
Xu
i s n o t an i d e a l o f P, t h e r e e x i s t s p ' < p such t h a t p ' E X and p' E Q 0 0 > r (X) = r ( X ) , where r and r a r e t h e associated rank f u n c t i o n s .
since ro(X (J p)
But p ' < p says t h a t p ' i s l i s t e d b e f o r e p.
Hence
XU p '
E
Q,
contradicts the
choice o f p .
5 THE GREEDY HEURISTIC FOR ORDERED SYSTEMS As another a p p l i c a t i o n o f t h e D i l w o r t h completion we now d e r i v e a worst case bound f o r the performance o f the greedy a l g o r i t h m f o r c e r t a i n ordered systems by redu c t i o n t o a s i m i l a r r e s u l t o f K o r t e and Hausmann [14]
f o r independence systems.
An ordered system i s a f i n i t e non-empty c o l l e c t i o n Q o f i n t e g r a l vectors i n such t h a t x 6 y
ED^^
and y
E
Q implies x
E
Q.
IN E
Matroids on ordered sets and the greedy algorithm Let b
E
lNE be a bounding v e c t o r f o r Q and w : P
127
IR an a d m i s s i b l e w e i g h t f u n c t i o n ,
-f
where P i s t h e o r d e r e d s e t o f j o i n - i r r e d u c i b l e elements o f t h e d i s t r i b u t i v e l a t t i c e D(b).
Again, we i d e n t i f y each x
E
Q w i t h i t s i d e a l o f j o i n - i r r e d u c i b l e elements.
Theorem 8: L e t Q b e t h e i n t e r s e c t i o n o f k supermatroids Q,,-..,Qk v e c t o r s o f t h e i n t e g r a l p o l y m a t r o i d s IP1,...
,Bk.
4 be t h e Q the optimal
Furthermore, l e t xg e
s o l u t i o n w i t h r e s p e c t t o t h e s p e c i a l greedy a l g o r i t h m and x* solution.
of integral
E
Then:
P r o o f : As i n p r o o f o f Theorem 7, n o t e t h a t t h e greedy a l g o r i t h m y i e l d s t h e same r e s u l t as t h e greedy a l g o r i t h m w i t h r e s p e c t t o t h e i n t e r s e c t i o n o f t h e k m a t r o i d s 0 0 0 Q,, ...,Q,, where Qi i s t h e D i l w o r t h c o m p l e t i o n o f Qi on Po. B u t f o r t h e l a t t e r , (8.1) i s e x a c t l y t h e r e s u l t o f K o r t e and Hausmann. Theorem 8 has a l s o been proved b y G i r l i c h and Kowalow [13]
f o r so-called
We w i l l end by showing t h a t
' s e p a r a b l e d i s c r e t e l y concave' o b j e c t i v e f u n c t i o n s .
t h o s e o b j e c t i v e f u n c t i o n s a r e induced by a d m i s s i b l e w e i g h t f u n c t i o n s . A function f :
NE + R i s d i s c r e t e l y concave i f f(f(x) t f(z)) 6 f(y)
Furthermore, f i s s e p a r a b l e i f f o r e v e r y e f(x) =
c fe(xe) ecE
We w i l l assume f e ( 0 ) = 0 f o r a l l e
E
for all x E
E,
-
fe(x
-
1).
y 4 z E NE
.
(8.2)
t h e r e e x i s t s f e : OU
f o r every x
+ll? such t h a t (8.3)
Ernt.
E.
Consider t h e f i x e d c o o r d i n a t e f u n c t i o n f e : N w(x) = f e ( x )
6
Thus, f o r e v e r y b
E
+
R.
For every x
E
N, x a 1, l e t
INE, f i s induced on t h e d i s t r i b u -
t i v e l a t t i c e D(b) by t h e w e i g h t f u n c t i o n w i f f i s separable. We c l a i m t h a t w i s a d m i s s i b l e i f f i s d i s c r e t e l y concave.
Consider x
E
IN, x a 1.
Then, by (8.2), w(x t 1 )
-
w(x) = f e ( x t 1)
-
2fe(x) t fe(x
-
1)
4 0 . The greedy a l g o r i t h m f o r supermatroids o f p o l y m a t r o i d s i s a l s o known as a ' g r a d i e n t ' a l g o r i t h m since, a t e v e r y step, t h e v e c t o r a l r e a d y c o n s t r u c t e d i s i n c r e a s e d b y one u n i t i n t h e d i r e c t i o n y i e l d i n g t h e b i g g e s t i n c r e a s e o f t h e o b j e c t i v e function.
We remark t h a t f o r g r a d i e n t a l g i r t h m s , Theorem 8 may f a i l a l r e a d y
i n t h e case k = 1 i f t h e d i s c r e t e l y concave f u n c t i o n f i s n o t separable.
U.Faigle
128
REFERENCES Birkhoff, G . , L a t t i c e Theory (A.M.S. Colloq. Publ., Providence, 3rd ed. 1967). Boruvka, O . , 0 j i s t m problemu minimalnim, Prace Moravske Prirodovedecke Spolecnosti 3 (1926) 37-53. Crapo, H . H . ,
Geometric d u a l i t y and the Dilworth completion, Proc. I n t . Conf.
on Combinatorics, Calgary, (Gordon and Breach, New York, 1970) 37-46.
Crawley, P . , and Dilworth R . P . , Algebraic Theory of L a t t i c e s , (Prentice-Hall, Englewood C l i f f s , N.J., 1973). Croitoru, C . , A n a n a l y s i s of t h e greedy algorithm f o r p a r t i a l l y ordered s e t s , Discr. Appl. Math. 4 (1982) 113-117. Dilworth, R.P., Dependence r e l a t i o n s i n a semimodular l a t t i c e , Duke. Math. J . 11 (1944) 575-587. Dunstan, F.D.J., Ingleton, A.W., and Welsh, D.J.A., Supermatroids, Proc. Conf. Comb. Math., (Math. I n s t . Oxford, 1972) 72-122. Edmonds, J . , Submodular functions, matroids and c e r t a i n polyhedra, Proc. I n t . Conf. on Combinatorics, Calgary. (Gordon and Breach, New York, 1970) 69-87. Edmonds, J . , Matroids and t h e greedy algorithm, Math. Programing 1 (1971) 127-1 36. Faigle, U., The greedy algorithm f o r p a r t i a l l y ordered s e t s , Discr. Math. 28 (1979) 153-159. Faigle, U., 26-51.
Geometries on p a r t i a l l y ordered sets, J . Comb. Th. B 28 (1980)
Gale, D., Optimal assignments i n an ordered s e t : an a p p l i c a t i o n of matroid theory, J . Comb. T h . 4 (1968) 176-180. G i r l i c h , E . , and Kowaljow, M.M., Nichtlineare d i s k r e t e Optimierung, (Akademie-Verlag, B e r l i n , 1981). Korte, B . , and Hausmann, D., An a n a l y s i s of the greedy h e u r i s t i c f o r independence systems, Ann. Discr. Math. 2 (1978) 66-74. Korte, B . , and L o v k z , L., Mathematical s t r u c t u r e s underlying greedy algorithms, ( P r e p r i n t Univ. Bonn, Reoort No. 81189-0R, 1981). Lovgsz, L . , Matroid matching w i t h some a p p l i c a t i o n s J. Comb. Th. B 28 ( 1980) 208- 236. McDiarmid, C.J.H., Rado's theorem f o r polymatroids, Math. Proc. Camb. P h i l . SOC. 78 (1975) 263-281.
Annals of Discrete Mathematics 19 (1984) 129-134 0 Elsevier Science PublishersB.V. (North-Holland)
129
AN ALGORITHM FOR THE UNBOUNDED MATROID INTERSECTION POLYHEDRON A. Frank*and E. Tardos Research I n s t i t u t e f o r Telecomunication, Budapest, Hungary.
An a l g o r i t h m i c r e l a t i o n , between r e s u l t s o f Edmonds, Cunningham, McDiarmid and G r o f l in-Hoffman, i s discussed.
INTRODUCTION Throughout t h e paper we suppose two matroids M1 and M2 ( w i t h o u t loops) on a f i n i t e groundset E w i t h rank f u n c t i o n s rl, r 2 and a non-negative weight f u n c t i o n
w on E.
L e t us denote the maximum c a r d i n a l i t y o f a comnon independent s e t i n A
by r ( A ) . I t i s known t h a t r ( A ) = min(r,(X) + r 2 ( A - X ) ) proved t h e Matroid Polyhedron I n t e r s e c t i o n Theorem:
THEOREM 1.
[Z].
I n [2]
Edmonds a l s o
The 1i n e a r system
x a 0, d e f i n e s t h e convex h u l l
x(A)
Q
min(rl(A),r2(A))
for
A c E
(11
P o f conmon independent sets o f M1 and M2 and (1) i s
t o t a l l y dual i n t e g r a l . (A l i n e a r system Ax 4 b i s c a l l e d t o t a l l y dual i n t e g r a l o r TDI i f the l i n e a r programming dual min(yb: y
0, yA = w ) has an i n t e g r a l optimum f o r each i n t e g r a l
w whenever t h e optimum e x i s t s .
A b a s i c f e a t u r e o f T D I systems i s t h a t they
d e f i n e a polyhedron whose f a c e t s c o n t a i n i n t e g e r p o i n t s [4,8]). Edmonds [3]
a l s o provided a good a l g o r i t h m f o r o p t i m i z i n g a l i n e a r o b j e c t i v e
over P and f o r producing an optimal s o l u t i o n t o t h e l i n e a r p r o g r a m i n g dual. Fulkerson [6]
proposed t o i n v e s t i g a t e an unbounded polyhedron i n connection w i t h
matroid intersections.
Denoting by Pk the convex h u l l o f k-element comnon inde-
pendent s e t s o f M1 and M2 Fulkerson conjectured and l a t e r Cunningham [l] and McDiarmid [g]
independently proved t h a t Pk
+ Rk
A C- E l . (2) showed t h a t the l i n e a r system i n ( 2 ) i s T D I .
= { x : x(A) a max(0,k-r(E-A))
F i n a l l y , Grb'flin and Hoffman
p]
for
'* Research p a r t i a l l y supported by Sonderforschungsbereich 21 (DFG) I n s t i t u t
Operations Research, U n i v e r s i t a t Bonn.
fur
A. Frank and E. Tardos
130
The o r i g i n a l p r o o f o f G r o f l i n and Hoffman r e l i e s on t h e concept o f l a t t i c e p o l y hedra and does n o t seem t o p r o v i d e an a l g o r i t h m f o r f i n d i n g t h e optimal s o l u t i o n s i n t h e corresponding primal and dual l i n e a r programs.
The purpose o f t h i s note
i s t o present a c o n s t r u c t i v e p r o o f f o r t h e Groflin-Hoffman theorem by d e s c r i b i n g such an a l g o r i t h m . For a subset A S E and a weighting w on and w(A) = z(w(e): e o f subsets, F
E
h
A).
E,
XA denotes t h e i n c i d e n c e v e c t o r o f A
For a number x l e t x+ = max(0,x).
3 i s c a l l e d w-minimal i n 7:
Given a f a m i l y 7
i f w(F) s w ( X ) f o r each X
E
9.
PROOF AND ALGORITHM Without l o s s o f g e n e r a l i t y we can suppose t h a t rl(E)
= r 2 ( E ) = r ( E ) = k.
hen
t h e theorem o f G r o f l i n and Hoffman mentioned i n t h e I n t r o d u c t i o n i s as f o l ows.
THEOREM 2. [7]
For every i n t e g r a l weight f u n c t i o n w
>,
0 t h e dual p a i r o f
inear
programs min(wx: x(A)
>,
k-r(E-A)) = max(zArE y ( A ) ( k - r ( E - A ) ) :
(3)
Y(A) a 0, zAsE Y(A)XA=W) have i n t e g r a l optimum s o l u t i o n s .
Proof and algorithm.
Since an o p t i m a l i n t e g r a l v e c t o r i n t h e l e f t - h a n d s i d e o f
( 3 ) corresponds t o a comnon base of H1 and M2, t o prove Theorem 2 we have t o f i n d a comnon base 6 and an i n t e g r a l v e c t o r y which p r o v i d e e q u a l i t y i n ( 3 ) . By complementary slackness, t h i s i s e q u i v a l e n t t o showing t h a t y ( A ) > 0 i m p l i e s ( l B n A [ = ) x(A) = k-r(E-A) subset o f E-A.
t h a t i s B-A i s a maximal c a r d i n a l i t y common independent
Such a s e t A i s c a l l e d admissible ( w i t h r e s p e c t t o B).
Thus our
purpose i s t o f i n d a common base B and a f e a s i b l e v e c t o r y so t h a t y ( A ) > 0 o n l y
ifA i s admissible. I n [!] we proved t h e f o l l o w i n g v e r s i o n o f Theorem 1 .
LEMMA 3.
Given M1,M2,w,
a comnon base B i s w-minimal i f and o n l y i f t h e r e a r e
weights w1 ,w2 such t h a t w1+w2=w and B i s a wi-minimal Moreover, i f w i s integer-valued, The p r o o f o f t h i s lemma i n t h e w-minimal
[q
wi
base o f Mi i = 1,2.
can be chosen integer-valued.
i s by d e s c r i b i n g an a l g o r i t h m which provides both
B and t h e r e q u i r e d weight s p l i t t i n g w1,w2.
s t a r t s w i t h these data and c o n s t r u c t s y from them.
The present method
An algorithm for the unbounded matroid intersection polyhedron
... <
L e t p1 <
p,
and q1 <
... <
r e s p e c t i v e l y and s e t po = qo =
qfi be t h e d i s t i n c t v a l u e s o f t h e w e i g h t s w1,w2, Arrange t h e elements o f E i n t o a two-dimen-
-m.
s i o n a l a r r a y so t h a t x E E i s i n e n t r y ( i , j ) i f wl(x)
= pi and w ( x ) = q
2 Note t h a t t h e r e may be e n t r i e s w i t h more t h a n one element i n them. (i,j), i f p. i
s e t Aij = { v E E: wl(v) + q J > 0 and pmi l t qj-l
LEMMA 4.
If (i,j)
Proof.
L e t A1 = i v
AIU A2
=
E-Aij
< 0.
i s c r i t i c a l , t h e s e t AiJ
E: wl(v)
E
<
j' F o r an e n t r y
C a l l an e n t r y ( i , j ) c r i t i c a l w2(v) > q j } . The key o b s e r v a t i o n i s t h e f o l l o w i n g
3 pi,
and s i n c e ( i , j )
131
i s admissible.
and A2 = Cv
pi)
E
E: w2(v)
i s c r i t i c a l , A 1 f l A2 =
0.
Then < q.1. J What we show i s t h a t
B n A h i s a maximal c a r d i n a l i t y independent s u b s e t o f Ah i n Mh ( h = 1,2). i n d i r e c t l y , t h e r e e x i s t s an element v i n Mh t h e n t h e r e i s an element u
E
E
Ah-B such t h a t ( B n Ah)
B-Ah such t h a t B
S i n c e wh(v) < wh(u) t h e wh-weight o f B
+
v
-
+
v
-
+
If, v i s independent
u i s a base o f M.,
u i s s t r i c t l y smaller than t h a t o f
B, a c o n t r a d i c t i o n . /
otherwise
0 and y ( A . . ) > 0 i m p l i e s t h a t ( i , j ) i s c r i t i c a l , and then, by 1J y(A)=w(E) f o r i s a d m i s s i b l e . The f e a s i b i l i t y o f y, t h a t i s E , i m m e d i a t e l y f o l l o w s by a p p l y i n g t h e n e x t t r i v i a l lemna f o r cij = (pi+qj)
Obviously y(A) Lemma 4, Aij e
LEMMA 5.
L e t C = ( c .) be an m by n m a t r i x . Then f o r 1 6 s 6 m y 1 6 t 4 n iJ s t Cst = ' i = 1 ' j = l (Cij+Ci-lj-l-Ci-lj-'ji-l)
where coj and c 10 . i s meant t o be 0. By now t h e p r o o f o f Theorem 2 i s complete.
//
To i l l u s t r a t e t h e method c o n s i d e r t h e f o l l o w i n g example w i t h two g r a p h i c a l m a t r o i d s on e i g h t elements a
b
c
d
e
f
g
h : t h e weights
1
3
1
5
5
0
8
6
+.
A. Frank and E. Tardos
132
The comnon base and t h e weight s p l i t t i n g provided by t h e a l g o r i t h m i n
F]:
B = {a,e,d,f}
wl:
w2:
a
b
c
d
5
5
3
3
-4
-2
-2
2
e
f
3
-
2
g 2
2
h
5
3
3
3
The array:
-4
-2
F i n a l l y , we remark t h a t S c h r i j v e r DO]
2
3
proved a theorem f o r polymatroids
analogous t o the r e s u l t o f G r o f l i n and Hoffman. extends t o polymatroids as w e l l .
I t can be shown t h a t our method
An algorithm for the unbounded matroid intersection polyhedron
133
REFERENCES Cunningham, W.H., An unbounded m a t r o i d i n t e r s e c t i o n polyhedron, L i n e a r Algebra and I t s Appl. 16 (1977) 209-215. Edmonds, J . , Submodular f u n c t i o n s , matroids and c e r t a i n polyhedra, i n : Guy, R. e t a1 (eds.), Combinatorial S t r u c t u r e s and t h e i r A p p l i c a t i o n s (Gordon and Breach, New York, 1970) 69-87.
.,
Edmonds, J . , Matroid i n t e r s e c t i o n , Annals o f D i s c r e t e Math. 4 (1979) 39-49. Edmonds, J . , and Giles, R., A min-max r e l a t i o n f o r submodular f u n c t i o n s on graphs, Annals o f D i s c r e t e Math. 1 (1977) 185-204. Frank, A., A weighted m a t r o i d i n t e r s e c t i o n algorithm, Journal o f Algorithms 2 (1981) 328-33. Fulkerson, D.R., Blocking and a n t i b l o c k i n g p a i r s o f polyhedra, Math. P r o g r a m i n g 1 (1971) 108-194. G r o f l i n , A., and Hoffman, A.J., (1981) 188-194.
On m a t r o i d i n t e r s e c t i o n s , Combinatorica 1
Hoffman, A.J., A g e n e r a l i z a t i o n o f max-flow min-cut, Math. Programming 6 (1974) 352-359. McDiarmid, C.M., Blocking, Anti-blocking, and p a i r s o f matroids and p o l y matroids, J . Combinatorial Theory B 25 (1978) 313-325. Schri j v e r , A.,
Polyhedral Combinatorics, (John Wiley) t o appear.
This Page Intentionally Left Blank
Annals of Discrete Mathematics 19 (1984) 135-146 0 Elsevier Science Publishers B.V. (North-Holland)
135
ALGEBRAIC FLOWS
A.M.
Frieze
Department o f Computer Science & S t a t i s t i c s Queen Mary C o l l e g e U n i v e r s i t y o f London M i l e End Road London E l 4NS, G.B. We c o n s i d e r a n a t u r a l g e n e r a l i s a t i o n o f t h e maximum v a l u e f l o w problem, where f l o w v a l u e s a r e elements o f an o r d e r e d d-monoid. Assuming c o n s e r v a t i o n o f f l o w a t v e r t i c e s and c a p a c i t y c o n s t r a i n t s on t h e a r c s we a r e a b l e t o p r o v e a MaxFlow Min-Cut Theorem u s i n g a f l o w augmenting p a t h a l g o r i t h m . I f t h e monoid i s weakly c a n c e l l a t i v e t h e n we can make t h e a l g o r i t h m s polynomial l y bounded. INTRODUCTION We c o n s i d e r h e r e t h e problem s t u d i e d b y Hamacher [3]:
we a r e g i v e n
a d i g r a p h D = (V,A) w i t h v e r t i c e s V and a r c s A S V
x
V which (l.la)
i s l o o p l e s s and symmetric a t o t a l l y o r d e r e d commutative d-monoid (H,*,6) w i t h i d e n t i t y e a set
i.e.
H t o t a l l y o r d e r e d by
binary operation
*
c and an a s s o c i a t i v e ,
comnutative
satisfying
(1)
a 6 b i m p l i e s a*c 6 b*c f o r a l l a,b,ceH.
(2)
a
<
b i m p l i e s t h e r e e x i s t s c > e such t h a t a*c = b f o r a,b
a c a p a c i t y f u n c t i o n c:A
+
H such t h a t c ( u )
>
e for u
E
A.
2 s p e c i a l v e r t i c e s s and t. An s - t f l o w i s a f u n c t i o n f : A
+
E
H.
(1.lc) (l.ld)
H satisfying A
(1.2a)
f(v:V) = f(V:v) f o r v # s,t
(1.2b)
e
where f o r s e t s X,Y 5 V
(l.lb)
6
f ( u ) s. c ( u ) f o r u
X : Y = {(v,w) f(S) =
E
A:v
34
E
X,w
E
E
Y } and f o r S 5 A
f(v,w)
(V,W)ES N a t u r a l l y v:V i s an a b b r e v i a t i o n o f { v } : V . The v a l u e v a l ( f ) o f f l o w f i s d e f i n e d t o be f ( s : V ) . The problem o f f i n d i n g t h e s - t f l o w t h a t maximises v a l ( f ) i s a n a t u r a l
136
A . Y Frieze
g e n e r a l i s a t i o n o f t h e c l a s s i c a l maximum f l o w problem o f Ford and Fulkerson examples can be found i n [3].
[ -and
Hamacher made t h e p o i n t t h a t i n p r a c t i c e , minimum
capacity cuts are more important than maximun flows, which stresses t h e importance o f a Max-Flow Min-Cut Theorem. Unfortunately, 1.2 i s n o t q u i t e r e s t r i c t i v e enough t o make a ' s e n s i b l e ' problem and f u r t h e r r e s t r i c t i o n s a r e required.
Indeed we can see from the f o l l o w i n g ex-
ample t h a t the given d e f i n i t i o n o f f l o w leaves problems:
2 Figure 1 Note t h a t 1.2b i s s a t i s f i e d and y e t f ( s : V ) = f ( V : t ) as one
Here H = (Rt,max,<). might expect.
We w i l l consider 2 e s s e n t i a l l y d i f f e r e n t ways o f overcoming t h i s
problem.
cuts As usual a s e t X c_ V such t h a t s capacity c(X:x).
E
X, t
E
f
= V-X generates a c u t X:!
which has
Note t h a t i f i n the above examole we assume f ( u ) = c ( u ) f o r each
arc, then where X = (s,1,2,3)
we have v a l ( f ) = 2
>
0 = c(X:R) and so there i s no
Max-Flow Min-Cut Theorem f o r a r b i t r a r y flows. Hamacher d e a l t w i t h t h i s problem by p u t t i n g a r e t u r n a r c ( t , s ) and reDlac rig 1.2b by
f ( x : R ) = f(2:X) and assuming t h a t (ti,*)
for a l l
xcv
(1.3)
has a weakly c a n c e l l a t i v e p r o p e r t y (see Section 3
.
Thus
t o make any progress we must r e s t r i c t our a t t e n t i o n t o flows w i t h a d d i t i o n a l properties. I n Section 2 we consider a c l a s s o f flows f o r which we are a b l e t o prove, cons t r u c t i v e l y , a Max-Flow Min-Cut Theorem w i t h o u t making any e x t r a assumptions about the d-monoid H.
Unfortunately, t h i s a l g o r i t h i s n o t (proved t o be) p o l y n a n i a l l y
bounded. I n Section 3 we consider a d i f f e r e n t class o f flows and assume
H i s weakly cancel-
l a t i v e so t h a t we again have a Max-Flow Min-Cut Theorem and t h i s time a polyn o m i a l l y bounded algorithm f o r c o n s t r u c t i n g a maximum flow.
137
Algebraic flows DECOMPOSABLE FLOWS F o r a f l o w f l e d D ( f ) be t h e d i g r a p h ( V , A ( f ) )
where A ( f ) = { u
E
A: f ( u )
>
el.
A
f l o w f i s a P-flow i f (2.la)
D ( f ) i s a s i m p l e p a t h P ( f ) from s t o t ( p l u s i s o l a t e d v e r t i c e s )
H such t h a t f ( u )
there exists a = a ( f )
= a
for
u
E
A ( f ) and (2.lb)
f ( u ) = e f o r u E A-A(f).
A f l o w f i s decomposable i f t h e r e e x i s t P-flows fl,f 2,...,fk ( k = k ( f ) ) such t h a t f = fl*f2* ...f k ( i . e .
f ( u ) = fl(u)*f2(u)*
...f k ( u )
for u
E
A).
The m a i n r e s u l t o f t h i s s e c t i o n i s t h e f o l l o w i n g Theorem 2.1.
The maximum v a l u e o f a decomposable s - t f l o w i s equal t o t h e minimum
c a p a c i t y o f a c u t s e p a r a t i n g s and t. T h i s w i l l be proved u s i n g a f l o w augmenting p a t h argument, b u t f i r s t we need t o i n t r o d u c e some n o t a t i o n and p r o v e some s i m p l e lemnas. For a
E
H
Lemma 2.1.
dom(a) = { b a
E
E
H: a*b = a } and f o r a f l o w f dom(f) = d o m ( v a l ( f ) ) .
dom(b) and c > b i m p l i e s a
E
dom(c)
Proof. L e t c = b*d, then a*c = a*b*d = b*d = c Lemma 2.2. ___ Proof.
I f f i s a decomposable f l o w t h e n v a l ( f ) + f ( u ) f o r a l l u
E
A.
S t r a i g h t f o r w a r d b y i n d u c t i o n on k ( f ) .
( n o t e t h a t o u r example o f F i g u r e 1 shows t h i s i s n o t t r u e i n g e n e r a l f o r d-monoid flows). F o r a decomposable f l o w l e t K = K ( f ) = [k(f)]
Lemma 2.3.
L e t f be a decomposable f l o w .
where f o r p o s i t i v e i n t e g e r n
I f t h e r e e x i s t s u e A and s e t s 1,J
I c J c K such t h a t (1)
fi(4 > e
(2)
f I ( U ) = fJ(U)
for i
E
L
=
J-I
then v a l ( f ) = v a l ( f M ) where M = K-L. Proof.
L e t a = f L ( u ) = v a l ( f L ) by ( 1 ) .
L e t b = fI(u)
and c = val(f,,,).
Then
'4.M. Frieze
138
b
f M ( u ) 4 c b y Lemna 2.2.
a
E
dom(b) by ( 2 ) .
A p p l y i n g Lemna 2.1 g i v e s
a c dom(c). Lemma 2.4.
L e t f b e a decomposable f l o w and X:! val(f)*f(X:x)
Proof. arc i n
a cut.
Then
(2.2)
= f(X:!)
E q u a t i o n 2.2 h o l d s f o r P - f l o w s u s i n g t h e f a c t t h a t an s - t p a t h has one more
X:g
t h a n i t has i n
2:X.
The t r u t h o f 2.2 i n g e n e r a l f o l l o w s b y combining
t h e s e p a r a t e e q u a t i o n s f o r each p a t h . N o t e t h a t 2 . 2 i s e s s e n t i a l l y Hamacher's c o n d i t i o n and g e n e r a l l y speaking one has t o add e x t r a c o n d i t i o n s t o 1.2b i n o r d e r t h a t 2.2 o r 1.3 h o l d s . C o r o l l a r y 2.5.
L e t f b e a decomposable f l o w and X : R a c u t .
Then
v a l ( f ) 6 c(x:ji)
(2.3)
Proof. -
C o r o l l a r y 2.6.
I f f i s a decomposable f l o w t h e n
f(s:V) = f(V:t) Proof.
Put X = V - { t ;
i n 2.2.
We d e f i n e n e x t the i n c r e m e n t a l graph G ( f ) = ( V , E ( f ) ) f.
F o r (v,w)
E
V
x
V l e t i>(V,W) = (w,v)
E ( f ) = I u = (v,w)
-3
V
x
w i t h respect t o a given flow
then V):
( a ) v = t and w = s and (b) u
E
A,f(u)
{x:x*f(u)
<
c ( u ) and
= c ( u ) ~ n dom(f) =
D
or (c)
~ ( u )E A and f ( p ( u ) )
# dom(f) i
The s e t of a r c s EF of G ( f ) d e f i n e d i n ( b ) a r e c a l l e d f o r w a r d a r c s and t h e s e t o f a r c s EB d e f i n e d i n ( c ) a r e c a l l e d backward a r c s . A s i m p l e p a t h f r o m s t o t i n G ( f ) i s c a l l e d a f l o w augmenting Dath w i t h r e s p e c t t o f.
We n e x t p r o v e
139
Algebraic j k w s Lemma 2.5.
L e t f be a decomposable f l o w f o r which G ( f ) has no f l o w augmenting
paths.
Then f i s a maximum f l o w .
Proof.
Let X = I v
t
f
E
V:
E
v i s r e a c h a b l e f r o m s by a p a t h i n G ( f ) } .
Then s
E
X and
by assumption.
For u
X:g l e t g ( u )
E
. g(u)
dom(f) be such t h a t f ( u ) * g ( u ) = c ( u )
E
can r e a c h a v e r t e x o f X.
Note a l s o t h a t s i m i l a r l y u
e x i s t s e l s e we
x:X implies f ( u )
E
E
dm(f).
Thus V a l ( f ) = Val( f ) * f (
i:x)*g( x :2)
= f(X:X)*g(X:2) = c(X:R)
Thus f i s a maximum f l o w and X generates a minimum c u t . P r o v i n g t h e converse r e s u l t i . e .
t h a t g i v e n a f l o w augmenting p a t h we can a c t u a l l y T h i s i s i n e f f e c t why we
augment t h e f l o w has proved somewhat more d i f f i c u l t . a r e l o o k i n g a t decomposable f l o w s .
Lemma 2.6.
I f f i s a decomposable f l o w and G ( f ) has a f l o w augmenting p a t h then
t h e r e i s a decomposable f l o w
Proof.
Let
+
w i t h Val(+) > v a l ( f ) .
P b e a f l o w augmenting p a t h w i t h r e s p e c t t o f and l e t t h e a r c s X o f P
be d i v i d e d i n t o f o r w a r d a r c s XF and backward a r c s XB ( i t s i m p l i f i e s t h i n g s s l i g h t l y t o r e f e r t o t h e a r c s i n XB as t h e y a r e i n A i n s t e a d o f i n
.
XF l e t g ( u ) t dom(f) b e such t h a t f ( u ) * g ( u ) = c ( u
u
E
e
= min(min(g(u): u
E
XF), m i n ( f ( u ) : u
E
e 6
For
Let
XB).
As expected we a r e g o i n g t o c o n s t r u c t a f l o w f f o r wh ch Val(;) as
p(A)).
= v a l ( f ) * e > Val ( f )
dom(f) b y c o n s t r u c t i o n .
A problem a r i s e s f o r u
E
XB i f we want t o ' s u b t r a c t '
e
from f ( u ) i . e . choose f ( u )
such t h a t f ( u ) * e = f ( u ) , as one expects t o do i f one f o l l o w s an analogous proced u r e t o t h e c l a s s i c a l r e a l case.
The problem a r i s e s f r o m t h e p o s s i b i l i t y o f t h e r e
b e i n g s e v e r a l choices f o r f ( u ) and i t i s n o t c l e a r ( t o t h e a u t h o r ) which, i f any, maintain conservation o f flow. cedure t h a t we now d e s c r i b e . Phase 1.
T h i s we hope w i l l j u s t i f y t h e r a t h e r complex p r o There a r e 3 phases t o t h e update o f f.
A t t h e end o f t h i s phase, t h e decomposition o f f w i l l have been amended
( b u t n o t f i t s e l f ) so t h a t f o r each u
E
XB
t h e r e e x i s t s r = r ( u ) such t h a t
e
= f,(u)*f2(u)*
...*f r ( u ) .
(2.3)
140
A.M. Frieze
Suppose t h e r e e x i s t s u
Define x,y
f
[PI
(u)
>
F
H by
F
XB f o r which 2.3 f a i l s , then f o r some p 6 k ( f ) we have
6 = f
IP-11 x necessarily.
Now renumber f
P+ 1
,..., f k as
( u ) * x and x*y = f
[PI
( u ) which w i l l be p o s s i b l e as
fp+2,... , f k + l t o leave a gap f o r a new fp+l.
Replace a ( f p ) by x and add a new P-flow fp,l
Let Q = P(fp).
t o t h e decomposition
w i t h P ( f p + l ) = Q and a ( f p + l ) = y.
C l e a r l y 2.3 now holds f o r u w i t h r = p t l and i f 2.3 h e l d f o r u ' = u b e f o r e t h i s change, i t w i l l s t i l l h o l d b u t r ( u ' ) may have increased by 1.
We s h a l l use the term s p l i t t i n g f Phase 2.
P
u s i n g x,y t o denote the above c o n s t r u c t i o n .
A t the end o f t h i s phase t h e d e c m p o s i t i o n o f f w i l l have been amended
so t h a t t h e r e i s a sequence al ,a2,. such t h a t f o r each u
. . ,a P
o f mmbers o f H - { e l
XB t h e r e e x i s t s a permutation
E
o f t h e n o n - i d e n t i t y members o f t h e sequence
...,f r ( u ) ,
fl(u),f2(u),
t o al,a2,.
. . ,a P
Suppose then t h a t XB = {u, ,u2,.
u
E
{ U ~ , U ~ , . . . , U ~ - ~ } ,which
(thus
r =
e
.. ,ul}
r(u), which i s i d e n t i c a l
* * . ..ap).
= al a2
and assume i n d u c t i v e l y t h a t 2.4 holds f o r
holds t r i v i a l l y f o r m = 2.
We show now how t o extend
2.4 t o i n c l u d e .u, Suppose then t h a t t h e n o n - i d e n t i t y members o f t h e sequence f (urn),f 2( urn) ,. . . fr(um), r = r ( u m ) are bl,b2
,..., b
9'
We then i t e r a t i v e l y do t h e f o l l o w i n g : i f al = bl:
cl:= al;
i f a,
l e t al = ai*b,;
.,
bl:
continue w i t h a2,...ap,b2,... cl:= bl;
bq
.
f o r i:= 1 t o m-1 l e t j ( i ) be such t h a t al
corresponds t o f . ( u ) i n the g i v e n permutation o f t h e n o n - i d e n t i t y J(i) i members o f f l ( u l ) ; f o r j d j ( l ) , . . . j (m - 1 ) ) s p l i t f j using ai,bl; continue w i t h ai,a 2,...ap,b2,...bq. i f al
bl:
l e t bl = bi*al;
s p l i t t h e P-flow f . such t h a t f . ( u ) J ~m u s i n g bi,al; continue w i t h a2,...ap,bi,b2,...bq.
cl:=
corresponds t o bl,
al;
' A f t e r a t most p+q-1 i t e r a t i o n s o f t h e above we w i l l have produced a sequence
Algebraic flows c1,c2, bl
...c P' and
,. . .bq.
141
have exhausted one ( o r b o t h ) o f t h e sequences al,
L e t dl
,. . .dq'
...a P o r
denote t h e remainder o f t h e unexhausted sequence.
...bq
convenience assume t h a t bl,
For
g e t s exhausted f i r s t , t h e o t h e r case i s s i m i l a r .
We now f i n d t h a t f o r i = 1 ,...,m-1 t h e n o n - i d e n t i t y members o f t h e sequence fl(ui)
,... f r ( u .i ) ,
r = r(ui)
a r e a p e r m u t a t i o n o f c1 ,... c p,,dl,...d
n o n - i d e n t i t y members o f t h e sequence fl(um),
and t h a t t h e q' r = r ( u m ) a r e a permuta-
...fr(um),
and f u r t h e r t h a t d, *d *.. .*dq, E dom(c1*c2* ...c ) . Note t h a t -*.cP' P' t h e P-flows corresponding t o dl, ...d are d i s t i n c t from those corresponding t o q' as t h e f o r m e r have n o t been a f f e c t e d by t h e above procedure. We can cl,.. . c P' then a p p l y Lemma 2.3 t o remove t h e f l o w s corresponding t o d, ,. and we f i n d dq' t h a t 2.4 h o l d s w i t h u, i n c l u d e d .
t i o n o f cl,
..
Phase 3.
L e t al
,... a P be
as i n 2.4 and l e t S = {al
t h e f o l l o w i n g : l e t I = {i:a ( f i )
= a].
one u n i t o f f l o w a l o n g each p a t h P(fi)
,... aPI .
For each a
E
S do
C o n s t r u c t an i n t e g r a l f l o w g by sending f o r i E I.
L e t m b e equal t o t h e number
Augment g by an amount m u s i n g t h e f l o w augmenting
o f t i m e s a occurs i n al,..
path P t o c r e a t e a new i n t e g r a l f l o w h .
Decompose h i n t o a s e t o f f l o w s o f v a l u e
1 a l o n g paths from s t o t as f o l l o w s : f i n d a Dath Q from s t o t u s i n g o n l y a r c s u Decrease h b y 1 on each a r c o f Q.
' f o r which h ( u ) > 0.
S t o r e Q.
t h e r e i s no p a t h from s t o t w i t h p o s i t i v e h f l o w i n a l l i t s a r c s . be t h e paths s t o r e d . paths Q1,
...Q
q
Replace t h e P-flows fi f o r i
E
Repeat u n t i l L e t Ql,
...0'q
I b y t h e s e t o f P-flows w i t h
and f l o w v a l u e a.
I t s h o u l d b e c l e a r t h a t t h e above 3 phase procedure does i n f a c t augment f t o a flow
?- w i t h
Val(;)
= val(f)*e.
To complete t h e p r o o f o f Theorem 2.1 we must show t h a t we need o n l y augment a f i n i t e number o f times.
T h i s i s n o t d i f f i c u l t because a f t e r an augmentation a l o n g
a p a t h P e i t h e r a f o r w a r d a r c o f P becomes s a t u r a t e d o r a backward a r c o f P becomes f l o w l e s s .
Thus i f we a p p l y t h e obvious analogue o f t h e D i n i c A l g o r i t h m
[l] , t h e same arguments can be used t o show t h a t no more t h a n O ( 1 V I 4 ) augmentat i o n s a r e needed u n t i l G ( f ) has no f l o w augmenting paths. Complexity o f t h e a l g o r i t h m .
Although f i n i t e , t h e a l g o r i t h m above i s n o t p o l y -
nomial as t h e s i z e k ( f ) o f t h e decomposition seems t o b e capable t o growing exponentially.
The problem occurs i n Phase 2 where k c o u l d double (we can e a s i l y
ensure t h a t o n l y I A ( P - f l o w s a r e c r e a t e d f o r each a along a
E
E
S i n Phase 3 by r e d u c i n g h
Q by enough t o c r e a t e a t l e a s t one new h f l o w l e s s a r c ) .
142
A.M. Frieze
I n some cases t h e a l g o r i t h m can be made polynomial by ensuring t h a t a l l t h e paths P(fi),
i = 1,
a(fi)*a(f.)
...k
J
- i f P(fi) = P ( f J. ) we can r e p l a c e a(fi) by Thus i f we consider a c l a s s o f digraphs i n which t h e
are d i s t i n c t
and d e l e t e f j .
IVI
nunber o f s - t paths i s bounded by a polynomial i n
then t h e a l g o r i t h m becomes
polynanial.
ACYCLIC FLOWS Recall t h a t f o r a flow f s a t i s f y i n g 1.2 we d e f i n e A ( f ) = { u say t h a t a f l o w i s a c y c l i c i f t h e digraph D ( f ) = (V,A(f)) cycles.
A: f ( u ) > e l .
E
Note t h a t o u r 'problem' flow o f F i g u r e 1 i s n o t a c y c l i c .
t h a t t : V = V:s = Lemna 3.1.
We
has no ( d i r e c t e d ) We now assume
0 w i t h o u t any r e a l l o s s o f g e n e r a l i t y .
L e t f be an a c y c l i c f l o w and l e t
X:8 be a c u t s e p a r a t i n g s and t .
Then f(s:V)*f(X:x) Proof.
(3-3)
= f(X:R)
...
For t h e purposes o f the Lemma we assme V = {l, n l , s = 1 and t = n and
s i n c e D ( f ) i s a c y c l i c we can assume t h a t f ( i , j )
e implies i c j .
augment A w i t h those a r c s ( i , j ) where 1 6 i < j d n and (i,j)
I
A.
We t e m p o r a r i l y
We p u t
f ( i , j ) = e f o r such a r c s and note t h a t f i s s t i l l an a c y c l i c f l o w .
ii)
L e t p = p ( x ) = max( i:iE. x ) , q = q(X) = min( i:i E Case l : q > p. case.
Thus X = 11
,... p ) , R
= {p+l,-.. n l .
Note t h a t
f(f:X)
= e i n this
We v e r i f y 3.1 by i n d u c t i o n on p.
I f p = 1 then s:V = X : i and so t h e r e i s n o t h i n g t o prove. Suppose then we have v e r i f i e d t h i s case f o r
1x1
p and suppose now t h a t
c
1x1
= p.
...p- 11 then L e t Y = {l, f(Y:Y) = f(Y:p)*f(Y:R) = f(p:ii)*f(Y:f)
using 2 . l b
= f(X:l)
and t h e i n d u c t i o n s t e p i s e a s i l y completed. Case 2: q
<
p.
We proceed by i n d u c t i o n on p-q.
Case 1 provides t h e base p-q
Now f ( s : V ) * f ( R:X) = f ( s : V ) * f ( i i - t q l : X ) * f ( q : X ) = f(s:V)*f(R-{q):X
u {q})*f(q:X)
<
0.
Algebraic .flows
s i n c e f ( R - { q } : = q ) = e.
143
Also f (X:ii) = f ( X : q ) * f (X :R-Iq}) = f(q:X)*f (q:R-{q})*f (X:R-{ql)
Thus
since f(X:q) = f(V:q) = f(q;V).
f(X:R) = f(q:X)*f(X
u
{q}:ii-{ql).
Thus 3.1 h o l d s i f f(s:V)*f(X-{q}:XU
I q } ) = f(X
u {q}:l-{ql)
B u t t h i s can be assumed i f we use i n d u c t i o n on p-q,
s i n c e i f p(X) > q(X) we have
P(X) = P ( X U { q ( X ) } ) and q(X U { q ( X ) I ) > q ( X ) . Note t h a t t h e c o n c l u s i o n s o f C o r o l l a r i e s 2.5 and 2.6 t h u s h o l d f o r a c y c l i c f l o w s . Flow augmenting paths a r e d e f i n e d e x a c t l y as i n S e c t i o n 2, ( e x c e p t t h a t we can r e - d e f i n e EF = { u
6
1 dom(f)}
A: c ( u ) - f ( u )
which has t h e advantage o f b e i n g
s i m p l e r and computable i n O( ( A ] ) t i m e ) . Lemna 3.2.
I f f i s an a c y c l i c f l o w and G ( f ) has no f l o w augmenting p a t h s t h e n f
i s a maximum f l o w . P r o o f ( i d e n t i c a l t o Lemna 2.5). We now assume t h a t o u r d-monoid obeys t h e weak c a n c e l l a t i v e r u l e a*b = a*c i m p l i e s b = c o r a*b = a f o r a,b,c
E
H.
(3.2)
Then i t can be shown t h a t t h e r e e x i s t s a n o t h e r o r d e r e d s e t ( I , & ) and a s u r j e c t i v e function in: H
+
A satisfying a 4 b implies in(a)
Q
(3.3a)
in(b)
in(a*b) = max(in(a),in(b))
(3.3b) (3.3c)
i n ( a ) < i n ( b ) i m p l i e s a*b = b a*b = a*c and i n ( a ) = i n ( b ) = i n ( c ) i m p l i e s b = c I t f o l l o w s t h a t t h e element c d e f i n e d i n l . l ( b ) ( 2 ) i s unique.
t h i s b y a-b and extend t h e d e f i n i t i o n o f Note t h a t i n ( a - b ) = i n ( a ) f o r a For i
E
I l e t H ( i ) = {a
E
>
-
t o a-a = e.
b.
H:in(a) = i]
L e t K = {iE I : ( H ( i ) l = l}. P r o o f s o f a l l these r e s u l t s can be f o u n d i n Zimmermann [5].
(3.3d)
We s h a l l denote
144
A.M. Frieze
Given a f l o w augmenting path P and XB,XF and 6 as d e f i n e d i n Lemma 2.5 we l o o k i n t h i s case a t the much more s t r a i g h t f o r w a r d way o f updating t h e f l o w : SIMPLE UPDATE Let
f(a) = f(a)*e
a
E
XF
= f(a)-e
a
E
XB
= f(a)
a C P
We assume t h a t we s t a r t t h e a l g o r i t h m w i t h f ( a ) = e f o r a r A.
Now
does n o t i n f a c t guarantee t h a t f i s a flow, l e t alone a c y c l i c .
SIVPLE UPDATE
We n e x t d e f i n e
a quasi-flow f : A + H t o be one t h a t s a t i s f i e s 1.2a and in(f(a)) 6 in(f)
where i n ( f ) = i n ( v a l ( f ) )
f(v:V) = f(V:v) for a l l v
E
(3.4a) (3.4b)
V such t h a t i n ( f ( v : V ) ) = i n ( f ) o r i n ( f ( V : v ) ) ) = i n ( f ) .
f(s:V) = f ( V : t )
(3.4c)
I t i s easy t o prove Lemma 3.3.
I f f i s a q u a s i - f l o w then a f t e r SIMPLE UPDATE f i s a l s o a q u a s i - f l o w .
(Some proofs w i l l be o m i t t e d because they a r e obvious o r a s i m i l a r r e s u l t has been proved i n Hamacher.
Indeed Hamacher used SIYPLE UPDATE b u t assumed t h a t
H ( i ) had i t s own i d e n t i t y ei and l e t a-a = ei f o r a
H(i).
This means t h a t f
remains a f l o w b u t a t t h e 'expense' o f i n t r o d u c i n g ei). Thus from now on f l o w augmenting oaths a r e d e f i n e d i n terms o f quasi-flows, Lanma 3.3.
I f f i s a q u a s i - f l o w and i f G ( f ) has no f l o w augmenting paths then
t h e r e e x i s t s a c u t X:? Lemna 3.4.
such t h a t v a l ( f ) = c ( X : j ) .
L e t f be a q u a s i - f l o w .
Then t h e r e e x i s t s an a c y c l i c f l o w f " such
that val(f) = val(f"). Proof. -
F i r s t d e f i n e f ' by i f ( f ( a ) d dorn(f)) o r ( i n ( f )
f'(a) = f(a) = e (we recognise i n ( f )
E
K and f ( a ) = v a l ( f ) )
otherwise E
K by v a l ( f ) # e and v a l ( f ) * v a l ( f ) = v a l ( f ) ) .
I t i s easy t o see t h a t f ' i s a f l o w because o f 3.4b.
However f ' may n o t be a c y c l i c .
Note t h a t v a l ( f ' ) = v a l ( f ) .
Algebraic flows
I f in(f)
E
145
K f i n d an s - t path Q i n D ( f ) and l e t f " be t h e P-flow w i t h path Q and
value v a l ( f ) .
Such a path e x i s t s by 3.4b and 3 . 4 ~ .
Otherwise i f D ( f ' ) has a c y c l e C l e t f"(a)-e f o r a
C.
E
e
= m i n ( f ' ( a ) : a E C).
Replace f " ( a ) by
One can show f a i r l y e a s i l y t h a t f ' remains a f l o w and t h a t We continue u n t i l we o b t a i n an a c y c l i c f l o w f " .
v a l ( f ' ) i s unchanged.
We can again use t h e O i n i c a l g o r i t h m t o search f o r f l o w augmenting paths and again because a f t e r augmentation along a path
P, one forward a r c o f P becomes
s a t u r a t e d o r one backward a r c o f P becomes f l o w l e s s , t h e a l g o r i t h m w i l l run i n We are thus l e d t o c l a i m t h e f o l l o w i n g r e s u l t :
0 ( l V l 4 ) time. Theorem 3.1.
The maximum v a l u e o f an a c y c l i c f l o w i s equal t o t h e minimum cap-
a c i t y o f a c u t i f t h e d-monoid i s weakly c a n c e l l a t i v e . Other a l g o r i t h m s e.g.
Malhotra, Kumar and Maheshwari [4]
use augmenting paths i n a
l e s s d i r e c t manner and i t i s worth checking t h a t they do n o t c r e a t e problems: these a l g o r i t h m s proceed i n a sequence o f stages. f i n d a flow
7
The aim o f each stage i s t o
i n the l a y e r e d subgraph LG(f) = (V,E(f))
s - t paths i n D ( f ) .
The f l o w
f
made up o f t h e s h o r t e s t
i s chosen t o s a t u r a t e each s - t path i n L G ( f ) .
These a l g o r i t h m s have s t r a i g h t f o r w a r d a l g e b r a i c analogues where we add and subt r a c t and compare as if H was t h e s e t of r e a l s (we never need t o compute a-b where a
i
b).
f,
Having computed
a new f l o w f ' i s computed by
f ' ( a ) = f(a)*f(a)
a
E
EF
(3.5a)
f'(a) = f(a)-f(a)
a
E
EB
(3.5b)
We need t o check t h a t 3.4 h o l d s f o r f ' .
We note f i r s t t h a t i t can e a s i l y be
shown t h a t i n t h e a l g e b r a i c analogue o f t h e a l g o r i t h m o f [4]
that f' satisfies
I t f o l l o w s from V : s = t : V = 0 and t h e d e f i n i t i o n o f E ( f ) t h a t f '
3 . 4 ~ i n LF(g).
s a t i s f i e s 3 . 4 ~i n G also, and t h a t f'(s:V)
= f(s:v)*i(s:V)
(3.6)
We can consider two cases: Case 1: i n ( f ' )
>
in(f).
By considering o n l y those arcs a
E
E ( f ) f o r which i n
( f ( a ) ) = i n ( f ) we see t h a t they must b e forward a r c s and these a r e t h e o n l y arcs t h a t need be considered i n confirming 3.4a and 3.4b, which f o l l o w s as f i s a f l o w .
Note t h a t t h i s n e c e s s a r i l y includes t h e case i n ( f ' ) Case 2: i = i n ( f ' ) = i n ( f u .
E
K.
I n t h i s case H ( i ) i s e i t h e r an ordered group o r
t h e p o s i t i v e cone o f an ordered group
[g
and by consider'ing t h e same s e t o f arcs
146
A.M. Frieze
as i n the f i r s t case we can reduce t o t h e group case which i s e s s e n t i a l l y the same as the r e a l case. I t i s suggested t h a t one works w i t h q u a s i - f l o w s u n t i l no more f l o w augmenting
paths can be found.
Only then do we reduce t h e q u a s i - f l o w t o an a c y c l i c flow.
I t o n l y remains t o check t h a t we can do t h i s i n O(lVllAl) time.
We o u t l i n e next
how t h i s can be done. We use d e p t h - f i r s t search on D ( f ) , s t a c k i n g v e r t i c e s as they a r e v i s i t e d and removing them f r a n t h e s t a c k a f t e r a l l neighbours o f a v e r t e x have been v i s i t e d . I f t h e n e x t neighbour of t h e v e r t e x c u r r e n t l y b e i n g v i s i t e d i s on the stack then a c y c l e has been found.
I n O ( l V 1 ) time we can examine the cycle. reduce the f l o w
i n it, remove one ( o r more) a r c s from D(f)
and r e s t a r t t h e search a t t h e t a i l o f
t h e f i r s t a r c ( i n t h e o r d e r i n which used i n t h e search) removed.
We c o n t i n u e
u n t i l no more cycles a r e found i n t h i s manner and t h e search f i n i s h e s .
To bound t h e t o t a l t i m e taken we a p p o r t i o n t h e work done i n t o work done ( i ) between f i n d i n g cycles and r e s t a r t i n g t h e search, ( i i ) t r a v e r s i n g arcs between f i n d i n g cycles t h a t do n o t l i e on t h e n e x t c y c l e found and ( i i i ) t r a v e r s ng arc between f i n d i n g cycles t h a t l i e on t h e n e x t c y c l e found. For each c y c l e found t h e time spent doing ( i ) and ( i i i ) i s O( I V l ) and as no more than
A
cycles can be found, because we d e l e t e a t l e a s t one a r c a f t e r f n d i ng
one.
REFERENCES
ti1
D i n i c , E.A., A l g o r i t h m f o r s o l u t i o n o f a problem o f maximum f l o w i n a network with power e s t i m a t i o n , S o v i e t y Mathematics Doklady 11 (1970) 12771280.
t21
Ford, L.F., and Fulkerson, D . R . , Press, Princeton, 1962).
c3-J
Hamacher, H., Maximal a l g e b r a i c f l o w s : a l g o r i t h m s and examples, i n : Pape, U. (ed.), D i s c r e t e S t r u c t u r e s and Algorithms (Hansser, Munich, 1980) 153-166.
c41
Kumar, M.P., and Maheshwari, S . N . , An algorithm f o r Malhotra, V.M., f i n d i n g maximun f l o w s i n networks, I n f o r m a t i o n Processing L e t t e r s 7 (1978) 277-278.
M
Zimmermann, U., Linear and combinatorial o p t i m i z a t i o n i n ordered a l g e b r a i c s t r u c t u r e s , Annals o f D i s c r e t e Mathematics 10 (North-Holland P u b l i s h i n g Co., Amsterdam, 1981).
Flows i n networks ( P r i n c e t o n U n i v e r s i t y
O(l V I 3 )
Annals of Discrete Mathematics 19 (1984) 147-164 0 Elsevier Science Publishers B.V. (North-Holland)
147
OF RECENT RESULTS
LINEAR ALGEBRA I N DIOIDS: A SURVEY
M. Gondran
M. Minoux
D i r e c t i o n des Etudes e t Recherches EDF 1 avenue du General de G a u l l e 921 41 C1 amart FRANCE
C e n t r e N a t i o n a l d ' E t u d e s des T e l ~ c m u n i c a t i o n sPAA/TIM 38-40 r u e du General L e c l e r c 92131 I s s y l e s Moulineaux FRANCE
T h i s paper i s i n t e n d e d as a survey o f a whole s e t o f r e c e n t r e s u l t s c o n c e r n i n g some v e r y g e n e r a l a l g e b r a i c s t r u c t u r e s c a l l e d d i d i d s . It i s shown t h a t t h e most i m p o r t a n t concepts and p r o p e r t i e s o f c l a s s i c a l l i n e a r a l g e b r a (such as: s o l u t i o n o f l i n e a r and n o n l i n e a r e q u a t i o n s and systems, l i n e a r dependence and independence, d e t e r m i n a n t s , eigenvalues and eigenvectors) may be extended, i n some way, t o dioi'ds. The u s e f u l ness and a p p l i c a b i l i t y o f t h i s new t h e o r e t i c a l framework i s i l l u s t r a t e d by means o f a few t y p i c a l examples o f problems which, once f o r m u l a t e d i n t h e a p p r o p r i a t e a l g e b r a i c s t r u c t u r e , can b e g i v e n n a t u r a l and i l l u m i n a t i n g i n t e r p r e t a t i o n s : p a t h problems i n graphs, h i e r a r c h i c a l c l u s t e r i n g , p r e f e r e n c e a n a l y s i s problems.
1.
INTRODIJCTION
C l a s s i c a l a l g e b r a i c s t r u c t u r e s , such as f i e l d s and l a t t i c e s , appear, a t f i r s t s i g h t , t o be unrelated.
However, b o t h may be c o n s i d e r e d as s p e c i a l i n s t a n c e s o f
more g e n e r a l a l g e b r a i c s t r u c t u r e s c a l l e d : dio'ids. I n t h i s paper, we survey t h e m a i n r e s u l t s o b t a i n e d , o v e r t h e p a s t few y e a r s , i n t h e s t u d y o f dioi'ds,
by which i t i s shown t h a t t h e m o s t i m p o r t a n t concepts and
p r o p e r t i e s o f c l a s s i c a l l i n e a r a l g e b r a may be extended t o dio'ids:
solution o f
l i n e a r and n o n l i n e a r equations, l i n e a r dependence and independencc d e t e r m i n a n t s , e i g e n v a l u e s and e i g e n v e c t o r s . I n S e c t i o n 2 , we i n t r o d u c e t h e m a i n concepts and d e f i n i t i o n s .
The s o l u t i o n of
l i n e a r e q u a t i o n s and systems i s s t u d i e d i n S e c t i o n 3 and t h e g e n e r a l i z a t i o n t o n o n l i n e a r e q u a t i o n s i s discussed i n S e c t i o n 4.
S e c t i o n 5 i s devoted t o problems
o f l i n e a r dependence and independence and S e c t i o n 6 t o e i g e n v a l u e problems and t h e i r applications.
2
THE DIOIDS
A d i o ' i d (S,
a,@)i s
a s e t S w i t h two i n t e r n a l c o m p o s i t i o n laws @ and
6
M. Gondrnn and M. Minoux
148
such t h a t : the o p e r a t i o n @ ('ladd") g i v e s t h e s e t S a s t r u c t u r e o f
(A1 1
comnutative mono'id (closure, commutativity, a s s o c i a t i v i t y ) w i t h n e u t r a l element
E;
the operation @ ("multiply") gives the s e t S a structure
o f monoi'd ( c l o s u r e , a s s o c i a t i v i t y ) , w i t h n e u t r a l element e ( u n i t ) ; moreover
E
6
i s absorbing ( a
E
(A21
= ~ , v aQ S ) and
Q i s r i g h t and l e f t d i s t r i b u t i v e w i t h r e s p e c t t o @ ;
t h e preorder r e l i t i o n
( r e f l e x i v i t y , t r a n s i t i v i t y ) induced
by @ ( c a n o n i c a l p r e o r d e r i n g ) , and d e f i n e d by: a S b w X e S : a = b i s a p a r t i a l ordering, i . e . a
3
b and b
satisfies:
a+a
-
Algebraic s t r u c t u r e s s a t i s f y i n g ( A l )
@ c
= b (antisymnetry).
( A 3 ) are u s u a l l y r e f e r r e d t o as semi-rings
Shimbel 1954, Cuninghame-Green
and have been s t u d i e d by many authors (see e.g.
1960, 1962, Y o e l i 1961, G i f f l e r 1963, Cruon and Herve 1965, Peteanu 1967, Robert and Ferland 1968, Carre 1971, e t c . ) .
However, s i n c e t h e term "semi-ring"
naturally
suggeststhat "almost a l l the p r o p e r t i e s o f a r i n g a r e f u l f i l l e d " , we f i n d i t a more n a t u r a l and convenient terminology t o r e s t r i c t t h i s term t o t r i p l e s
(S,
0 ,@ )
f o r which t h e @ o p e r a t i o n i s c a n c e l l a t i v e (hence can be symmetrized)
and thus isomorphic t o t h e p o s i t i v e cone o f a r i n g . new term a p p l i c a b l e t o more general s t r u c t u r e s .
T h i s e x p l a i n s t h e use o f a
The name 01010 i t s e l f was f i r s t
suggested by K u n t a a n n (1972). Remark Since a @
E
= a M a f S ) t h i s i m p l i e s , by axiom ( A 3 ) , t h a t :
a b E Thus,
E
&a's)
i s t h e unique l e a s t element o f S .
As a consequence, suppose we have an
equation o f t h e form: a @ b t h i s implies and b = -
> a and
E
=
~
a b, and s i n c e b i s antisymmetric, t h i s i m p l i e s a =
E
E.
Note t h a t axiom ( A 3 ) may be unnecessary f o r t h e study o f c e r t a i n classes o f problems.
This axian i s nevertheless i m p o r t a n t t o prove uniqueness o f s o l u t i o n s
f o r epuations i n dio'ids. The s t r u c t u r e s
(R+U {+-I,
max, min),
(RU I + - } ,
min, + ) , ([O,l],
max, x ) ,
Linear algebra in dioids @?,
+,
x ) f o r i n s t a n c e , a r e dio’ids.
Gondran
3
-
149
F o r a d e t a i l e d r e v i e w o f such examples, c f .
Minoux 1979 Ch. 3 and Zimmerman 1981.
SOLVING LINEAR EQUATIONS AND SYSTEHS I N DIOIDS
L e t (S, @ t h e form
, @)
be a
dio’id and
suppose we have t o s o l v e a l i n e a r e q u a t i o n o f x = a
0
x @ b
(1)
(where a,b E S a r e g i v e n ) . We n o t e t h a t ( 1 ) reduces t o f i n d i n g a f i x e d p o i n t f o r t h e ( l i n e a r ) mapping: rdx : f ( x ) = a @ x @ b. A p p l y i n g t o ( 1 ) t h e s u c c e s s i v e a p p r o x i m a t i o n scheme: xo =
(2)
E
Xk+l = a
@
@
’k
(3)
l e a d s by i n d u c t i o n t o : x ~ =+ ( e~ @ a @ a’
w h e r e v k : ak = a @ a @
...
@
...
@ ak) @ b
@ a (k times).
Convergence o f ( 2 ) - ( 3 ) i s t h u s s t r o n g l y r e l a t e d t o t h e convergence o f : a(k) = e
0
a
0
For m o s t examples o f p r a c t i c a l imoortance, t o p o l o g i c a l p r o p e r t i e s on dio’ids: sufficient.
a*
0
... 0
ak .
i t i s n o t necessary t o i n t r o d u c e
f i n i t e convergence ( d i s c r e t e t o p o l o g y ) w i l l be
Thus we a r e l e d t o :
Definition 1 = a(p)
An element a e S i s c a l l e d p - r e g u l a r i f :
We say t h a t an element a E S i s r e g u l a r i f t h e r e e x i s t s a p such t h a t a i s pr e g u 1a r . The d e f i n i t i o n above a l s o i m p l i e s :
= a(’)
(Ur
>
0).
A s p e c i a l case f o r p - r e g u l a r elements a r e p - n i l p o t e n t elements i . e . such t h a t ap = E . Assuming p - n i l o p t e n c y i s s t r o n g e r t h a n p - r e g u l a r i t y b u t sometimes necessary when o t h e r p r o p e r t i e s (such as c o m m u t a t i v i t y o f @ ) a r e l a c k i n g .
T h i s i s t h e case,
f o r i n s t a n c e , f o r t h e g e n e r a l i z e d p a t h a l g e b r a s s t u d i e d b y Minoux 1976.
M. Gondran and M. Minoux
150
Property 1 For each p - r e g u l a r e l a n e n t a r S, t h e r e e x i s t s a * € S c a l l e d quasi-inverse o f a and such t h a t : a* = l i m a(‘) k-w
= ,(PI
= a(P+l)
...
Property 2 Let a e S be p - r e g u l a r w i t h quasi-inverse a*, and consider t h e f o l l o w i n g l i n e a r equation : x = a @
(4)
b
x @
then :
(i)t h e successive approximation scheme ( 2 ) - ( 3 ) converges i n a t most p t 1 steps t o : a* @ b s o l u t i o n o f ( 4 ) ( i i ) moreover, a* @ b i s t h e unique minimum s o l u t i o n o f ( 4 ) .
As a consequence o f Property 2 a* i s recognized as t h e (unique) minimum s o l u t i o n o f b o t h equations : x = a
6
x @ e and : x = x @ a
0
e.
Example 1 Suppose t h a t ( S ,
0 , 0 )i s
a d i o i d i n which @ and @ a r e idempotent.
Then
any element i s 1 - r e g u l a r since: a(’)=,@
a
0
0
a 2 = e e a
a = e
0
a=a(’).
This applies, i n p a r t i c u l a r , t o a l l d i o i d s which a r e d i s t r i b u t i v e l a t t i c e s .
Example 2 I n t h e case o f t h e d i o i d
(R,
Min, + ) (where
E
=
i s 0-regular and t h e quasi-inverse i s : a* = e. n o t r e g u l a r s i n c e : a ( k ) = Min {O, a, Za,
...
+-
and e = 0) any element a a 0
However, i f a
ka) = ka and a ( k )
<
0, then a i s
+ -m
when k
+ t-.
Property 2 can be extended t o systems o f l i n e a r equations i n t h e f o l l o w i n g way. L e t (5,
0 ,0 )be
a dio’id
and (Mn(S),
@
, 0 )t h e
d i o i d o f n x n square
m a t r i c e s on S. Consider t h e l i n e a r system over M,(S): X = A @
where A and
B a r e given.
X @ B
(5)
151
Linear algebra in dioids
Then we have:
Property 2 ' I f A has a q u a s i - i n v e r s e A*, (i)
x
(ii)
Moreover, A* @
=A*
@
then:
B i s a solution o f (5). B i s t h e minimum s o l u t i o n o f ( 5 ) .
I n p a r t i c u l a r i t has been shown b y many a u t h o r s t h a t a g r e a t number o f pathf i n d i n g problems i n graphs amount t o s o l v i n g such systems on a p p r o p r i a t e d i d i d s . Consider a graph G = [XI
U]
and a s s o c i a t e t o each a r c ( i , j )
4
U o f G an element
L e t A = ( a . .) be t h e g e n e r a l i z e d adjacency m a t r i x o f t h e graph
sij E S.
1J
d e f i n e d by:
-E
sij
a i j-
we n o t e : p =
r.
= { jE X / ( i , j )
. . 1 ...,i ) 9
(i1,i2,
E
if (i,j) E U otherwise
E U] ( t h e s e t o f successors o f i ) and f o r any p a t h
we d e f i n e t h e w e i g h t w(p) by:
The f o l l o w i n g s t a t e s t h e c o n d i t i o n s under which A i s r e g u l a r and g i v e s t h e minimum s o l u t i o n o f t h e system: X = A
0
X @ A
(6)
Theorem 1 (Gondran 1975, Minoux 1976) Suppose t h a t one o f t h e f o l l o w i n g assumptions i s s a t i s f i e d . (a)
a l l t h e elementary c i r c u i t s i n G have p - r e g u l a r w e i g h t and e i t h e r p = 0 o r Q i s commutative
(b)
a l l t h e elementary c i r c u i t s i n G have p - n i l p o t e n t weight.
Then m a t r i x A i s r e g u l a r , w i t h q u a s i - i n v e r s e A*,
(i) At = A @ A* = A* (ii)
(At)ij
=
6
and we have:
A i s t h e minimum s o l u t i o n o f ( 6 )
z ~ ( p1 J. . )where t h e sum extends o v e r a l l t h e paths uij frm i t o j
i n G. ( i i ) shows t h a t many p a t h problems can be reduced t o computing At o r a row ( o r a column) o f At which i s e q u i v a l e n t t o s o l v i n g a l i n e a r system i n an a p p r o p r i a t e dio'id
( c f . Table 1 ) .
As a consequence, t h e main a l g o r i t h m s f o r s o l v i n g p a t h
M. Gondran and M.Minoux
152
problems i n graphs may be viewed as extensions o f well-known a l g o r i t h m s o f c l a s s i c a l l i n e a r algebra.
For i n s t a n c e t h e f o l l o w i n g a l g o r i t h m g e n e r a l i z e s
F l o y d ' s a l g o r i t h m f o r canputing a l l s h o r t e s t paths i n a graph:
Connectivity
t0,1}
max
Path enumeration
P(X*)
U
Mu1t i - c r i t e r i a problems
P(RP)
active vectors i n t h e union
Maximal capacity
Ti
I
k - t h Shortest Path
1
1
R U {+-I
=
cone o f
m in
k s m a l l e s t terms o f two v e c t o r s
iik
I
a c t i v e vectors i n t h e Sum
Maximal r e l i a b i l i t y
ta
I
0 -s a < 13
R
Path numbering Markov chains
o r IN
{ a I O < a d l 3
Network r e l i a b i l i t y Regular language generation (Kleene)
+ k s m a l l e s t terms o f sums o f couples
polynomials w i t h variables s e t o f words
X
t
X
t
X
symmetrical difference
I Boolean
1
max
I
" 1
concatenation
Gauss-Jordan a l g o r i t h m . Computation o f the m a t r i x A+
from 1
n
* a
akk
F o r aij
i f i j f r o m l G n
+
aij
@ aik
@
tt
@ akj
A t the end, the o u t p u t m a t r i x i s A'. @are o(1).
I
X
Table 1
For K -
1
I
min
sequence o f the ordered sequence sequence o f the elements o f R w i t h n - s m a l l e s t terms n - s m a l l e s t terms o f two sequences o f sums o f couples amplitude Q
n-optimal paths
1
l a t i n multiply
max
Shortest path
1
m in
The complexity i s O(n3), assuming@ and
I
I
Linear algebra in dioids
153
I n t h e p a r t i c u l a r case where 3 i s a t o t a l o r d e r r e l a t i o n o v e r S w i t h e as t h e l a r g e s t element, we g e t t h e f o l l o w i n g g e n e r a l i z a t i o n o f D i j k s t r a ' s a l g o r i t h m ( c f . Gondran 1975):
Greedy a l g o r i t h m {Computation o f t h e f i r s t row o f A+)
...,n},
a)
S={2,3,
b)
For a l l i e ni+n. 1
c)
t
n1=e,
n i = = E i i 2 2 n n ;
j = l , k = l
rJ. n S
nj @ aji
D e f i n e j E S by TI. = J S + S - j , k + k t 1.
iEs
ni
I f k = n, end { t h e v e c t o r TI i s t h e f i r s t row o f At}, -
e l s e go t o b ) .
The c o m p l e x i t y i s O(n2). For f u r t h e r d e t a i l s about a l g o r i t h m s , see Gondran & Minoux 1979 Chapter 3. We end up t h i s s e c t i o n by m e n t i o n i n g a r e c e n t r e s u l t about e l i m i n a n t s ( K . A b d a l i and D. Saunders).
Given a m a t r i x A E Mn(S), t h e e l i m i n a n t o f A, denoted by \ A ( ,
i s t h e element o f S d e f i n e d i n t h e f o l l o w i n g way: - F o r n = 1 and 2, t h e e l i m i n a n t can be computed e x p l i c i t l y by: la1 = a
-
For n
and
= d + c a * b
3, t h e v a l u e i s s p e c i f i e d i n
terns o f a smaller order eliminant:
where
a ~ j +, 1
al 1 b.. = 'J
l,
1.
ai+l ,j+l
x = A @ x@b where b and x a r e column v e c t o r s . except t h e
ith e n t r y
Denoting b y ei t h e row w i t h a l l e n t r i e s €
equal t o e, we can t h e n s t a t e :
P r o p e r t y 3 ( K . A b d a l i , D. Saunders 1983) The s o l u t i o n o f ( 7 ) i s g i v e n by
(7)
M. Condran and M. Minoux
154
1
x.
4
:I
=
SOLVING NONLINEAR EQUATIONS IN DIOIDS
We now turn to show how t h e concept of p-regularity can be used t o s o l v e nonlinear equations in dio'ids. Suppose we want t o s o l v e the nonlinear (quadratic) equation: x = a @ x2 @ e For a
E
S, d e f i n e
a (k)2 = e @ a @ 2a2 @ 5a3 Q ( t h i s i s the formal developnent o f : e -
...
@ vkak
Je) 2a
where Vk:
and where f o r w
E.
( t h e so-called Catalan numbers)
t N
)k(l:
vk =
N,
va = a
0
...
a @
@ a
(V
times)
Then we have:
Property 4 (Gondran 1979) For each p-regular ejement, a of a) defined by:
E
S, t h e r e e x i s t s a*/'
(PI2 a*/'
=
Tim a
= a
( c a l l e d : quasisquare-root
(P+1)2 -= a
...
k-
As a consequence of the existence of quasi-square-roots, proved :
t h e following can be
Property 5 (Gondran 1979) l o ) I f a i s r e g u l a r , a*" 2')
If b and a
6
i s t h e minimum s o l u t i o n of (9).
c a r e r e g u l a r and i f @ i s carmutative then:
(b* @ c ) @ [(b*'
@ a @ c)*/']
x = a @ x2 @ b Q x @ c
e x i s t s and i s a s o l u t i o n o f :
155
Linear algebra in dioids (moreover, i t i s c o n j e c t u r e d t h a t i t i s t h e u n i q u e minimum s o l u t i o n o f ( 1 0 ) ) More g e n e r a l l y , c o n s i d e r a n o n l i n e a r ( p o l y n o m i a l ) e q u a t i o n o f t h e form:
0
x = f(x) = a
0
xn
e
As i n t h e q u a d r a t i c case, f o r each p - r e g u l a r element a d e f i n e a*/n (quasi-nth
(11) &
S, i t i s p o s s i b l e t o
r o o t o f a ) , minimum s o l u t i o n o f ( 1 1 ) by: a*/n = , ( ~ ) n = a ( P + l ) n
...
where, Yk:
Q u a s i nth r o o t s may be used t o s o l v e polynomial e q u a t i o n s i n die-ids, more g e n e r a l than (11).
I n p a r t i c u l a r , i t i s p o s s i b l e t o show t h a t , i f b and a @ cn-' a r e
r e g u l a r , and i f @ i s commutative, then: (b* @ c ) @ [(b*n
@ a @ cn-l)*'']
e x i s t s , and i s a s o l u t i o n o f : x = a @ xn @ b Z (J
5
L I N E A R DEPENDENCE AND INCEPENDENCE I N D I O I D S .
I n t h e f o l l o w i n g , we c o n s i d e r t h e d i o ' i d (Mn(S), on S w i t h zero:
z
x @ c .
BIDETERMINANTS
0 ,0 )o f
n x n square m a t r i c e s
=
*"]and
E....E
u n i t element: E = [ : e . : j
We f i r s t g e n e r a l i z e t h e concept o f d e t e r m i n a n t f o r a m a t r i x A = ( a . . ) E M n ( S ) . 1J
LetG,
(resp. : 8,) be t h e s e t o f a l l odd ( r e s p : even) p e r m u t a t i o n s o f
{l,Z,.
.. ,n}.
of
the product:
U)
For any g i v e n p e r m u t a t i o n
'A(')
0
E dl
= a ~ , o ( ~@ ) a2,u(2)
u G 2 we @
* * *
n o t e wA(u) ( t h e " w e i g h t "
@ an,u(n)*
Definition 2
I;;;!
F o r A E Mn(S) we c a l l b i d e t e r m i n a n t o f A t h e p a i r A(A) =
where
M. Gondran and M. Minoux
156
@ A2(A)=
Note t h a t t h e permanent o f A i s perm (A) = nl(A)
uA(u). C ul"a 2
d e f i n i t i o n o f b i d e t e r m i n a n t f i r s t appears i n Kuntzmann (1972).
The
A number o f w e l l -
known p r o p e r t i e s i n c l a s s i c a l l i n e a r a l g e b r a r e a d i l y extend t o b i d e t e n i n a n t s i n general dio'ids such as: lo)
i f A has a column ( o r a row) w i t h a l l elements = A&A) =
al(A)
2')
E,
then
E
I f a column ( o r a row) o f A i s a l i n e a r combination o f o t h e r columns ( o t h e r rows) then: al(A)
= n2(A) ( i n c l a s s i c a l l i n e a r algebra,
i m p l i e s d e t ( A ) = 0 s i n c e d e t ( A ) = Al(A) I n o r d e r t o extend 2')s
-
t h i s obviously
h2(A)).
we now i n t r o d u c e a more general d e f i n i t i o n o f l i n e a r
dependance:
D e f i n i t i o n 3 (Gondran and Minoux 1978) p n-vectors A' A2,. J1 c { l , ... ,p;, (Vj E
J1uJ 2
. . ,Ap
i n Sn a r e s a i d t o be l i n e a r l y dependent i f f t h e r e e x i s t s
J 2 c il,. . . ,p l ( J , n
# 0) such t h a t :
I f no such J1 , J 2 , )
z
JeJl
e x i s t , A1 ,A2..
J2 = 13). and A . # E J x 0 AJ = j ~ JX . @ AJ J 2 J
(12)
.Ap a r e l i n e a r l y independent.
Now, we s t a t e t h e main r e s u l t s r e l a t i n g b o t h n o t i o n s o f dependence and bidetermi nant:
Theorem 2 (Gondran and Minoux 1978) Suppose t h a t t h e columns o f A a r e l i n e a r l y dependent.
Then al(A)
= a2(A) under
each o f the f o l l o w i n g assumptions: (i)
@ and @ a r e c a n c e l l a t i v e
(ii)
a @ b = a o r b (Ya,b € 5 ) and
(iii)
a @ b = a or b
a @ b = i n f (a,b)
I
6
i s cancellative
(Va, b E S )
and t h e A . i n (12) s a t i s f y : A . J J
> perm ( A ) ( V j ) .
The converse o f the above r e s u l t can be proved i n t h e f o l l o w i n g case:
Theorem 3 (Gondran and Minoux 1978) We assume t h a t , Va, b
r S a @ b
= a
or b and
@ i s invertible.
Linear algebra in dioids
Then, i f A r M n ( S ) i s such t h a t al(A)
157
= a2(A) t h e columns ( a n d t h e rows) o f A a r e
1 i n e a r l y dependent. I n t h e case o f d i d i d s which a r e d i s t r i b u t i v e l a t t i c e s ,
nl(A)
however, t h e c o n d i t i o n
i s g e n e r a l l y n o t s u f f i c i e n t f o r g e t t i n g a dependence r e l a t i o n on
= A2(A)
t h e columns o f A.
1; :; lj
Consider, f o r i n s t a n c e , t h e m a t r i x :
(IR'U
i n t h e dio'id
{+=I, Max, M i n )
We check t h a t : A (A) = w ( U ) = M i n {11,14,101
=
10
a2(A) = wA(u2) = M i n {12,15,10)
=
10.
1
Thus, nl(A)
A 1
= a2(A) = perm (A) = 10, b u t no dependance r e l a t i o n between t h e
columns ( o r the rows) o f A can b e found w i t h A ~ , A ~ , x z~ 10.
I n view o f t h i s c o u n t e r example, t h e c o n d i t i o n s f o r a m a t r i x t o b e s i n g u l a r (columns l i n e a r l y dependent) i n l a t t i c e - d i o i d s have t o b e g i v e n a s l i g h t l y d i f f e r e n t form. We f i r s t d e f i n e t h e s k e l e t o n o f a m a t r i x A, as t h e 0-1 m a t r i x S(A) = (sij) sij
= 1 i f aij
where:
>
perm (A)
s . . = 0 otherwise
75 we say t h a t a 0-1 n x n m a t r i x B = ( b . .) i s p a r t l y - b a l a n c e d i f and o n l y i f t h e 1J
s e t o f i t s columns J may be p a r t i t i o n e d i n t o J, and J 2 such t h a t :
( b a l a n c e d m a t r i c e s have been s t u d i e d b y Berge 1972).
Then we have:
Theorem 3 ' (Minoux 1982) Assume t h a t a @ b = a o r b and a @ b = i n f (a,b).
ThenA i s s i n g u l a r i f and
o n l y i f S(A) i s p a r t l y balanced. Theorem 3' above a l l o w s t o g i v e a new c h a r a c t e r i z a t i o n o f balanced m a t r i c e s of Berge (1972) i n terms o f l i n e a r dependence and independence i n t h e d i o ' i d ( { O , l l ,
M. Gondran and M. Minoux
158 Max, Min)
.
L e t A be a 0-1 m a t r i x , I and J the s e t o f i t s rows and columns. For K c J and K L c I , we denote by AL t h e submatrix d e f i n e d by t h e subset o f columns ( r e s p o f rows) K (resp. : L ) . For any K c J ,
& ( K ) denotes t h e subset o f rows i E I i n
which t h e r e e x i s t a t l e a s t two t e r n s equal t o 1 and belonging t o columns i n
K.
Then we can s t a t e :
Corol l a r y 1 The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t and c h a r a c t e r i z e a balanced m a t r i x A: (i)
Every subset K c J can be p a r t i t i o n n e d i n t o K, and K2 i n such a way t h a t :
(ii)
4Kc
J, L
= 6 ( K ) t h e submatrix
K AL has i t s columns dependent i n t h e dioi'd
( t O , l I , Max, Min) (iii) For any 0-1 v e c t o r b t h e l i n e a r program: z = Max c x subject t o Ax < b x,o has an optimal i n t e g e r s o l u t i o n f o r a l l c . Condition ( i ) corresponds t o t h e d e f i n i t i o n , and c o n d i t i o n ( i i i ) has been given by Berge (1972).
Condition ( i i ) i s t h e new c h a r a c t e r i z a t i o n i n terms o f l i n e a r
dependence and independence i n dio'ids, which can be recognized as a formal analogue t o a well-known c h a r a c t e r i z a t i o n of unimodular m a t r i c e s due t o GhouilaHouri (1962).
6 A
EIGENVP.LUE PROBLEMS AND APPLICATIONS S i s c a l l e d an eigenvalue o f AcMn(S) i f and o n l y i f t h e r e e x i s t s V
Sn
(eigenvector) such t h a t : A@v = x k ' v . There i s a s t r o n g r e l a t i o n between eigenvalues and l i n e a r dependence i n general d i d i d s as shown by:
Theorem 4 (Gondran and Minoux 1978) i s eigenvalue o f Ae Mn(S) i f and o n l y i f t h e 2n x 2n m a t r i x
Linear algebra in dioids
' e
E
159
' e
E
has i t s columns dependent. By considering h as a v a r i a b l e , each term o f t h e bideterminant o f A ( h ) may be
considered as a polynomial i n A. Pl(A) then
P(A) =
]:l[
Thus i f we note
= Al(A(h))
and P2(h) = A2(A(h))
w i l l be c a l l e d t h e c h a r a c t e r i s t i c b i p o l y n m i a l o f A.
A f i r s t important property o f t h e c h a r a c t e r i s t i c bipolynomial i s t h e generalizat i o n t o dio'ids
o f the Caley-Hamilton theorem which reads:
Theorem 5 (Stranbing 1983, Gondran 1983) The m a t r i x A i t s e l f solves the c h a r a c t e r i s t i c equation P1(A) = P2(A). On t h e o t h e r hand, combining Theorem 2 and Theorem 3 leads t o t h e f o l l o w i n g g e n e r a l i z a t i o n o f a c l a s s i c a l r e s u l t i n o r d i n a r y l i n e a r algebra:
Theorem 6 L e t @ be such t h a t a @ b = a o r b and
6
invertible.
Then, h i s eigenvalue
o f A EMn(S) i f and o n l y i f h i s s o l u t i o n o f the c h a r a c t e r i s t i c equation: P1(X) = P2(X). Other c o n d i t i o n s f o r the existence o f eigenvalues and eigenvectors have been given by Gondran and Minoux (1977) such as:
Theorem 7 If a @ b = a (i)
or b,
i f @ i s idempotent and commutative, then:
A* (quasi i n v e r s e o f A) e x i s t s ;
M. Gondrun and M. Minoux
160 (ii)
every >
t
5 i s an eigenvalue of A;
( i i i ) J ( h ) ( t h e s e t of a l l eigenvectors f o r h ) i s t h e moduloid generated by those 6 1 @ ui vectors of the form where ui = (
[ATi
'A(')
denotes t h e i t h column o f A*,
and C i i i s the set of a l l c i r c u i t s origina-
ting a t i in t h e graph associated w i t h A ) . As an i l l u s t r a t i o n , we g i v e below a number o f examples where a p p l i c a t i o n of t h e
above r e s u l t s may lead t o some natural and illuminating i n t e r p r e t a t i o n s i n terms of eigenvalues and eigenvectors in dio'ids. 6.1
Jobshop scheduling
( Cuninghame-Green 1960, 1962)
The determination of the steady s t a t e behaviour of a set of machines (processing jobs with precedence c o n s t r a i n t s only) reduces t o finding an eigenvector i n t h e dioid (R, Max, +) ("schedule algebra") 6.2
Routing problems
The determination of a minimum r a t i o (cost/time) c i r c u i t i n a graph ( c f . Dantzig B l a t t n e r & Rao 1967) i s equivalent t o finding the eigenvalue and t h e associated eigenvector f o r t h e c o s t matrix i n the dioid ( R U {+-I, Min, + ) .
6.3
Hierarch i cal c 1us t e r i ng
Let A be the distance matrix of n o b j e c t s t o be c l a s s i f i e d . ( c f . Gondran 1977) t h a t i n the d i o i d ( R U {+-I, Min, Max):
.
Then i t can be shown
each level of the c l a s s i f i c a t i o n (simple-linkage c l u s t e r i n g ) corresponds t o a n
eigenvalue of A;
.
the p a r t i t i o n of t h e n o b j e c t s obtained a t any given level X , corresponds t o a
generator of the set 6.4
J (? ) .
Preferences analysis (Gondran 1979a)
Let A = ( a . .) be a preferences matrix: a i j i s equal, f o r example, t o t h e number 1J
of judges who prefer i t o j . Now, l e t us i n t e r p r e t eigenvalues and eigenvectors of A in some dio'ids. I f t h e dio'id i s (R', +, x ) , the eigenvector associated w i t h the l a r g e s t eigenvalue gives an average order, and we f i n d again the well known method of Berge 1958. If t h e dioid i s (R'U !+-I, max, x ) , the matrix A admits a unique eigenvalue which corresponds t o a c i r c u i t yo o f G such t h a t :
Linear algebra in dioids 1
1
m
n(v,) x
= W(Y0)
161
= max W ( Y ) Y
where n(Y) and w(y) a r e r e s p e c t i v e l y t h e number o f a r c s and t h e p r o d u c t o f t h e a r c s v a l u a t i o n s o f t h e c i r c u i t Y.
T h i s c i r c u i t determines t h e s e t o f o b j e c t s f o r
which t h e consensus i s t h e w o r s t . When G i s s t r o n g l y connected, we o b t a i n A by a v a r i a n t i n O(nm) o f an a l g o r i t h m o f KARP ( c f f o r i n s t a n c e Gondran Minoux 1979, Chap. 3 ) .
I f t h e d i o i d i s (R'
u
(+-I, max, min) t h e eigenvalues and t h e e i g e n v e c t o r s o f
t h e u n i q u e base a s s o c i a t e d w i t h each e i g e n v a l u e d e f i n e a f a m i l y o f p r e o r d e r s whose e q u i v a l e n c e c l a s s e s f i t i n t o each o t h e r .
ACKNOWLEDGEMENTS We thank t h e r e f e r e e s f o r t h e i r c o n s t r u c t i v e comments which h e l p e d improve t h e f i r s t v e r s i o n o f t h i s paper.
REFERENCES
[l] A b d a l i , K.S., Saunders, D.B., T r a n s i t i v e c l o s u r e and r e l a t e d s e m i r i n g p r o p e r t i e s v i a e l i m nants" (1983) To appear. [2] Berge, C.,
La t h g o r i e des graphes e t ses a p p l i c a t i o n s (Dunod, P a r i s ) .
[3]
Berge, C.,
Balanced M a t r i c e s , Mathematical Programming 2, 1 (1972) 19-31.
[4]
Carre, B.A., An Algebra f o r network r o u t i n g problems, 7 (1971) 273-294.
[5]
Cruon, R., Herve Ph. Quelques probl&nes r e l a t i f s 3 une s t r u c t u r e a l g 6 b r i q u e e t a son a p p l i c a t i o n au problgme c e n t r a l de l'ordannancement, Revue F r . Rech. Op. 34, (1965) 3-19.
[6]
Cuninghame-Green, R.A., Process s y n c h r o n i s a t i o n i n a s t e e l w o r k s - a problem o f f e a s i b i l i t y , in:Banbury and M a i t l a n d ( e d s . ) , Proc. 2nd I n t . Conf. on O p e r a t i o n a l Research, ( E n g l i s h U n i v e r s i t y Press, 1960) 323-328.
J. I n s t . Maths. A p p l i c s
[7] Cuninghame-Green, R.A., D e s c r i b i n g i n d u s t r i a l processes w i t h i n t e r f e r e n c e and a p p r o x i m a t i n g t h e i r s t e a d y - s t a t e b e h a v i o u r , O p e r a t i o n a l Research Q u a r t . 13, 1 (1962) 95-100. [8]
Cuninghame-Green, R.A., Minimax Algebra: L e c t u r e Notes i n Economics and Mathematical Systems ( S p r i n g e r Verlag, 1979).
[9]
D a n t z i g , G.B., B l a t t n e r , W.D., and Rao, M.R., F i n d i n g a c y c l e i n a g r a p h w i t h minimum c o s t t o t i m e r a t i o w i t h a p p l i c a t i o n t o a s h i p r o u t i n g problem", i n : T h g o r i e des graphes, Proc. o f t h e I n t . Symp., Rome, I t a l y , (Dunod, P a r i s 1967)
M. Gondran and M. Minoux
162 77-83.
[lq
Ghouila-Houri , A . , C a r a c t g r i s a t i o n des m a t r i c e s totalement unimodulaires, C.R.A.S. P a r i s , tome 254 (1962) 1192.
[ll] G i f f l e r , B., Scheduling general p r o d u c t i o n systems u s i n g schedule algebra, Naval Research L o g i s t i c s Q u a r t e r l y , v o l . 10, no. 3 (1963).
[lq
Gondran, M., Path algebra and alqorithms, i n : Combinatorial Programing, (B. Roy Ed.), Reidel (1975).
[13]
Gondran, M., L ' a l g o r i t h m e g l o u t o n dans l e s algebres de c h m i n s , B u l l e t i n de l a D i r e c t i o n Etudes e t Recherches, EDF, S e r i e C, 1, (1975a) 25-32.
[14]
Gondran, M., Eigenvalues and eigenvectors i n h i e r a r c h i c a l c l a s s i f i c a t i o n , i n : Barra, J.R., e t a1 (Eds.), Recent Developments i n S t a t i s t i c s (North Holland Publishing Company, 1977) 775-781.
[15]
Gondran, M., Les Clements p-re'guliers dans l e s d i d i d e s , D i s c r e t e Mathematics, 25, (1979) 33-39.
[16]
Gondran, M., Ualeurs propres e t vecteurs propres en analyse des preferences, Note EDF HI-3199 (1979a). Gondran, M.,
Le the'orsme de Cayley-Hamilton dans l e s dio'ides, note EDF (1983)
[l8]
Gondran, M., Minoux, M. , Ualeurs propres e t vecteurs propres dans l e s s m i modules e t l e u r i n t e r p r e t a t i o n en t h e o r i e des graphes, B u l l e t i n de l a D i r e c t i o n Etudes e t Recherches, EDF, S e r i e C, 2, (1977) 25-41.
[19]
Gondran, M., Minoux, M. , L'independance l i n e a i r e dans l e s dio'ides, B u l l e t i n de l a D i r e c t i o n Etudes e t Recherches, EDF, S e r i e C, 1, (1978) 67-90.
[20]
Gondran, M.,
[Zl]
Johnson, S.C., 241-243.
Minoux, M.,
Graphes e t Algorithmes ( E y r o l l e s , P a r i s , 1979).
H i e r a r c h i c a l c l u s t e r i n g schmes, P s y c h m e t r i c a 32, (1967)
[Z!] Kuntnann, J . , Theorie des reseaux graphes (Dunod, P a r i s , 1972). [23]
Minoux, M . , S t r u c t u r e s algebriques generalisees des p r o b l m e s de cheminement dans l e s graphes: theoremes, algorithmes e t a p p l i c a t i o n s , Revue Fr. Automatique, Infonnatique, Rech. Op., Vol. 10, 6, (1976) 33-62.
[24]
Minoux, M., Linear dependence and independence i n l a t t i c e d i o i d s , Note I n t e r n e CNET ( 1982)-
[Zq
Peteanu. V., An algebra o f the optimal path i n networks, Mathematica 9, (1967) 335-342.
[26]
Robert, P., Ferland, J., G e n e r a l i s a t i o n de 1 ' a l g o r i t h m e de Warshall, Revue Fr. I n f o m a t i q u e Rech. Op. 2 (1968) 71-85.
[27]
Shimbel, A,, S t r u c t u r e i n communication nets, Proc. Symp. on I n f o r m a t i o n Networks, P o l y t e c h n i c I n s t i t u t e o f Brooklyn. (1954) 119-203.
[28]
Stambing, H., A combinatorial p r o o f o f the Cayley-Hamilton theorem, D i s c r e t e Mathematics 43 (1983) 273-279.
[29]
Yoeli, M.,
A note on a g e n e r a l i z a t i o n o f boolean m a t r i x theory, h e r . Math.
Linear algebra in dioids
Monthly 68 (1961) 552-557. [30] Zimmermann, U., Linear and combinatorial optimization i n ordered algebraic s t r u c t u r e s , Annals of Discrete Mathematics 10 (North Holland, 1981).
163
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Annals of Discrete Mathematics 19 (1984) 165-182 0 Elsevier Science Publishers B.V. (North-Holland)
165
ALGEBRAIC FLOWS AND TIME-COST TRADEOFF PROBLEMS
H.W. Hamacher and S. T u f e k c i Department o f I n d u s t r i a l and Systems E n g i n e e r i n g University o f Florida, Gai n e s v i 11e, F l o r i d a 32611 U.S.A. I n t h i s paper we i n t r o d u c e a p r o j e c t c r a s h i n g model where t h e problem i s f o r m u l a t e d as an a l g e b r a i c o p t i m i z a t i o n model. By e x p l o i t i n g t h e u n d e r l y i n g network s t r u c t u r e o f t h e problem, t h e model i s t r a n s f o r m e d i n t o a sequence o f a l g e b r a i c network f l o w problems. An e f f i c i e n t a l g e b r a i c maximal f l o w a l g o r i t h m i s implemented t o o b t a i n t h e a l g e b r a i c m i n i m a l c u t s a t each s t e p t o d e t e r m i n e t h e a c t i v i t i e s t o b e m o d i f i e d . Different selections o f algebraic structures y i e l d d i f f e r e n t o b j e c t i v e f u n c t i o n s which have i n t e r e s t i n g meanings i n r e a l l i f e situations.
1
INTRODUCTION
The c l a s s i c a l t i m e - c o s t t r a d e o f f problem (CTCTP) i s a w e l l s t r u c t u r e d l i n e a r programming problem.
T h i s problem has been s t u d i e d b y s e v e r a l r e s e a r c h e r s [I, 4,
5, 9, 10, 11, 12, 13, 14, 151.
I n a l l t h e s e s t u d i e s , t h e u n d e r l y i n g network
s t r u c t u r e o f t h e model i s e x p l o i t e d .
F u l k e r s o n [4]
and K e l l e y [8] f o r m u l a t e d t h e
problem as a l i n e a r p r o g r a m i n g problem where t h e d u a l o f t h e problem possesses a network f l o w s t r u c t u r e .
P h i l l i p s and Dessouky [12]
f o r m u l a t e d t h e problem as a
network f l o w problem where a t each i t e r a t i o n a minimal c u t i s sought t o d e t e r m i n e t h e a c t i v i t i e s t o b e crashed.
T u f e k c i [15]
i n d i c a t e d t h a t i n t h e f l o w network
suggested by P h i l l i p s and Dessouky, t h e l o c a t i o n o f a minimal c u t can e a s i l y be o b t a i n e d by u s i n g a l a b e l i n g a l g o r i t h m .
T h i s a l g o r i t h m u t i l i z e s t h e maximum f l o w
v a l u e s o b t a i n e d i n t h e p r e v i o u s s t e p as a s t a r t i n g f e a s i b l e f l o w i n t h e succeeding step.
A l l t h e s e a l g o r i t h m s a s s p e a s i n g l e l i n e a r o b j e c t i v e f u n c t i o n which r e p r e s e n t s t h e t o t a l a d d i t i o n a l d i r e c t c o s t based on a g i v e n s e t o f a c t i v i t y d u r a t i o n s . Moore, e t a l . ,
[lo]
suggested t h a t i n many r e a l l i f e p r o j e c t s , m i n i m i z a t i o n o f
t h e a d d i t i o n a l d i r e c t c o s t may n o t n e c e s s a r i l y r e f l e c t t h e t r u e o b j e c t i v e o f t h e management.
F u r t h e r , t h e y c l a i m t h a t t h e management i s g e n e r a l l y f a c e d w i t h
m u l t i p l e objectives.
They i n t u r n propose a goal programming approach f o r a
mu1 t i - c r i t e r i a p r o j e c t c r a s h i n g model. I n t h i s paper we show t h a t t h e c l a s s i c a l TCTP as w e l l as c e r t a i n m u l t i - o b j e c t i v e
H. W.Hamacher and S. Tufekci
I66
TCTP can be t r e a t e d w i t h i n a u n i f i e d framework, i f t h e problem i s modeled as an a l g e b r a i c o p t i m i z a t i o n problem.
By t a k i n g t h e u n d e r l y i n g network s t r u c t u r e o f
t h e model i n t o c o n s i d e r a t i o n the problem i s converted i n t o a sequence o f a l g e b r a i c network f l o w problems.
Recent a l g o r i t h m i c developments i n a l g e b r a i c f l o w s and
a l g e b r a i c minimal c u t s [6,
71
enable us t o s o l v e t h i s problem v e r y e f f i c i e n t l y .
I n s e c t i o n two we i n t r o d u c e t h e generalized TCTP w i t h a l g e b r a i c o b j e c t i v e function.
We a l s o show t h a t the c l a s s i c a l TCTP and the m u l t i o b j e c t i v e TCTP a r e
s p e c i a l cases o f t h i s g e n e r a l i z e d problem.
Section t h r e e i n t r o d u c e s t h e a l g o r i t h m
f o r the a l g e b r a i c time-cost t r a d e o f f problem.
An example problem i s provided i n
I n s e c t i o n f i v e we prove t h e v a l i d i t y o f t h e presented a l g o r i t h m .
section four.
Section s i x concludes t h i s work.
2
TIWE-COST TRADEOFF PROBLEM MITH ALGEBRAIC OBJECTIVE FUNCTION
I n what f o l l o w s we assume t h a t G i s an a c y c l i c a c t i v i t y - o n - a r c p r o j e c t network w i t h node set ties.
N, r e p r e s e n t i n g t h e events and a r c s e t A, r e p r e s e n t i n g t h e a c t i v i -
For convenience we assume t h a t node one represents t h e beginning o f the
p r o j e c t and node n represents t h e end o f the p r o j e c t . we associate two numbers aij
and bij
(aij
6 bij)
normal d u r a t i o n o f t h e a c t i v i t y , r e s p e c t i v e l y .
With each a c t i v i t y
(i,j)E
A
c a l l e d the crash d u r a t i o n and For g i v e n d u r a t i o n s dij,
(i,j) E A
we can f i n d the d u r a t i o n o f the p r o j e c t , T(d) by u s i n g a standard CPM technique
[81. I f T i s a g i v e n a l l o w a b l e p r o j e c t d u r a t i o n , then a c t i v i t y d u r a t i o n s d = (dij), ( i , j ) E A are c a l l e d a f e a s i b l e s o l u t i o n w i t h r e s p e c t t o time T, i f aij 6 dij 4 b i j f o r a l l (i,j) E A and T(d) 6 T. s e t w i t h respect t o time .T, i.e., F(T) = Idij:aij I f we d e f i n e d . . = bij, 1J
duration.
6 dij
V(i,j),
6
bij,
By F(T) we denote t h e f e a s i b l e
( i , j ) E A , T(d) c< T I .
then T(d) = Tn i s c a l l e d t h e normal p r o j e c t
Similarly, f o r d.. = a
V(i,j), T(d) = T C d e f i n e s t h e crash d u r a t i o n ij’ For an a r b i t r a r y p r o j e c t d u r a t i o n T, T C s. T < Tn some a c t i v i t i e s IJ
o f the p r o j e c t .
must be crashed t o achieve the d e s i r e d p r o j e c t l e n g t h . some a c t i v i t y d u r a t i o n s from dij
= bij
to dij
r e q u i r e s a d d i t i o n a l resources, and money.
<
bij.
That i s , we have t o change I n t h e TCTP t h i s crashing
Thus, t h e TCTP i s d e f i n e d as f i n d i n g
t h a t s e t of a c t i v i t y d u r a t i o n s d E F(T), which y i e l d s t h e minimal c o s t . I n our approach the c o s t s are elements o f an ordered s e t H provided w i t h an a l g e b r a i c s t r u c t u r e (H,*,&).
By t a k i n g t h e c o s t elements f r a n such a s e t we g e t
a u n i f i e d treatment o f d i f f e r e n t o b j e c t i v e f u n c t i o n s based on t h e choice o f (H,*,,c).
The p r a c t i c a l importance o f t h i s approach w i l l be made c l e a r e r i n
section four.
167
Algebraic flowsand time-cost tradeoff problems D e f i n i t i o n 1.
(H,*,$)
i s c a l l e d a t o t a l l y o r d e r e d commutative group (TOCG) i f f
a 6 b "a*c Note t h a t s i n c e a*b = a*c -b=c a
6
(H.2) i s t o t a l l y o r d e r e d
(2.1)
(H,*) i s a c o m n u t a t i v e group
(2.2)
b*c ( c o m p a t i b i l i t y r u l e )
V a,b,c
E
(2.3)
(H,*) i s a group, t h e c a n c e l l a t i o n r u l e h o l d s ; i . e . , Y a,b,c 0 H . We denote w i t h H, t h e s e t o f a l l elements a
Q
H with
>, E .
H = LRm, s e t o f m-component v e c t o r s o v e r R ,
Example 1.
*
i s t h e componentwise
26
a d d i t i o n , and 4 i s t h e l e x i c o g r a p h i c a l o r d e r i n g o f IRm, ( t h a t i s , a = b o r ai [-
Example 2.
<
H =
bi f o r i = m i n { j : a j # b j l ,
R, *
Y
+,be
= (R t o H =
b
RT.
i s t h e m u l t i p l i c a t i o n , and 6 i s t h e n a t u r a l o r d e r i n g o f r e a l s .
n o t a TOCG ( s i n c e 2.3 i s n o t s a t i s f i e d ) . Note t h a t t h i s system i s -
H
H.
R, t h e n (H,*,<)
I f we change
becomes a TOCG.
I n o r d e r t o f i n d a measure f o r t h e c o s t o f c r a s h i n g a c t i v i t i e s , we want t o compose t h e m o u n t (bij Cij
6
H.
-
d . . ) e R t , by which an a c t i v i t y ( i , j )
i s crashed, w i t h t h e c o s t
1J
To f a c i l i t a t e t h i s we i n t r o d u c e an e x t e r n a l c o m p o s i t i o n 0 : IR
x
H
-f
H.
Now, t h e a l g e b r a i c TCTP can be f o r m u l a t e d as f o l l o w s : f o r any g i v e n T w i t h T' 4 T 6 T
n solve
(ATCTP)
min{(i:j)rA
-
(bij
dij)u
cijldaF(T)l.
I n o r d e r t o s o l v e t h i s problem i n t h i s g e n e r a l form we need s e v e r a l axioms about t h e external composition El. (2.4.a)
a U ( B 0 a ) = (a.B)Oa (a+B)
aa
= (am)
aa(a*b) =
(am) *
with
= -a, (2.4.d),
(2.4.b)
(BW)
(mob)
Y
a , M ,
a,beH
(2.4.c)
E
(2.4.d)
lna = a
(2.4.e)
Ooa =
By u s i n g (2.4.b)
*
(2.2.d)
and t h e c a n c e l l a t i o n r u l e i t i s
easy t o show t h a t ( - a ) o a = -(am) Moreover, we need
(2.4.f)
H W.Harnacher and S.Tufekci
168
ir
<
6
saOC 6
a 6 b -aoa I f we r e s t r i c t t h e elements
2
6
EM.,
4 a,BER,
nab,
4 a&+,a,lxH
e @ t o a subring ( R , + , - )
r e a l s , and
=
*
H = R,
o f the f i e l d o f r e a l s then
H).
s t a n d i n g we denote t h i s s t r u c t u r e s i m p l y by
R
(2.5.b)
a module o v e r R (whenever t h e r e i s no misunder-
we c a l l t h e s t r u c t u r e (R,O,H,*,s)
Example 3.
(2.5.a)
CaH,.
i s the a d d i t i o n o f r e a l s , 4 i s t h e n a t u r a l o r d e r i n g o f
i s t h e m u l t i p l i c a t i o n of r e a l s .
The c o r r e s p o n d i n g ATCTP becomes t h e c l a s s i c a l TCTP (CTCTP) w i t h s i n g l e l i n e a r o b j e c t i v e f u n c t i o n and l i n e a r c o n s t r a i n t s . and K e l l e y [8], and T u f e k c i
[la
As f i r s t i n d i c a t e d by F u l k e r s o n [4]
and l a t e r by M o r l o c k and Neumann Dl],
P h i l l i p s and Dessouky e2],
t h e d u a l o f t h i s problem possesses a f l o w network s t r u c t u r e .
In
F u l k e r s o n ' s and K e l l e y ' s f o r m u l a t i o n s t h e dual problem becomes a minimum c o s t f l o w problem, whereas i n P h i l l i p s and Dessouky's and T u f e k c i ' s work t h e problem s o l v e d i s a minimum c u t problem.
= N,
F o r a g i v e n f l o w network G(N,A) and X
(x,R)
=
($,X)
= {(i,j)
((i,j)
let
2
= N\X and
: iaX, jeR, ( i , j )
E
A1
and :
ief,
jcX,
(i,j) € A } .
L e t t h e source and s i n k nodes b e denoted by 1, and n, r e s p e c t i v e l y .
n
6.
f
then the s e t ( X J )
U(x,X)
i s called a cut.
If 1
~l
X and
A t each i t e r a t i o n o f t h e
a l g o r i t h m t h e d u r a t i o n s a r e decreased a l o n g a c u t w h i c h r e q u i r e s minimum a d d i t i o n a l cost.
Example 4.
R = ZZ = s e t of i n t e g e r s
g r o u p of m-component v e c t o r s o v e r R w i t h l e x i c o g r a p h i c a l
(H,*,<)
ordering
a: s c a l a r
multiplication.
The c o r r e s p o n d i n g ATCTP i s c a l l e d a l e x i c o g r a p h i c a l TCTP (LTCTP). t h e c o s t element c . . e H a r e v e c t o r s 1J
I n t h i s case
Algebraic flows and time-cost tradeoff problems
169
The f o l l o w i n g i s a nonexhaustive l i s t of p o s s i b l e l i n e a r o b j e c t i v e f u n c t i o n s
.. .k.
which may b e i n c l u d e d as t h e p t h component o f t h e LTCTP, p = 1 ,2,. 1.
2.
c ( i , j ) = c!., t h e i n c r e m e n t a l d i r e c t c o s t of r e d u c i n g a c t i v i t y ( i , j ) by one P 1.J u n i t . The corresponding o b j e c t i v e f u n c t i o n p r o v i d e s t h e t o t a l a d d i t i o n a l d i r e c t c o s t over t h e normal d i r e c t c o s t f o r a g i v e n schedule i d . ) . iJ c ( i , j ) = 1, t h e corresponding o b j e c t i v e f u n c t i o n p r o v i d e s t h e t o t a l amount P o f r e d u c t i o n on a l l a c t i v i t i e s . T h i s o b j e c t i v e f u n c t i o n ensures t h e s e l e c t i o n o f a l e a s t number o f a c t i v i t i e s t o be crashed f o r t h e same c o s t i f used i n conjunction w i t h t h e o b j e c t i v e f u n c t i o n defined i n ( 1 ) . i m p o r t a n t f r o m t h e c o n t r o l p o i n t o f view.
T h i s may b e e x t r e m e l y
I t p r o v i d e s a schedule which
r e q u i r e s a s m a l l e r number o f a c t i v i t i e s t o be m o n i t o r e d under a c r a s h program.
3.
( i , j ) = P . ., where Pij i n d i c a t e s t h e p r e f e r e n c e ( r a n k i n g ) o f a c t i v i t i e s t o P 1J b e crashed. T h i s o b j e c t i v e f u n c t i o n p r o v i d e s p r e f e r r e d a c t i v i t i e s t o be c
crashed ahead o f o t h e r n o t so d e s i r a b l e a c t i v i t i e s .
4.
( i , j ) = r . . where r . i s t h e amount o f a r e s o u r c e consumed by a u n i t ij P 1J r e d u c t i o n i n a c t i v i t y ( i , j ) which i s n o t i n c l u d e d i n o t h e r o b j e c t i v e f u n c t i o n s . c
To i n t r o d u c e t h e d i f f e r e n c e between CTCTP and LTCTP we p r e s e n t t h e p r o j e c t n e t work i n F i g u r e 1.
I n t h i s F i g u r e c1 ( i , j )
= a d d i t i o n a l incremental d i r e c t cost
per u n i t o f reduction i n a c t i v i t y duration, c2 ( i , j ) ranking o f a c t i v i t y ( i , j ) . f o u r t e e n have l e n g t h 35. cut (X,j)
1 and c
3 (i,j)
= preference
Assume t h a t i n t h i s network a l l t h e paths a r e o f equal
l e n g t h and thus a r e a l l c r i t i c a l . units.
=
That i s , a l l t h e paths from node one t o node
Now assume t h a t t h e d e s i r e d p r o j e c t l e n g t h i s 27 t i m e
T h e r e f o r e t h e CTCTP w i l l proceed as f o l l o w s : f i r s t t h e a c t i v i t i e s on t h e w i l l be m o d i f i e d b y s i x u n i t s , where X = 11) and
=
N\X.
Note t h a t
a l a b e l i n g a l g o r i t h m w i l l always p r o v i d e a m i n i m a l c u t w i t h t h e s m a l l e s t number o f nodes i n X .
A f t e r t h i s c r a s h i n g , t h e p r o j e c t l e n g t h i s reduced t o 29.
second i t e r a t i o n t h e CTCTP w i l l s e l e c t X = il, 2, 3, 4, 5, 6 , 7 ? and Now a l l t h e a c t i v i t i e s on
(X,f) w i l l
?
be crashed by two more t i m e u n i t s .
t h i s s t e p t h e p r o j e c t d u r a t i o n i s 27 and t h e a l g o r i t h m stops. d i r e c t c o s t r e q u i r e d f o r t h i s schedule i s 6(1+1+1+1+1+1)
t
I n the = N\X.
After
The a d d i t i o n a l
2(1+1+1+1+1+1) = 48.
The t o t a l amount o f t i m e r e d u c t i o n s on a l l t h e a c t i v i t i e s i s 6 ( 6 )
+
6 ( 2 ) = 48.
H. W. Hamacher and S. Tufekci
1 70
The amount o f p r i o r i t y consumed by the crashed a c t i v i t i e s i s 6(3t3t2+2+1+1)
t
2(4+4+6+6+2+2) = 120.
Figure 1.
An example p r o j e c t network. b..
1J
-
ai
,
( V a l u a t i o n of arcs:
r c l ( i J), c 2 ( i , j ) ,c3(i ,j)]
Now consider the same problem w i t h l e x i c o g r a p h i c a l o b j e c t i v e f u n c t i o n w i t h ck ( i , j ) ,
k = 1, 2, 3, as d e f i n e d over each a r c .
w i l l s e l e c t (X,f) w i t h
2
= I14) and
X = N\X as t h e lexmin cut.
I n the n e x t step, LTCTP w i l l s e l e c t
f
= {13,
141 and X
t i e s (11, 13) and (12, 13) w i l l be reduced by two time u n i t s . LTCTP w i l l s e l e c t
I
Hence a c t i v i t y
Now t h e p r o j e c t d u r a t i o n i s reduced
(13, 14) w i l l be reduced by f i v e time u n i t s . t o 30.
I n t h e f i r s t i t e r a t i o n , LTCTP
=
N\g and a c t i v i -
I n the t h i r d step
= {il,13, 141 and X = N\? and a c t i v i t i e s (8, l l ) , (9, 11),
(10, 11) and (12, 13) w i l l be reduced by one t i m e u n i t .
A t t h i s p o i n t the
p r o j e c t d u r a t i o n i s reduced t o 27 and t h e a l g o r i t h m w i l l stop.
Note t h a t by
u s i n g LTCTP we have the otpimal value o f t h e f i r s t component o f t h e v e c t o r valued o b j e c t i v e f u n c t i o n as 6(5)
+
2(3t3)
+ 2(2)
+
1(1+1+1+3) = 48, t h e same as the
CTCTP.
The second component i s l ( 6 )
CTCTP.
F i n a l l y , the t h i r d component i s t h e t o t a l amount o f p r i o r i t y consumed by
t 4(1) = 14 as compared t o 48 i n
t h i s crash schedule i s g i v e n by 5(1) t 2 ( 2 + 2 ) + 1(3+3+3+2) = 24 which i s a l s o less than t h e CTCTP r e s u l t .
ALGEBRAIC FLOWS AND AN ALGORITHM FOR SOLVING THE ALGEBRAIC TIME-COST TRADEOFF PROBLEM
3
I n t h i s s e c t i o n we present an a l g o r i t h m f o r s o l v i n g t h e a l g e b r a i c time-cost t r a d e o f f problem by g e n e r a l i z i n g an a l g o r i t h m o f Tufekci [15] concept o f a l g e b r a i c minimal c u t s as discussed i n Hamacher [6,
and by u s i n g t h e
71.
Algebraic flows and time-cost tradeoff problems
171
F i r s t we summarize some r e s u l t s of t h e a l g e b r a i c f l o w t h e o r y : l e t II, u: A two f u n c t i o n s ( c a l l e d
lower and
-+
H be
upper c a p a c i t y f u n c t i o n s , r e s p e c t i v e l y ) a s s i g n i n g
t o each ( i , j ) € A elements a ( i , j ) , u ( i , j ) E H. I n what f o l l o w s t h e o p e r a t i o n " - ' I w i l l r e p r e s e n t t h e i n v e r s e o f t h e group o p e r a t i o n * (see 2.2.d), i.e., a-b = a*(-!-). A ( f e a s i b l e ) a l g e b r a i c f l o w ( w i t h r e s p e c t t o L and u ) i s a f u n c t i o n f : A
-+
H
satisfying: L(i,j) 6 f(i,j)
B
W (i.j)
u(i,j)
E
A
and f(x,N) (Here f ( x , N ) =
( X , h of t h e a l g e b r a i c f l o w . f(X,f)
-
f(f,X)
-
f(N,x) =
f(x,y)
v
i f x = 1
-v
if x = n
(
0
x EN
otherwise
*
and f ( N , x ) =
(YJkA Note h e r e t h a t s i n c e (H,*)
f(y,x).)
v i s va l e d t h e
9
i s cancellative, the required
= 0 which i s necessary i n t h e case o f semigroups w i t h o u t c a n c e l -
l a t i v e law, can be r e p l a c e d by t h e c o n s e r v a t i o n l a w i n ( 3 . 2 ) . Analogous t o t h e case o f t h e c l a s s i c a l network f l o w t h e o r y (e.g.,
[2])
71
we can p r o v e [6,
Ford, F u l k e r s o n
the following result:
A l g e b r a i c max f l o w - m i n c u t theorem.
v i s t h e v a l u e o f a maximal a l g e b r a i c f l o w i f
and o n l y i f t h e r e e x i s t s a s u b s e t X o f t h e node s e t N w i t h 1
E
X and n
E
f:
= N\X
such t h a t
v
=
* u(x,y) (X,Y)E(X,f)
- *
II(y,x) (Y,X)dX,X)
.
I n short max v =
We c a l l such a c u t
m i n - IU(X,X) lsX,nrX
(X,A)u(f,X)
-
II(A,x)I.
an a l g e b r a i c minimal c u t .
I n t h e a p p l i c a t i o n o f t h e a l g e b r a i c f l o w t h e o r y t o t h e t i m e - c o s t t r a d e o f f problems we a r e m a i n l y i n t e r e s t e d i n i d e n t i f y i n g an a l g e b r a i c minimal c u t .
Such a c u t
(and s i m u l t a n e o u s l y a maximal a l g e b r a i c f l o w ) can be found b y t h e f o l l o w i n g a l g o r i t h m which i s a s t r a i g h t f o r w a r d g e n e r a l i z a t i o n o f Ford and F u l k e r s o n ' s [2] l a b e l i n g a l g o r i t h m f o r c l a s s i c a l network f l o w s . Maximum a l g e b r d i c f l o w a l g o r i t h m . ___ Start:
1).
A
-+
H l o w e r and upper c a p a c i t y , r e s p e c t i v e l y .
f: A
-f
H a f e a s i b l e a l g e b r a i c f l o w w i t h v a l u e v.
k,u:
S t a r t i n g f r o m node 1 s u c c e s s i v e l y l a b e l nodes y f r o m a l r e a d y l a b e l i y nodes
H. W.Hamacher and S. Tufekci
172 x if
2).
(x,y) E. A and f ( x , y )
< u(xly)
("forward arcs"),
(y,x) E A and f ( y , x )
> e(y,x)
("backward a r c s " )
or if
If i n t h i s l a b e l l i n g process n is l a b e l l e d t h e n go t o s t e p f o u r .
Otherwise
go t o s t e p t h r e e .
3).
D e f i n e X t o b e t h e s e t o f l a b e l l e d nodes and
Node n cannot b e l a b e l l e d . t h e s e t o f u n l a b e l l e d nodes.
Now,
(X,x)u(R,X)
x
i s an a l g e b r a i c minimal c u t
(and f i s a n a l g e b r a i c maximal f l o w ) .
4).
Use t h e l a b e l s t o i d e n t i f y an augmenting p a t h P f r o m 1 t o n. 61 = miniu(x,y)
-
f(x,y)
62 = m i n [ f ( x , y )
-
n(x,y)
1 I
(x,y)
forward arc i n P j
(x,y)
backward a r c i n P }
Define,
63 = 1 n i n { 6 ~ , 6 ~ j ,
and change f by,
f(x,y)
=
1
f(x,y)
*
h 3 , i f (x,y)
f(x,y)
-
A3,
d3,
P
backward a r c i n P
, otherwise.
f(x,y)
Set v = v
i f (x,y)
forward arc i n
go t o s t e p one.
The presented l a b e l i n g a l g o r i t h m i s o n l y one p o s s i b i l i t y t o s o l v e a l g e b r a i c f l o w problems.
[7],
One c o u l d a l s o use g e n e r a l i z a t i o n s o f more e f f i c i e n t a l g o r i t h m s [3],
b u t a p r e s e n t a t i o n o f t h e s e a l g o r i t h m s i s beyond t h e scope o f t h i s paper. The i d e a o f t h e subsequen-
An example o f t h i s a l g o r i t h m w i l l be performed l a t e r .
t l y d e s c r i b e d a l g o r i t h m f o r s o l v i n g t h e a l g e b r a i c t i m e - c o s t t r a d e o f f problem i s
t o s o l v e a sequence of a l g e b r a i c maximum f l o w problems.
The r e s u l t i n g a l g e b r a i c
minimum c u t s a r e used t o change t h e a c t i v i t y t i m e s d. ..
Following t h e idea o f
1J
T u f e k c i [15]
t h e a l g e b r a i c maximum f l o w f a t t h e end o f i t e r a t i o n i i s used as
s t a r t i n g f l o w i n t h e beginning o f t h e n e x t i t e r a t i o n i t 1 . A l g o r i t h m f o r t h e a l g e b r a i c TCTOP.
I n what f o l l o w s we assume t h a t t h e r e a d e r i s
f a m i l i a r w i t h t h e CPM method f o r p r o j e c t networks.
L e t Ei and Li r e p r e s e n t t h e
e a r l i e s t r e a l i z a t i o n t i m e and l a t e s t r e a l i z a t i o n t i m e o f e v e n t i, r e s p e c t i v e l y . Then t h e s l a c k s S . . f o r each a c t i v i t y ( i , j ) E A a r e d e f i n e d b y Sij 1J
(see F o r d and F u l k e r s o n , [3]). Start: -
= L. J A c t i v i t y ( i , j ) i s c r i t i c a l i f f S . . = 0. 1J
d.. = b.. 1J
f(i,j)
1J
=
E
V
( i , j ) r A.
M l a r g e element o f H, e.g.,
M =
* ( i ,j )EA
c(i,j)
- E. 1
d..
1J
Algebraic flows and time-cost traa'eoff problems
173
1 ) . Use t h e CPM method t o compute f o r each a c t i v i t y ( i , j ) t h e s l a c k s Sij l e n g t h o f a l o n g e s t p a t h w i t h r e s p e c t t o t h e d u r a t i o n s dij.
(STOP).
t h a n o r equal t o t h e d e s i r e d p r o j e c t l e n g t h , 2).
and t h e
If T i s less
Otherwise go t o s t e p 2.
Define, u(i,j):
=
and L(i,j):
IE c(i,j)
i f d.. > a.
M
i f dij
= a
i f Sij
>
IJ
c(i,j)
=
ij
and Sij
= 0
and S . . = 0 ij 1J
0
i f d.. < b.. 1J 1J i f dij = b . .
1J
3 ) . Use t h e c u r r e n t a l g e b r a i c f l o w f ( i , j ) ,
Y ( i , j ) E A as f e a s i b l e s t a r t i n g f l o w
f o r f i n d i n g an a l g e b r a i c maximum f l o w f ' ( i , j ) c o r r e s p o n d i n g a l g e b r a i c minimal c u t
4).
If v ' a
M
then
(STOP).
(X,!)
w i t h f l o w v a l u e v ' and a
(J ( x , X ) .
The t o t a l c o m p l e t i o n t i m e cannot be d i m i n i s h e d w i t h -
o u t v i o l a t i n g t h e c r a s h d u r a t i o n o f a t l e a s t one a c t i v i t y , i.e., If v' <
M t h e n denote, A1: = { ( i , j ) 6 ( X , i )
5).
A2: = { ( i , j ) E
(R,X)
A3: = { ( i , j )
(X,R)
~1
I
Sij
= 0, d . . > a . . )
I I
Sij
= 0 and dij
Sij
>
1J
1J
<
b..) 1J
03.
Define, A1:
= min{dij
-
minIbij
-
A *: =
a.. 1J dij
A ~ =: minIS..
1J
A : = min(Al,A2,A3)
6).
T = Tc.
I I I
(i,j)
>
0.
Q
All
(i,j) E Apl ( i , j ) e A33
(3.6) and s e t
Redefine, [d:: dij:
-
A i f ( i , j ) E A1
d . . t A if ( i , j ) E A2
=
I d ij f(i,j)
= f'(i,j),
(3.7)
otherwise
v
(i,j)
E
A
T : = T - A . I f T i s l e s s t h a n o r equal t o t h e d e s i r e d p r o j e c t l e n g t h ,
Otherwise go t o s t e p one.
(STOP).
H.W.Hamcher and S. Tufekci
174
We note here t h a t f o r R = Z t h e a l g o r i t h m solves a t most Tn
-
f l o w problems and performs t h e same number of CPM a l g o r i t h m s .
Tc a l g e b r a i c maxThe complexity o f
t h e CPM a l g o r i t h m f o r an n node a c y c l i c network i s O(n2) and t h e complexity o f t h e best algebraic max-flow problem i s O(n3) (see [7] ) . Therefore t h e o v e r a l l complexi t y o f t h e proposed a l g o r i t h m i s O((T,
-
Tc)n3).
The v a l i d i t y o f t h e a l g o r i t h m i s
proved i n Section 5.
4 AN EXAPPLE - THE LEXICOGRAPHICAL TCTP Here we implement t h e a l g o r i t h m provided i n t h e previous s e c t i o n t o t h e a l g e b r a i c s t r u c t u r e d e f i n e d i n Example 4 of Section 2. Consider t h e p r o j e c t network g i v e n i n F i g u r e 2.
The c o s t v e c t o r c ( i , j )
i s repre-
sented as a column v e c t o r over each a r c .
The numbers b e f o r e and a f t e r c(i,j) represent t h e crash d u r a t i o n and normal d u r a t i o n o f t h a t a c t i v i t y , r e s p e c t i v e l y .
Note t h a t t h e f i r s t component o f t h e c o s t v e c t o r represents incremental d i r e c t costs.
The second component i s 1 f o r each a c t i v i t y i n d i c a t i n g t h e d e s i r e t o
minimize t h e c o n t r o l on t h e crashed a c t i v i t i e s .
c(i,j)
L a s t l y , t h e t h i r d component o f
represents t h e p r i o r i t y r a n k i n g o f each a c t i v i t y .
Note t h a t an a c t i v i t y
w i t h smaller r a n k i n g i s p r e f e r a b l e t o an a c t i v i t y w i t h higher ranking.
L e t Ei,
and Li f o r each node i E N represent t h e e a r l i e s t r e a l i z a t i o n t i m e and l a t e s t r e a l i z a t i o n time o f event i, determined by CPM method, r e s p e c t i v e l y .
F i g u r e 2 . An example p r o j e c t network f o r LTCTOP. aij, Start:
d.. = b. 1J
1J’
f = -
( V a l u a t i o n o f arcs:
g(i,j),bij)
0
Applying the CPM technique we g e t t h e l a b e l s as shown i n F i g u r e 3.
Algebraic flows and time-cost tradeoff problems
F i g u r e 3. Label (Ei,Li)
on each node by CPM.
175
(Valuation o f arcs:
d . . = b .) 1J iJ The a c t i v i t y s l a c k s may be found by u s i n g Sij = LJ. - Ei - d i j . T h i s y i e l d s S45 = 8, S58 = 10, S48 = 20 and o t h e r Sij = 0. The corresponding f l o w network and t h e upper a r c c a p a c i t i e s a r e d e p i c t e d i n F i g u r e 4 below. are L(i,j) =
0for
F i g u r e 4.
(The l o w e r c a p a c i t i e s
a l l (i,j)).
I n i t i a l Flow Network f o r LTCTOP.
Since t h e c u r r e n t f l o w i s f
E
0,we
(Arc valuations: l ( i , j ) )
can l a b e l t h e nodes 1, 2 , 5, 7, 8 and i d e n t i f y
t h e augmenting p a t h P = 11, 2 , 5, 7, 81. Ey(x,y)
Set f(1,2)
= f(2,5)
-
f(x,y)l
H. W.Hamacher and S. Tufekci
176
The n e x t augmenting p a t h w i l l be P = (1, 3, 5, 7, 8 } w i t h E = 5, = l e x m i n { g ( x , y ) -3
-
f(x,y)}
=
(X,Y)EP
Set f(1,3)
= f(3,5)
=
[I], [I, 1,I]!-[
The t h i r d augmenting p a t h i s P = {l,3, 6, 7, 81 w i t h E = Ll = l e x m i n -3
Set f(1,3)
=
I] 11, f(3,6)
=
f(6,7)
=
f(7,8)
=
[I.
=
Next f l o w augmenting p a t h i s P = (1, 4, 6, 7, 81
D u r i n g t h e f o l l o w i n g l a b e l l i n g a l l nodes g e t l a b e l e d e x c e p t node 8. lexmax i s a t hand w i t h
I![
, R
=
a c t i v i t y (7,8) by mint8-3,10,201
=
= 5.
{8}, and X
=
Thus a
N\x ( s e e F i g u r e 5 ) .
Now t h e p r o j e c t d u r a t i o n i s 21.
CPM method t h e new s l a c k v a l u e s a r e o b t a i n e d as f o l l o w s : S45
=
8, S58
Crash
By u s i n g =
5, S48 =I5
and o t h e r S . . = 0. 1J
The c o r r e s p o n d i n g f l o w network f o r t h i s i t e r a t i o n w i t h updated f l o w bounds i s d e p i c t e d i n F i g u r e 5 below.
Note t h a t t h e lexmax f l o w o f t h e l a s t i t e r a t i o n i s
used as a f e a s i b l e l e x f l o w f o r t h i s i t e r a t i o n . C o n t i n u i n g w i t h l a b e l l i n g we o b t a i n t h e p a t h P = 11, 4, 6, 7, 8 } w i t h
Algebraic flows and time-cost tradeoff problems
Figure 5
Maximal l e x f l o w i n i t e r a t i o n 1 w i t h updated f l o w bounds. (Arc v a l u a t i o n & ( i , j ) ,
For each ( i , j ) E P s e t f ( i , j )
f(4,6)
=
11,
f(6,7)
=
= f(i,j)
I]
and
t
!(i,j),
u(i,j))
s3. Now -f(1,4)
f(7,8)
=
r:].
=
Next a t t e m p t of l a b e l l i n g s t o p s s h o r t of l a b e l l i n g node 8. t h e nodes i n
177
1 a r e 7 and 8. A = Al
Thus X = 11, 2, 3, 4, 5, 6,},
= min{d57 = min(5-2,
-
a57' d67
8-5,
-
a67' '58'
When l a b e l l i n g s t o p s
1
= {7,
8},
'48'
5, 151 = 3.
S i n c e t h e d e s i r e d p r o j e c t l e n g t h i s 19 reduce t h e p r o j e c t l e n g t h by 2 more u n i t s by r e d u c i n g a c t i v i t i e s (5,7) d67 = 8
-
2 = 6.
and (6,7) by two u n i t s .
Set d57 = 5 - 2 = 3 and
Since t h e d e s i r e d p r o j e c t l e n g t h T = 19 i s achieved, t h e
algorithm terminates.
I f we were t o c o n t i n u e c r a s h i n g t h e n we should have
crashed t h e two a c t i v i t i e s above by t h r e e t i m e u n i t s .
I n such a case, t h e bounds
on t h e network must be updated and t h e l a b e l i n g a l g o r i t h m must c o n t i n u e .
5
VALIDITY OF
THE ALGORITHN
We prove t h e v a l i d i t y of t h e a l g o r i t h m by i n d u c t i o n on t h e number o f i t e r a t i o n s . F o r e v e r y p r o j e c t t i m e T and a c t i v i t y d u r a t i o n s d G(T) = { ( i , j )
I
Sij
=
ij
we denote
0).
(5.1)
R W. Hamacher and S. Tufekci
118
G(T) contains a l l c r i t i c a l paths w i t h r e s p e c t t o p r o j e c t t i m e T, i.e.,
a l l paths
P satisfying
1
d(P) =
(5.2)
d 1J .. = T
( i ,j )eP S t a r t = I t e r a t i o n 0: L e t T = Tn be t h e normal p r o j e c t time.
Obviously d .
ij
= bij
m i n i m i zes t h e o b j e c t i v e f u n c t i o n
* (i,j)EA I t e r a t i o n I: L e t T = T~
-
( b . .-d. . ) 0 c 1J
1J
6, 6 s u f f i c i e n t l y small.
I n order t o achieve a p r o j e c t time of Tn G(Tn) by 6 time u n i t s .
ij
-
6 we have t o crash a l l paths P i n
This can be done by f i n d i n g an a r b i t r a r y c u t ( X , j )
and crash a l l a c t i v i t y d u r a t i o n s by 6, i.e.,
G(Tn)
in
by d e f i n i n g
* C... Therefore a c u t The c o s t increase i s 6 o c ( X , R ) where c(X,R) = (i,j)e(X,i) IJ w i t h minimal value c(X,x) w i l l y i e l d a minimal c o s t increase. Such a c u t can be found by applying a maximal a l g e b r a i c flow a l g o r i t h m t o the lower and upper capacities I.(i,j) "6 sufficiently
minimal c u t ( X , x )
= 0 and u ( i , j )
E
1J
s m a l l " means t h a t d . .
1J
-
6 >, a.
1.j
Note t h a t
f o r a l l ( i , j ) E (X,X).
I f the
c o n t a i n s an a r c ( i , j ) w i t h d . - = a . . then c(X,?) > M and a l l 1J
c u t s i n G(T) have a v a l u e >c M. (i,j)
= c . . Y ( i , j ) f G(T), r e s p e c t i v e l y .
1J
Then we can f i n d a p a t h P w i t h dij
P which shows t h a t Tn = TC and t h e a l g o r i t h m stops.
= aij
for a l l
On t h e o t h e r hand i t
should be guaranteed t h a t no p a t h becomes c r i t i c a l which i s n o t contained i n G(Tn).
0<
Hence 6 6 min{S. . I S > 01. Therefore t h e a l g o r i t h m y i e l d s f o r a l l S w i t h IJ i j where c. i s d e f i n e d by ( 3 . 6 ) , optimal a c t i v i t y d u r a t i o n s d . ..
6 4 A,
1J
Iteration i
+
I t e r a t i o n ( i t l ) : Let T
<
Tn be a p r o j e c t time f o r which optimal
a c t i v i t y d u r a t i o n s d . . have been found by t h e a l g o r i t h m o f S e c t i o n 3. 1J
Consider a p r o j e c t time d(P) = T > T - 6 .
T
-
6,
A l l paths P i n G(T) s a t i s f y
6 s u f f i c i e n t l y small.
Therefore we have t o crash a t l e a s t one a c t i v i t y f o r each o f
d ! . .c< T - 6 . Again, t h e 1J ( i .J )Q problem i s t o f i n d a way o f c r a s h i n g which w i l l y i e l d a minimum c o s t increase i n
these c r i t i c a l paths i n o r d e r t o achieve d ' ( P ) =
the objective function. 6 c A3 = T
-
2
I f we choose d(P)
P n o n c r i t i c a l paths w i t h r e s p e c t t o T
(5.4)
(or e q u i v a l e n t l y , a3 = m i n I S - .IS.. > O } , as defined i n (3.6)), then t h e r e i s no 1 3 13 need t o crash d . f o r a r c s ( i , j ) which a r e n o t contained i n G(T). iJ
Algebraic flows and time-cost tradeoff problems
Ifwe f i n d i n G(T) a c u t (X,R)
such t h a t dij
-
6 3 aij
V(i,j)
179 E
(X,f)
then the
a c t i v i t y d u r a t i o n s d ! . defined by (5.3) w i l l y i e l d a p r o j e c t t i m e o f T ( d ! . ) 6 T 1J
S i n c e p a t h s i n G(T) can use a r c s of t h e c u t (X,R) can even become l e s s t h a n T
Example:
= (IR,+,c),
(H,*,&)
- 6.
1J
-
more t h a n once t h e l e n g t h d ' ( P )
6 y i e l d i n g an u n n e c e s s a r i l y h i g h c o s t i n c r e a s e :
T = 10, 6 = 1, G(T) c o n t a i n s t h e p a t h
P
= (1, 2, 5 ,
3, 4, 6 )
10 = T
dij
d(P)
d;j
d ' ( P ) = 8<9= 10 6
-
O b v i o u s l y we can add 1 u n i t o f t i m e t o t h e a c t i v i t y d u r a t i o n o f a c t i v i t y (5,3)
if d5,3 < b5,3. On t h e o t h e r hand i f d g Y 3 = b5,3 i n c r e a s i n g a c t i v i t y (5,3) above b g Y 3 w i l l n o t produce any s a v i n g s . T h e r e f o r e a c u t w i t h minimum v a l u e u(X,R)
-
k(R,X)
=
*
u(i,j)
(i,j)G(X,R)
* a(i,j) ( i ,j)e(R,x)
-
(5.5)
where u(i,j)
L(i,j)
y i e l d s a minimal c o s t i n c r e a s e .
=
=
f
I
c..
i f dij
>
aij
M
i f dij
= aij
cij
i f dij
<
E
i f dij
= bij
'J
b.. lJ
(5.7)
3
Note t h a t t h i s d e f i n i t i o n i s c o m p a t i b l e w i t h
(3.4) and (3.5) where we used t h e whole a c t i v i t y network as domain s e t . v a l u e u(X,R)
T
-
k(?,X)
>- M, t h e n by t h e same argumentation as i n t h e case above,
= TC, and t h e a l g o r i t h m s t o p s .
get d'(P) 4 T all 0
T -
4
6
6 A,
-
Otherwise we change dij
6 f o r a l l paths P i n G(T).
a c c o r d i n g t o (3.7) and
I f A i s computed by ( 3 . 6 ) ,
then f o r
d ! . i s t h e r e f o r e an o p t i m a l s o l u t i o n w i t h r e s p e c t t o p r o j e c t t i m e 1J
6.
I f the
H. W. Hamach er and S. Tufekci
180
Note f i n a l l y ,
t h a t a l l minimal c u t s (X,x)
a r e f o u n d by computing a maximal f l o w f .
S i n c e t h e c a p a c i t i e s change between two c o n s e c u t i v e i t e r a t i o n s o n l y on a r c s o f t h e c u t , and s i n c e f ( i , j )
= u(i,j)
Y (i,j) E
(X,f)
and f ( i , j )
= f.(i,j)
W (i,j) E
(y,X),
t h i s f l o w w i l l be f e a s i b l e i n t h e n e x t i t e r a t i o n and can t h e r e f o r e be used as s t a r t i n g f l o w i n t h e subsequent i t e r a t i o n .
6
CnNCLUSION
An a l g e b r a i c t i m e - c o s t t r a d e o f f a l g o r i t h m i s d e s c r i b e d i n t h i s paper. shown t h a t d i f f e r e n t o b j e c t i v e f u n c t i o n s (e.g.,
I t has been
sum o b j e c t i v e , and l e x i c o g r a p h i c a l
o b j e c t i v e ) can be handled by c o n v e r t i n g t h e c o r r e s p o n d i n g a l g e b r a i c o p t i m i z a t i o n model i n t o an a l g e b r a i c network f l o w problem.
Once t h e problem i s c o n v e r t e d i n t o
a n a l g e b r a i c f l o w problem, a n e f f i c i e n t F o r d and F u l k e r s o n [2]
type labeling
a l g o r i t h m may be employed t o d e t e r m i n e t h e a l g e b r a i c minimum c u t s . By f o r m u l a t i n g t h e problem as a l e x i c o g r a p h i c a l TCTP, t h e l e x i c o g r a p h i c a l t i m e c o s t t r a d e o f f a l g o r i t h m enables t h e p r o j e c t manager t o s e l e c t t h o s e c r i t i c a l a c t i v i t i e s f o r crashing which l e x i c o g r a p h i c a l l y optimize m u l t i p l e o b j e c t i v e functions. Selecting a smallest s e t o f a c t i v i t i e s f o r achieving t h e desired p r o j e c t d u r a t i o n a t a minimal d i r e c t c o s t g i v e s t h e manager a g r e a t e r f l e x i b i l i t y on e x e r c i s i n g b e t t e r c o n t r o l o f t h o s e a c t i v i t i e s which need c l o s e m o n i t o r i n g because o f c r a s h schedule.
ACKNOWLEDGEMENT The a u t h o r s w i s h t o express t h e i r a p p r e c i a t i o n t o t h e r e f e r e e s f o r many h e l p f u l comments c o n c e r n i n g t h e f i r s t d r a f t o f o u r p a p e r .
REFERENCES
[l] Elmaghraby, S.E., The d e t e r m i n a t i o n o f o p t i m a l a c t i v i t y d u r a t i o n i n p r o j e c t scheduling, J o u r n a l o f I n d u s t r i a l E n g i n e e r i n g , Vol. 19, No. 1, ( J a n u a r y 1968) 48-51. [2]
Ford, L . R . , and F u l k e r s o n , D.R., F l o w i n Networks, ( P r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n , New J e r s e y 1962)
.
[3]
F r i e z e , A.M.,
[4]
Fulkerson, D . R . ,
A l g e b r a i c f l o w s , Annals o f D i s c r e t e A p p l . Math.,
c u r r e n t volume.
A Network f l o w c o m p u t a t i o n f o r p r o j e c t c o s t curves, Managemert
Algebraic flows and time-cost tradeoff problems
181
Science, V o l . 21, No. 6, ( F e b r u a r y 1975) 718-722. c51
Goyal, S.K., A n o t e on ' A s i m p l e CPM t i m e - c o s t t r a d e o f f a l g o r i t h m ' , Management Science, Vol. 21, No. 6, (February 1975) 718-722.
FI
Hamacher,, H., D e t e r m i n i n g minimal c u t s w i t h a minimal number o f a r c s , Networks (1982) 493-504.
171
Hamacher, H. , Flows i n Regular M a t r o i d s , Mathematical Systems i n Economics, Vol. 69, (Oelgeschlager, Gunn and Hain, Cambridge, Mass., 1981). K e l l e y , J . E . , C r i t i c a l p a t h p l a n n i n g and s c h e d u l i n g : mathematical b a s i s , Operations Research, Vol. 9, No. 3, (May-June 1961) 296-320. Lamberson, L.R., and Hocking, R.R., Optimal t i m e compression i n p r o j e c t scheduling, Management Science, V o l . 16, No. 10, (June 1970) 8597-B606. Moore, L.J., T a y l o r , B.W., 111, Clayton, E.R., and Lee, S.M., A n a l y s i s o f m u l t i - c r i t e r i a p r o j e c t c r a s h i n g model, A I I E Transactions, (June 1978) 163169. Morlock, M., and Neumann, K. , E i n V e r f a h r e n z u r M i n i m i e r u n g d e r Kosten e i n e s P r o j e k t e s b e i vorgegebener P r o j e k t d a u e r , Angew. I n f . , Vol. 4, (1973) 135-140. P h i l l i p s , Jr.,S., and Dessouky, M.I., S o l v i n g t h e p r o j e c t t i m e / c o s t t r a d e o f f problem u s i n g t h e minimal c u t concept, Management Science, V o l . 24, No. 4, (December 1977) 393-400. Prager, W., A s t r u c t u r a l method o f computing p r o j e c t c o s t polygons, Management Science, V o l . 9, No. 3, ( A p r i l 1963) 394-404. Siemens, N., A s i m p l e CPM t i m e - c o s t t r a d e o f f a l g o r i t h m , Management Science, Vol. 17, No. 6, (February 1971) B354-8363. Tufekci, S., A f l o w p r e s e r v i n g a l g o r i t h m f o r t i m e - c o s t t r a d e o f f problem, A111 T r a n s a c t i o n s , Vol. 12, No. 3, (1982).
p6]
Zimmermann, U., L i n e a r and c o m b i n a t o r i a l o p t i m i z a t i o n i n o r d e r e d a l g e b r a i c s t r u c t u r e s , Annals o f D i s c r e t e Mathematics 10 ( N o r t h H o l l a n d Co., 1981).
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Annals of Discrete Mathematics 19 (1984) 183-200 0 Elsevier Science Publishers B.V. (North-Holland)
183
RANKING THE CUTS AND CUT-SETS OF A NETWORK
H.W. Hamacher M. Queyranne J.-C. P i c a r d Department of I n d u s t r i a l Dept. des Sciences A d m i n i s t r a t i v e s F a c u l t y o f Commerce and Systems Engineering Universitt! d u Qudbec % P o n t f e a l U n i v e r s i t y o f B r . Columbia University o f Florida M o n t d a l - Quebec H3C 3P8 Vancouver, V6T 1W5 Canada Canada Gainesvi 11e, FL 32611 U.S.A.
Given a graph G=(V,A) a & i s a s e t X,$) o f a r c s w i t h ? t a r t f o r a l l (x,y)c(X,X). A i n g p o i n t i n X and t e r m i n a l p o i n t i n c u t - s e t i s a c u t which does n o t i n c l u d e a n o t h e r c u t . We d i s cuss t h e problem o f r a n k i n g c u t s and c u t - s e t s i n G, i.e., t h e problem o f f i n d i n g t h e K b e s t c u t s and c u t - s e t s , where t h e v a l u e s o f t h e c u t s a r e elements o f a t o t a l l y o r d e r e d semigroup. We r e s e n t an O(K,n*) a l g o r i t h m f o r r a n k i n g t h e c u t s a x O(mK-P.n3) a l g o r i t h m f o r r a n k i n g t h e c u t - s e t s .
6
1.
INTRODUCTION
Consider a f i n i t e , d i r e c t e d graph G = (V,A) where V = { v ~ , v ~ , . . . , v ~ , v ~ +i s~ ~t h e s e t o f nodes, and A = {a,,
...,}a,
F o r convenience we o f t e n
i s t h e s e t o f arcs.
denote a r c s by a = (x,y) where x = t ( a ) , and y = h ( a ) a r e t h e tfl,
and
head o f
a,
respectively. I n G we d i s t i n g u i s h two nodes s,teV.
t = v
n+l '
Then an ( s , t ) - c u t ,
W.1.o.g.
we assume i n t h e f o l l o w i n g s = v
o r s i m p l y a c u t (X,$) (X,$)
0'
i s a s e t o f arcs
:= { a e A l t ( a ) E X , h ( a ) d }
(1.1)
induced by a s e t X o f nodes such t h a t sex, t t x :=
v-x.
A cut-set i s a minimal ( m i n i m a l i t y w i t h respect t o s e t inclusion) cut.
w i t h C t h e s e t o f a l l c u t s and w i t h I ) t h e s e t o f a l l c u t - s e t s . I) S C
(see F i g u r e 1,a).
subset o f t h e nodes, i . e . ,
We denote
I n general
Moreover c u t s C E C can be induced by more t h a n one C = (X,X)
= (X',x')
f o r X # X ' (see F i g u r e 1,b). 1
1
F i g u r e 1: a )
(X,$)
(XI,!)
w i t h X = {s,2}
satisfies: C E C but C 9
6 C w i t h X = {s,1,2,4}
# {s,1,2,4,5}
I);
= X'
b)
C = (X,X)
=
H. W. Hamacher el al.
184
two c u t s ( X , x )
We d i s t i n g u i s h c u t s by t h e i r i n d u c i n g s e t s , i . e . , d i f f e r e n t i f f X # X ' (though t h e arc sets (X,x)
I n F i g u r e l,b
(X,R)
and ( X ' , x ' )
and ( X ' , x ' )
are
m i g h t be t h e same).
are d i f f e r e n t cuts.
H, where H i s some s e t , t o t a l l y o r d e r e d by an t h e n c ( D ) i s a l s o d e f i n e d f o r a l l c u t - s e t s D e D and we can
I f we d e f i n e a f u n c t i o n c : C
o r d e r r e l a t i o n 4,
and ( X , X ' )
-t
c o n s i d e r t h e f o l l o w i n g problems: K Best Cut Problem:
(K-CP)
F i n d t h e K b e s t c u t s C1,
c(C1) 6
K Best Cut-Set Problem: (K-CSP)
\<
c(D1) 4
... \<
...,Ck
E
W C E C , C # C1, i = 1 ,..., K
c ( C k ) \< c(C)
F i n d t h e K b e s t c u t - s e t s D1,
... \<
C, i . e . ,
...,Dk € 2 7 ,
i.e.,
W DeD,D#Di,i
c ( D k ) 6 C(D)
= 1 , ...,K.
I n t h e f o l l o w i n g we d i s c u s s some examples where (K-CP) o r (K-CSP) can o c c u r .
Example 1:
L e t H = RI, c(c): =
i .e.,
I:
c:A
+
IR,
and d e f i n e
w c
c(a)
EC.
a d F o r K = 1 t h i s problem i s w e l l s o l v e d b y means o f network f l o w t h e o r y : i n t e r p r e t
s and t as s i n k and source, r e s p e c t i v e l y , and s o l v e a maximal f l o w problem w i t h capacity f u n c t i o n c.
From t h e c l a s s i c a l max f l o w
f o r t h e maximal f l o w v a l u e v
-
min c u t theorem [3]
we g e t
*
max'
vmax = m i n c(D) = m i n c ( C ) D d CEC T h a t means f o r K = 1 we do n o t have
Note t h a t a minimal c u t i s always a c u t - s e t .
t o w o r r y about t h e d i f f e r e n c e s between c u t s and c u t - s e t s , whereas f o r K have t o c o n s i d e r t h e s e d i f f e r e n c e s c a r e f u l l y .
1 we
>
Each maximal f l o w a l g o r i t h m
i d e n t i f i e s a minimal c u t o r c u t - s e t as soon as t h e maximal f l o w i s found and t h e e f f o r t t o s o l v e (1-CP) and (1-CSP) i s t h e r e f o r e O ( n 3 ) p6]. Many problems, t h e o r e t i c a l as w e l l as p r a c t i c a l ones, can be modeled by t h i s s p e c i a l case o f (1-CP), e.g.,
t i m e / c o s t t r a d e o f f problems p9, 241, b o o l e a n o p t i -
m i z a t i o n problems p5, 221, i n t e g e r programs [21], F o r an o v e r v i e w o f a p p l i c a t i o n s see [20].
o r s c h e d u l i n g problems [23].
In each o f t h e s e a p p l i c a t i o n s one
m i g h t be i n t e r e s t e d i n g e t t i n g i n f o r m a t i o n a b o u t t h e second, t h i r d , cut.
...,
k t h best
F o r i n s t a n c e , t h e f a i l u r e o f some a r c s m i g h t l e a d t o t h e i n a v a i l a b i l i t y o f
t h e minimal c u t .
O r t h e i n f o r m a t i o n a b o u t t h e v a l u e d i f f e r e n c e s c ( D ~ + ~- )c(Di)
Ranking the cuts and cut-sets of a network
185
may s u p p o r t d e c i s i o n s t o defend t h e minimal o r t h e b e s t k c u t s i n m i l i t a r y o r game s i t u a t i o n s . I f G i s a p l a n a r graph t h e n K-CSP can be s o l v e d by f i n d i n g t h e K s h o r t e s t paths
i n t h e d u a l graph G*
Example 2 :
[5].
N o t i c e t h a t K-CP cannot be s o l v e d by t h i s approach.
I f t h e e v a l u a t i o n o f c u t s i s depending on more t h a n one c r i t e r i o n we
can r a n k t h e c u t s i n t h e f o l l o w i n g way: f i r s t criterion.
f i r s t , rank t h e cuts according t o t h e
Secondly, rank c u t s h a v i n g t h e same v a l u e h t h r e s p e c t t o t h e
f i r s t c r i t e r i o n a c c o r d i n g t o t h e second c r i t e r i o n , e t c . . can b e modeled as f o l l o w s :
..., c,(a)).
a vector c ( a ) = (cl(a),
This ranking o f cuts
g i v e n a f u n c t i o n 5:A + I R T a s s i g n i n g t o each a r c a E A Find a ranking o f t h e cuts C
E
C where
and where t h e o r d e r i n g c(Ci) 6 C ( C ~ + ~i s) d e s c r i b e d by t h e l e x i c o g r a p h i c a l o r d e r ing.
(i.e.,
i = minIjl1
5 6
= (xl,
...,
j 6 m, x j
..., y),
xm) \< (yl,
=
y
<=>
y or -x = -
xi < y.1 f o r
# yjI).
For e v e r y a p p l i c a t i o n mentioned i n Example 1 more t h a n one c r i t e r i o n can be involved i n the evaluation o f cuts.
The problem o f f i n d i n g minimal c u t s w i t h a
minimal number o f a r c s [ll] and t h e l e x i c o g r a p h i c a l t i m e / c o s t t r a d e o f f [13] problem a r e problems o f t h i s t y p e which have been s t u d i e d as y e t .
Example 3.
L e t c:A + I R and d e f i n e c ( C ) = max c ( a ) . a d
A p p l i c a t i o n s o f t h i s problem a r i s e i f one asks f o r b o t t l e n e c k s i n c u t problems. I f c(x,y) i s , f o r i n s t a n c e , t h e f l o w o f evacuees d u r i n g a b u i l d i n g e v a c u a t i o n t h e b e s t c u t w i t h r e s p e c t t o t h i s e v a l u a t i o n a l l o w s an easy c o n t r o l o f t h e f l o w
(e.g.,
by m o n i t o r s ) .
F u r t h e r a p p l i c a t i o n s i n c l u d e boolean o p t i m i z a t i o n problems
w i t h rnax o b j e c t i v e , b o t t l e n e c k s c h e d u l i n g problems, e t c . . The examples discussed so f a r have i n common t h a t t h e e v a l u a t i o n s o f t h e c u t s a r e separable f u n c t i o n s , t h a t means t h e v a l u e of a c u t can be found by a c o m b i n a t i o n o f t h e values d e f i n e d on t h e a r c s a
E
A.
I n S e c t i o n 2 we w i l l s p e c i f y t h e c u t e v a l u a t i o n and d i s c u s s t h e s o l u t i o n o f (K-CP) and (K-CSP) f o r K = 1 u s i n g a l g e b r a i c f l o w t h e o r y .
I n S e c t i o n 3 an a l g o r i t h m o f
H. W. Hamacher et al.
186
p7,
Murty and Lawler
181 w i l l be a p p l i e d t o solve (K-CP) by an O(K.n4) algorithm.
Then we consider (K-CSP) where we present an O(mK-'.n3)
a l g o r i t h m and show t h a t
t h e problem o f f i n d i n g a K - t h s m a l l e s t c u t - s e t i s NP-hard.
2.
SPECIFICATION OF CUT EVALUATIONS AND SOLUTION FOR K = 1
I n order t o s p e c i f y the e v a l u a t i o n o f t h e c u t s C E C we i n t r o d u c e a f u n c t i o n c:A
+
H where (HI*,&)
i s an ordered semigroup.
O p t i m i z a t i o n problems i n semi-
groups were f i r s t considered by Burkard [l]. Since then many papers have appeared d e a l i n g w i t h combinatorial o p t i m i z a t i o n problems i n semigroups. overview the reader i s r e f e r r e d t o
[Z, 251.
For an
The d e f i n i t i o n s and r e s u l t s o f t h i s
s e c t i o n a r e from [9, 101 and w i l l be summarized t o have t h e paper s e l f - c o n t a i n e d . Def:
(H,*,G)
i s an ordered semigroup i f
(2.1)
(H,4) i s a t o t a l l y ordered set,
(2.2)
(H,*)
i s a commutative semigroup w i t h n e u t r a l element 0,
and (2.3)
a d
6 =>a* y 6 6
*
W a,B
Y
y
e H.
I n a d d i t i o n we assume t h e v a l i d i t y o f t h e reduction r u l e :
(2.4)
a c< $ => 3 y e H : a*y = 6 W a,B E H
and the (2.5)
weak c a n c e l l a t i o n r u l e a
*
$ = n
*
y
-i>
6[
= y Or a
Moreover we extend H by a symbol
-
c(C) =
*
8 = a]
satisfying a
We r e q u i r e t h a t c(x,y) a 0 f o r a l l (x,y) o f any c u t C E C by
(2.6)
*
e A.
-
W
cH.
a,B,y
and a
*
- -*
V a E H. Then we can d e f i n e t h e e v a l u a t i o n <
=
=
m
c(a)
a6C where
*
a€H '
a = a1
*
a2
* ... *
a
q
f o r a l l H ' = { a l , ...,a
9
c H.
I t i s easy t o see t h a t Examples 1-3 can be t r e a t e d i n t h e framework o f (2.1)-(2.6).
( 2 . 1 ) ensures t h a t (K-CP) and (K-CSP) a r e w e l l posed. a l l o w us t o s o l v e (1-CP), i.e.,
Properties (2.2)
i n an analogous way as we solved (1-CP) i n Example 1 o f Section 1 a t a l g e b r a i c flows i n G.
-
(2.5)
the problem o f f i n d i n g an a l g e b r a i c minimal c u t ,
-
by l o o k i n g
Ranking the cuts and cut-sets of a network L e t c:A
Def:
-t
Then f : A
187
W a E A. H be a f u n c t i o n w i t h c ( a ) 5 0 H i s c a l l e d a l g e b r a i c f l o w i f i t s a t i s f i e s t h e f o l l o w i n g two
+
properties: (2.7) 0
\i
f ( a ) 6 c(a)
V e s E
w xcv
= f(X,X)
( 2 . 8 ) f(X,R)
*
where f ( P , Q ) =
f(a)
P,Q E V .
for
a 4 P ,Q) I n t h i s f o r m u l a t i o n we assume t h e e x i s t e n c e o f an a r c ( t , s ) return arc
-
the so-called
I f we denote w i t h F t h e s e t o f
and c a l l f ( t , s ) t h e v a l u e o f f l o w f .
a l l a l g e b r a i c flows,
-
t h e n t h e f o l l o w i n g r e s u l t [9,
101 h o l d s :
A l g e b r a i c Max Flow-Min Cut Theorem max f ( t , s ) faF
= min c(C) = m i n
c a
*
c(a)
CeC a r c
It i s p o s s i b l e t o g e n e r a l i z e t h e w e l l known max f l o w a l g o r i t h m s t o a l g e b r a i c
flows:
i n t h i s way we g e t a n O(m2.n) l a b e l i n g a l g o r i t h m g e n e r a l i z i n g Edmonds'
and K a r p ' s a l g o r i t h m [6] and O(m.nz) and O(n3) l a y e r e d graph a l g o r i t h m s g e n e r a l i z i n g t h e a l g o r i t h m s of D i n i c [4]
I n a l l procedures i t s h o u l d
and Karzonov p 6 ] .
be n o t e d t h a t elementary o p e r a t i o n s i n c l u d e b i n a r y o p e r a t i o n s a comparisons w i t h r e s p e c t t o t h e o r d e r i n g
<
i d e n t i f i e d by t h e s e a l g o r i t h m s i s a c u t - s e t .
i n H.
*
6 i n H and
Moreover t h e minimal c u t
T h e r e f o r e t h e p r e c e d i n g approach
s o l v e s (1-CP) as w e l l as (1-CSP).
3.
AN 0 ( ~ . n 4 ) ALGORITHM FOR FINDING
THE K BEST CUTS IN A NETWORK
R e c a l l t h a t (K-CP) ranks t h e c u t s o f a network by c o n s i d e r i n g two ( o r more) p o s s i b l e d i f f e r e n t r e p r e s e n t a t i o n s (X,f) d i f f e r e n t cuts.
That i s (X,,R,)
and ( X ' , x ' )
w i t h X1 = {s,1,2,4)
o f F i g u r e 1 and ( X i , i i ) w i t h X1 = Is,1,2,4,5)
set
c
= (xl
,i)lequals
o f a cut C
E
C as two
i s a b e s t c u t i n t h e network
i s a 2nd b e s t c u t though t h e a r c
( x i $1;).
F o l l o w i n g Hammer (Ivanescu) p 5 ] ,
and P i c a r d and R a t l i f f [21]
we i n t r o d u c e now a
b i n a r y q u a d r a t i c a l g e b r a i c program which i s e q u i v a l e n t t o t h e problem o f f i n d i n g a best cut.
With each c u t (X,x) we a s s o c i a t e a c h a r a c t e r i s t i c v e c t o r
2 = ( Z . l i = 1, 1
...,n ) f 1
where i f node v -
8
X
R W.Hamocher et al.
I88
Note t h a t t h e r e i s no need t o s p e c i f y Boolean v a r i a b l e s f o r s = vo and t = Vn+l since vo
x
X, v
E f o r a l l cuts ( X , j ) . n+l c h a r a c t e r i s t i c vectors o f c u t s (X,X).
E
Thus Zo = 1 and Zn+l = 0 f o r a l l
Next we d e f i n e an e x t e r n a l o p e r a t i o n between a b i n a r y v a r i a b l e Z element a i n t h e semigroup
and an
E
{O,l}
-+
H i s equivalent
H by
O D a = 0 (3.2) 1 o a = a With (3.2) the problem o f f i n d i n g a b e s t c u t w i t h respect t o c:A
t o f i n d i n g an optimal s o l u t i o n o f t h e f o l l o w i n g B i n a r y Q u a d r a t i c A l g e b r a i c Problem
( 5QAp1
s.t. Z E 6 := { Z 6 I0,ll
n+2 : Zo = 1, Zn+,
= 01.
I n the f o l l o w i n g we w i l l have t o solve (3.3) under t h e a d d i t i o n a l c o n s t r a i n t s t h a t some v a r i a b l e s are f i x e d t o 0 o r 1 whereas o t h e r v a r i a b l e s a r e f r e e .
In
order t o see t h a t t h i s problem can be solved by f i n d i n g a best c u t i n a m o d i f i e d network we f i r s t r e w r i t e (3.3).
Define
1
c..
1J
if (vi,vj)
E
A
=
tj i,j = 0
if (vi,vj)
#
,..., n t l .
A.
Then t h e o b j e c t i v e f u n c t i o n o f (3.3) becomes
n+l n+l
(3.4)
*
*
(zi(l-zj))
i 3
Cij.
i=O j=O
We p a r t i t i o n N = i O , l ,
..., n,nt13
i n t o N = NoCJ
the sets of i n d i c e s i n N where the v a r i a b l e s Zi free, respectively. Then we w r i t e (3.4)
NIU
N'.
Here N,
N1. and N ' a r e
are f i x e d t o 0, f i x e d t o
, and
Ranking the cuts and cut-sets of a network n t l n+l J=fi
*
*
(zi.(l-zj)) 0
j=O
*
*
c 1. .J
(l-zj)
u c 1J ..
zi
oc.. 1J
* *
ieN' jeNO
189
The f i r s t p a r t o f t h e r i g h t - h a n d s i d e i s a c o n s t a n t and can be d i s r e g a r d e d i n t h e minimization.
The second p a r t can be i n t e r p r e t e d as t h e v a l u e o f a minimal c u t
i n a network which i s d e r i v e d f r o m G as f o l l o w s .
(3.5)
F o r a l l ?.
E
N ' do:
Set CSL = C(s,v,)
*
=
I f (s,va) 6 A and
Cia.
ieNl
Csa. > 0, t h e n add ( s , v a ) t o A. Set CLt
= C(V,,t)
*
=
6 A and
I f (v,,t)
Caj.
&No CLt
(3.6)
>
0, t h e n add (v,.t)
O m i t a l l v Q w i t h II
If ( s , t )
E
E
NoU N1 -
I s , t } and t h e a r c s i n c i d e n t t o such nodes.
A, t h e n d e l e t e a r c ( s , t ) f r o m G.
F o r each c h a r a c t e r i s t i c v e c t o r
*
t o A.
(zs(l-zL))D
WN' =
csa
*
=
1a s s o c i a t e d *
a€"
*
aeN ' i e N =
*
w i t h a c u t d n t h i s network we g e t
((l-ZaP ( l - Z R ) 0 Cia
*
ieN1
Cia)
(by the d e f i n i t i o n o f El)
1
*
ieNl LeN'
(1-Za)
17 Cia ( b y t h e c o m m u t a t i v i t y o f * )
H. W. Hamacher et al.
I90 and, analogously,
Hence the b i n a r y q u a d r a t i c a l g e b r a i c problem w i t h f i x e d v a r i a b l e s can be solved by f i n d i n g an a l g e b r a i c minimal c u t i n a m o d i f i e d network. (H,*,<)
= (R,+,e)
An example f o r
i s worked o u t i n F i g u r e 2.
1
1 1 F i g u r e 2:
F i n d i n g the best c u t i n the network o f F i g u r e 1 w i t h f i x e d v a r i a b l e s Z1 = 1, Z2 = 0 i n t h e characteristic vector
N0 =
cs3
{2,t}, = 1,
cs4
C3t = 9, Cqt Minimal c u t
N
1
=
Z.
{ ~ , lN } ’ ,= {3,4,5,6}
cs5
=
0,
= 0, CS6 = 1
=
0, Cgt = 0, Cst = 9 Therefore t h e
( I s ) , {3,4,5,6,tI).
optimal s o l u t i o n o f t h e f i x e d v a r i a b l e BQAP i s
z
= (1,0,0,0,0,0).
I n order t o solve (K-CP) we apply a technique developed by Murty p8] f o r assignment problems and formulated by Lawler c7] f o r a r b i t r a r y combinatorial optimizat i o n problems. We assume t h a t we have computed t h e k-1 b e s t s o l u t i o n s t h a t the f e a s i b l e s e t B o f ( 3 . 3 ) i s p a r t i t i o n e d i n t o
(3.7) where (3.8 (3.9 Init
B =
{Z1 ,...)-Zk-l
Bl ,k-1
.. . lJ BLYK-’
I 1 ,...,-Z k - l
(k 4 K) and
191
Ranking the cuts and cut-sets of a network s o l u t i o n i s t h e best, say
LJ’k-l, o f
the solutions Z
i,k-1
o f problems (3.3) w i t h
...,
B r e p l a c e d by BiSk-’ f o r i = 1, L. D e l e t e BJyk-T f r o m t h e p a r t i t i o n . L e t F = N ( ~ j ’ k - 1 TJ,k‘ ) denote t h e s e t o f f r e e v a r i a b l e s . I f F has p 3 1
u
-
elements, say F = {xl,.
,xp},
add p subsets B(l),...,B(p)
t o t h e p a r t i t i o n , as
f o l 1ows :
Z J Y k - l = 0 then x1
if -
Compute t h e b e s t s o l u t i o n i n each of t h e p newly c r e a t e d s e t s and r e a r r a n g e t h e p a r t i t i o n as (Bi’k-l:
i = l,...,L)
where t h e new v a l u e o f L i s L-l+p.
I n each i t e r a t i o n o f t h e r e s u l t i n g a l g o r i t h m we add a t most n s e t s Bi2k.
An
o p t i m a l s o l u t i o n o f t h e corresponding b i n a r y q u a d r a t i c a l g e b r a i c problem i s f o u n d by a p p l y i n g a maximal a l g e b r a i c f l o w a l g o r i t h m t o t h e m o d i f i e d network which has O ( n ) nodes.
T h e r e f o r e each i t e r a t i o n r e q u i r e s O(n4) elementary o p e r a t i o n s
and t h e c o m p l e x i t y o f t h e proposed a l g o r i t h m i s O(K.n4).
An example f i n d i n g t h e
4 b e s t c u t s i n t h e graph o f F i g u r e 1 i s worked o u t i n F i g u r e 3.
FINDING THE K BEST CUT-SETS Consider a t f i r s t t h e problem o f f i n d i n g two d i s t i n c t c u t - s e t s D1 and D2 such t h a t c(D1) 6 c ( D 2 ) and c(D2) 6 c(D) f o r a l l c u t - s e t s D d i s t i n c t f r o m D1 and D2. Since D1 does n o t c o n t a i n any c u t t h e r e must e x i s t a
E
D1 such t h a t a 9 D2.
Arc
a can be excluded f r o m any c u t - s e t w i t h f i n i t e c a p a c i t y by changing i t s c a p a c i t y to
-.
For a € D1, l e t D(a) denote a minimal a l g e b r a i c c u t w i t h r e s p e c t t o t h e
capacities
e
i
z1 =z2=1 Z3=l
z1=1 z2.0
s
s = {S,1,21 T = {t,3,4}
= Is,l) T = {2,tI
C(X4,X4)
Z ] =z2=1 Z3=O Z4=0
= 11
x4 = I S , l I
S = { S ,1,2,4,5)
T={t,3) c(Xi,iii)
= 3
2nd-best c u t zg=o z6=1
S= ( ~ , 1 , 2 , 5 } T= { t ,3,4 1 c ( X * , i8)=5 X ={~,1,2,5) 8
T= t,3,4,5
c ( xg ,i9) = I 2 xg={s,l ,2,8
I
zg=1
S={s,l,2,4,5,61 T= t,3
c ( X7,R7)=1 1
X7= Is,] ,2,4,5,6
B *2,B6,3,B
y 4
S={s,1,2,4,6] T=( t,3,51 C(xgyX6) = 11 X6={ s ,1,2,4,6)
( a l l var. f i x e d )
1
( a l l variables fixed) ( a l l variables fixed) F i g u r e 3:
t4
z1 =Z2=Z4=1 Z3=Zs=O
X ~ = { S , 1,2,4,5)
4th-best c u t
,,
W
F i n d i n g t h e f o u r b e s t c u t s i n t h e network o f F i g u r e 1
R
.F $:
P
3
!$
2 4
R
Ranking the cuts and cut-sets of a network
193
1
(4.1
Since t h e graph G = (V,A)
a E D1
such t h a t c,(D(a))
finite
-
D # D1,
i s n o t changed by (4.1),
D(a) i s a c u t - s e t .
c a ( D ( a ) ) f o r a l l a E D, and assume c,(D(a))
o t h e r w i s e t h e g i v e n network has a s i n g l e c u t - s e t . t h e r e e x i s t s , as n o t e d above, a E D1
c a ( D ( a ) ) > c,(D(a))
= c ( D ( a ) ) (because
second-best c u t - s e t .
a
-
D.
Choos? is
For every c u t - s e t
S i n c e a $ D, c(D) = ca(D) >/
D ( a ) ) and t h i s shows t h a t D ( B ) i s a
T h i s approach r e q u i r e s s o l v i n g a t most O(m) minimum
a l g e b r a i c c u t problems i n networks o f t h e same s i z e as G, and r e s u l t s i n an O(mn3) a l g o r i t h m . I n o r d e r t o improve t h e c o m p l e x i t y o f t h i s a l g o r i t h m we analyze t h e s i t u a t i o n i n which we f i n d a minimal a l g e b r a i c c u t D ( a ) w i t h r e s p e c t t o t h e c a p a c i t y f u n c t i o n
Suppose f i s a maximal a l g e b r a i c f l o w w i t h r e s p e c t t o c a p a c i t y c. f e a s i b l e w i t h r e s p e c t t o ca.
Obviously f i s
I n o r d e r t o f i n d D ( a ) we apply, f o r i n s t a n c e , a n
a l g e b r a i c l a b e l i n g a l g o r i t h m t o f and (G,ca), i.e., we i d e n t i f y s u c c e s s i v e l y R augmenting elementary paths p', ...,p f r o m s t o t where f o r a l l i = l , . . . , ~ (x,y)
(4.2)
(y,x)
forward a r c i n p
i
=>
f(x,y)
i
backward a r c i n p =>f(y,x)
<
ca(x,y)
> 0
Claim: A l l augmenting p a t h s c o n t a i n a r c a and do n o t c o n t a i n any
(4.3)
o t h e r a r c a ' e D1 , a ' # a.
Proof:
S i n c e f i s a maximal a l g e b r a i c f l o w i n (G,c),
f ( a ' ) = c(a') = ca(a') f o r
a l l a ' E D1, a ' # a. Hence a ' cannot be a f o r w a r d a r c i n some augmenting p a t h .
Ifa ' # a were a backward a r c o f an augmenting p a t h P t h e n h ( a ' ) labeled.
T h i s i s o n l y p o s s i b l e i f a r c a precedes a r c a ' i n P.
E
j1had been
Since P does n o t
c o n t a i n o t h e r f o r w a r d a r c s i n D1 than a and s i n c e t h e t e r m i n a l node o f P i s t
E
1, P would c o n t a i n a r c a f o l l o w i n g a r c a ' .
twice i n P
-
T h a t means t h a t a r c a would o c c u r
a c o n t r a d i c t i o n t o t h e d e f i n i t i o n o f an elementary path.
F i n a l l y , s i n c e D1 i s a c u t , any p a t h has t o c o n t a i n a t l e a s t one a r c o f D1, i . e . , a E P f o r a l l augmenting paths P. By (4.3) we can w r i t e each pi as
H.w.Hamacher et a[.
194
p
(4.4)
i
i
= (P,J’YSP
i Y
1
i i where x = t ( a ) , y = h(a), and p, and p are augmenting paths from s t o x and from Y y t o to, r e s p e c t i v e l y . Paths p i and pi can now be found independent from each o t h e r i n m o d i f i e d networks Y Gx and Gy. The node s e t o f G, is V x = tXIU{rt)} and t h e a r c s e t is Ax = Ca r A : t ( a )
E
X1} where we r e d e f i n e h(a) = t f o r a l l a € A x w i t h h ( a ) f X1.
A ) where V = {I, U { s l l and Ay = { a E A : h ( a ) E. Y’ Y Y where t ( a ) = s f o r a l l a E A w i t h t ( a ) E. X1 (see F i g u r e 4 ) . Y Analogously, Gy = ( V
F i g u r e 4:
2,)
Construction o f G, ( f o r x = v1 o r x = v2) and G ( f o r x = v3 o r v = v4) Y
By the c o n s t r u c t i o n o f Gx and Gy each augmenting p a t h p
i corresponds t o augmenting paths (px,x,t)
i = (px,x,y,py) i i i n (G,ca)
i n (Gx,ca) and (s,y,py)
i Conversely, each p a i r o f augmenting paths (p,,x,t)
i
i n (G ,c ) . Y a
i n (Gx,ca) and (s,y,py)
i ( G ,c ).corresponds t o an augmenting p a t h pi = (p,,x,y,p;). Y a i amount o f flow which can be sent along (px,x,t) and (s.y,p’), Y i i i E = min(rx.c:) can be sent along p
cl
i
in
and ci is t h e Y r e s p e c t i v e l y , then
If
.
Using an i n d u c t i v e argument we can t h e r e f o r e f i n d t h e value f a ( t , s ) o f a maximal a l g e b r a i c f l o w i n (G,ca) by computing maximal f l o w values f x ( t , s )
i n (Gx,ca) and
Ranking the cuts and cut-sets of a network fy(t,s)
i n (Gy,ca),
195
and = min(fx(t,s),f Y (t,s)).
fa(t,s)
(4.5)
The advantage o f t h i s procedure i s t h a t we can use f x ( t , s ) f o r a p o s s i b l e o t h e r a r c a ' E D,, a"
E
a ' # a, w i t h t f a ' ) = x and f (t,s) Y If
f o r a possibly e x i s t i n g arc
D1, a" f a, w i t h h ( a " ) = y.
t ( D 1 ) = {x E. X1 : x = t ( a ) f o r some a
h(D1) = { y e
x,
v
x E t(D,),
and f (t,s), Y
V y E h(D1),
In t h i s way we i d e n t i f y B by
Y a E D ] , by (4.5).
I
f a ( t y s ) = min{fa(t,s)
(4.8)
D1l
: y = h(a) f o r some a E D1}
then we have t o compute a l l values f,(t,s), i n order t o compute fa(t,s),
E
a E DII,
Then we can f i n d the 2nd best c u t - s e t by s o l v i n g an a d d i t i o n a l maximal a l g e b r a i c f l o w problem i n (G,cz). An a l t e r n a t i v e approach can use the a l g e b r a i c minimal cuts (X,R) o r (Y,p) i d e n t i f i e d w h i l e computing f x ( t , s ) and f ( t , s ) (where x = t ( i ) , y = h ( a ) ) . Y Following (4.5) we consider the f o l l o w i n g case a n a l y s i s .
Case 1:
fg(t,s) = fX(t,s), i.e.,
fx(t,s)
(X,x) corresponding a l g e b r a i c minimal c u t - s e t i n G,,
= c(X,R).
Define
x,
(4.9)
=
x, R,
=
RU
x1
Then the d e f i n i t i o n o f G, y i e l d s c(X2,12) = c(X,R) = f x ( t , s )
That i s , D2 = (X2,X,)
= fg(t,s).
i s a 2nd best cut-set. Case 2:
f-(t,s) a
= f (t,s)
set i n Gy, (4.10)
Y
i.e.,
< f
X
(t,s),
fy(t,s)
xp
=
(Y,p) corresponding a l g e b r a i c minimal cutDefine
= c(Y,Y).
vux,, x,
=
8.
Then, again, we conclude from t h e d e f i n i t i o n o f G Y c(X2,X2) = c(Y,p) = f (t,s) = f - ( t , s ) , i.e., D2 = (X2,R2) Y a i s a 2nd best cut-set.
196
H. W. Hamacher et al.
Since we can d e f i n e a 2nd b e s t c u t - s e t by (4.9) o r (4.10) we g e t t h e f o l l o w i n g result. (4.11)
I f t h e network G contains two o r more cut-sets, then t h e r e
Theorem:
e x i s t two best c u t - s e t s D1 = (X,,X,)
x1 n R,
crossing ( i . e . ,
=
(I o r ji,n
and D2 =
x2
=
(X2,f2)
t h a t are non-
0).
I n both a l t e r n a t i v e s we solve a t most It(Dl) U h(D1)l 6 n a l g e b r a i c f l o w problems t o f i n d a 2nd best c u t - s e t .
Hence t h i s procedure i s o f o r d e r O(n'+) compared w i t h
O(m.n3) = O(n5) of t h e f i r s t v e r s i o n . D1 = [(~,3),(2,3),(1,6)1, fs(t,s)
= ft(t,s)
f2(t,s)
=
f3(t,s)
= 9,
=
(Example see F i g u r e 5 ) .
h!D1) = ~ ~ , 2 , 1 1t,( D 1 ) = I 3 9 6 t I
m
5, f 1 ( t , s ) = 5 =
9
= minIf,(t,s),
f3(t,s)1 = 9
f(2,3)(t,s)
= minIf2(t,s),
f3(t,s)}
= 5
f(1,6)(t3s)
= minIfl(t,s),
fg(t,s)]
=
f
(t,s)
fg(t,s)
(5-3)
i
Hence
= (2,3) o r
a
= (1,6)
A minimal c u t - s e t i n (G,ca) F i W e 5:
5
i s i n b o t h cases D2 = I ( s , l ) , ( s , 3 ) 3
F i n d i n g t h e 2nd b e s t c u t - s e t i n t h e network o f F i g u r e 1.
In the f o l l o w i n g we show how t o g e n e r a l i z e t h e c a p a c i t y m o d i f i c a t i o n (4.1) t o t h e case o f f i n d i n g t h e K b e s t c u t - s e t s . Assume t h a t we have found K-1 best c u t - s e t s D1,...,DK-l {Dl
,. . .
I.
and l e t DK = 2)
Any c u t D E DK must be such t h a t , f o r each i = 1,.
e x i s t s an arc ai
E
Di
-
D, f o r otherwise D would n o t be minimal.
of arcs a minimal covering o f D1
,..., DK- 1 i f Rn Di
#
and i f no proper subset o f R s a t i s f i e s t h i s p r o p e r t y .
..,K-1
there
We c a l l a s e t R
(I f o r a l l i = 1
,...,K-1
There e x i s t a f i n i t e number number, say L, o f such minimal coverings. C l e a r l y L 4 6 mK-'. so we now assume t h a t we have a l i s t R1,R 2,...,RL o f a l l such minimal coverings. L e t D
Dn
(tl
Kj
such t h a t R . = 0, f o r j = 1, ..., L. K J I C l e a r l y OK = U DK,. although t h e DK. a r e i n general n o t d s j o i n t . L e t D K be a j=l J J j
denote t h e s e t of a l l c u t - s e t s D E D I
197
Ranking the cuts and cut-sets of a network minimum c u t - s e t i n DK,.
Then a K-th b e s t c u t - s e t DK i s s i m p l y a b e s t c u t - s e t
,...,
D F i n d i n t DK amounts t o f i n d i n g a minimal a l g e b r a i c c u t among OK ,D 1 Kp KL. T h i s can be achieved by f i n d i n g a m i n i s e t t h a t does n o t c o n t a i n any a r c i n R j* ma1 a l g e b r a i c c u t - s e t i n a network (G,cR.) where J (4.12)
Hence a K-th b e s t c u t - s e t can be found by s o l v i n g a t most 0 ( m K - l ) minimum
. . ,DK-l
a l g e b r a i c c u t - s e t problems, p r o v i d e d D1,.
are available.
This r e s u l t s i n
a 0(mK-l n 3 ) a l g o r i t h m f o r f i n d i n g t h e K b e s t c u t - s e t s i n a network w i t h n nodes and m a r c s . Note t h a t t h e above a l g o r i t h m , f o r f i x e d K, i s polynomial i n m and n.
However,
i t appears q u i t e i m p r a c t i c a l f o r l a r g e v a l u e s o f K, as i t i s e x p o n e n t i a l i n K. It
is unknown whether an a l g o r i t h m polynomial i n m, n
erroneous [14].
exists f o r finding
Note a l s o t h a t t h e O(K.n4) a l g o r i t h m proposed i n [12]
K best cut-sets.
is
A polynomial a l g o r i t h m i n m, n, and K would be a v a i l a b l e p r o v -
i d e d we can s o l v e t h e f o l l o w i n g problem: F i n d a b e s t c u t - s e t c o n t a i n i n g a g i v e n s e t I o f arcs. We now p r o v e t h a t t h e problem o f f i n d i n g Dl,..-,DKml
- is
2
K-th b e s t c u t - s e t
-
w i t h o u t knowing
That i s , an a l g o r i t h m p o l y -
= (IR,+,<).
NP--hard, even i f (H,*,<)
nomial i n m, n and log K e x i s t s f o r t h i s problem i f and o n l y i f
P
= NP.
L e t us
f i r s t f o r m u l a t e i t as a d e c i s i o n problem (see [ 8 ] ) :
K-TH SMALLEST CUT-SET
I'nstance: a d i r e c t e d graph G = (",A),
c a p a c i t y c ( a ) e 2'
f o r each a E A, p o s i t i v e
i n t e g e r s K and B. Q u e s t i o n : a r e t h e r e K o r more d i s t i n c t c u t - s e t s D As o t h e r k-th b e s t problems ([8]),
C_
A w i t h c a p a c i t y c ( D ) < B?
t h i s problem i s n o t known t o be i n NP.
(4.13)
Theorem:
Proof:
Use r e d u c t i o n f r o m t h e f o l l o w i n g NP-hard problem.
K - t h s m a l l e s t c u t i s NP-hard.
K-TH LARGEST SUBSET
Instance: a f i n i t e s e t U, w e i g h t w(u) B 2
+
f o r each u
E
U, p o s i t i v e i n t e g e r s K
198
H. W.Hamacher et al.
and L. Question: are t h e r e K o r more d i s t i n c t subsets U ' C U w i t h w ( U ' ) 3 L? To an instance(U,w,K,L)
o f K-TH LARGEST SUBSET a s s o c i a t e t h e f o l l o w i n g instance
o f K-TH SMALLEST CUT:
v
=
uU I s , t l
A = {(s,u)
:
6 U)
(we assume s,t IJ E
l+w(u) c ( a ) = Il
6 = I U I + w(U)
Ulu I ( u , t )
:
U
Q
Ul
f o r a = (s,u) f o r a = (u,t)
-
L.
There i s a one-to-one correspondence between subsets U ' and c u t - s e t s D = (S,T) d e f i n e d by S =
U'u { s l , w i t h c(S,T)
=
Ill1 +
w(U)
-
w(U'), and t h e r e s u l t f o l l o w s .
Note t h a t t h e r e i s a l s o a one-to-one correspondence between c u t - s e t s and c u t s i n t h e network used i n t h e above p r o o f . SMALLEST CUT problem i s a l s o NP-hard.
I t f o l l o w s t h a t t h e corresponding K-TH
I n c o n t r a s t w i t h the case o f cut-sets,
however, an a l g o r i t h m polynomial i n m,n and
K i s known f o r f i n d i n g t h e K-th best
c u t s i n a network, as was demonstrated i n Section 3 o f t h i s paper.
REFERENCES: Burkard, R.E., Kombinatorische Optimierung i n Halbgruppen in:Bulirsch, R . , O e t t l i , !4. and S t o e r , J. (eds.), Lecture Notes i n Mathematics 477 (Springer Verlag, B e r l i n , 1975). Burkard, R.E. and Zimnermann, U., Combinatorial O p t i m i z a t i o n i n L i n e a r l y Ordered Semimodules: A Survey, i n : Korte, B. (ed.), Modern Applied Mathematics : O p t i m i z a t i o n and Operations Research (Amsterdam, North-Hol land, 1982) 391-436. Flows i n Networks, (Princeton U n i v e r s i t y Ford, L.R. and Fulkerson, D.R., Press, Princeton, New Jersey, 1962). A l g o r i t h m f o r s o l u t i o n o f a problem o f maximum f l o w i n a Dinic, E.A., rietwork with power e s t i m a t i o n , S o v i e t Math. Dokl. 11 (1970) 1277-1280. Domschke, W. and Engele, G. , M o g l i c h k e i t e n der Transformation eines VerkehrF netzes i n e i n dazu uales Netz zum Zweck von Kapazitatsuntersuchungen, Angewandte I n f o r m a t i k 10 ( 1 978) 432-437. Edmonds, J. and Karp, R.M., T h e o r e t i c a l improvement i n a l g o r i t h m i c e f f i c i e n c y f o r network f l o w problems, Journal o f t h e A.C.M. 19 (1972) 248264.
Ranking the cuts and cut-sets of a network
[7]
Gabow, H.N., Two algorithms f o r generating weighted spanning t r e e s i n order, S I A M J. Comput. (1977) 139-150.
[8]
Garey, M.R. and Johnson, D.S., Computers and I n t r a c t a b i l i t y : A Guide t o t h e theory o f NP-completeness, (Freeman, San Francisco, 1979).
[9]
Hamacher, H., Maximal a l g e b r a i c flows: algorithms and examples, i n : Pape, U. (ed.), D i s c r e t e S t r u c t u r e s and Algorithms, (Hanser, Munchen-Wien, 1980) 153-166. Hamacher, H., Flows i n r e g u l a r matroids, Math. Systems i n Economics 69 (1 981 ) Oel geschl ager , Gunn & Hai n , Cambridge
.
Hamacher, H., Determining minimal c u t s w i t h a minimal number o f arcs, Networks 12 (1982) 493-504. Hamacher, H.W., An O(K.n') a l g o r i t h m f o r f i n d i n g t h e K b e s t c u t s i n a network, Operations Research L e t t e r s 1,5 (1982) 186-189. Hamacher, H. and Tufekci, S., Lexicographical time-cost t r a d e o f f problems, Methods o f Operations Research, 45 (1983) 257-268. Hamacher, H.W., Picard, J.C. and Queyranne, M., c u t s i n a network, Technical Note (1983).
On f i n d i n g t h e K b e s t
Hammer (Ivanescu), P.L., Some network f l o w problems solved w i t h pseudoBoolean programming, Oper. Res. 13 (1965) 388-399. Determining the maximal f l o w i n a network by t h e method o f Karzanov, A.V., preflows, S o v i e t Math. Dokl. 15 (1974) 434-437. Lawler, E.L., A procedure f o r computing t h e K b e s t s o l u t i o n t o d i s c r e t e o p t i m i z a t i o n problems and i t s a p p l i c a t i o n t o t h e s h o r t e s t path problem, Management Science 18 (1972) 401-405. Murty, K.G., An a l g o r i t h m f o r r a n k i n g a l l t h e assignments i n i n c r e a s i n g o r d e r o f cost, Operations Research 16 (1968) 682-687. P h i l l i p s , S. and Dessouky, M.I., S o l v i n g t h e p r o j e c t time/cost t r a d e o f f problem u s i n g t h e minimal c u t concept, Management Science 4 (1977) 393-400. Picard, J.C. and Queyranne, M., Selected a p p l i c a t i o n s o f maximum flows and minimum c u t s i n networks, INFOR (1982). Picard, J.C. and R a t l i f f , H.D., A graph t h e o r e t i c equivalence f o r i n t e g e r programs, Operations Research 21 (1973) 261 -269. Picard, J . C . and R a t l i f f , H.D., 5 (1975) 357-370.
Minimum cuts and r e l a t e d problems, Networks
Sidney, J.B., Decomposition algorithms f o r single-machine sequencing w i t h precedence r e l a t i o n s and d e f e r r a l costs, Operations Research 22 (1975) 283298. [24]
Tufekci, S. , A f l o w preserving a l g o r i t h m f o r time-cost t r a d e o f f problem, IIE Transactions, v o l . 14, # 2 (1982) 109-113.
199
200 cZ5]
H. W.Hamacher et al. Zimnermann, U, Linear and combinatorial o p t i m i z a t i o n in ordered a l g e b r a i c structures, Annals of D i s c r e t e Mathematics 10 (1981).
Annals of Discrete Mathematics 19 (1984) 201-214 0 Elsevier Science Publishers B.V. (North-Holland)
201
SHORTEST PATHS I N SIGNED GRAPHS
P. Hansen I n s t i t u t d’Economie S c i e n t i f i q u e e t de G e s t i o n , Lille, France, and F a c u l t e U n i v e r s i t a i r e C a t h o l i q u e de Mons, Belgium.
A l a b e l l i n g a l g o r i t h m i s proposed t o determine t h e s h o r t e s t signed paths between one v e r t e x and a l l o t h e r s i n a weighted signed graph. I t can be implemented t o t a k e O(mlog l o g D/d) t i m e and O(n + m + D/d) space where D and d denote t h e l a r t h e l a r g e s t and s m a l l e s t w e i g h t s o f t h e a r c s , r e s p e c t i v e l y . The case where t h e a d d i t i o n a l r e q u i r e m e n t t h a t t h e paths b e elementary i s imposed, which i s NP-complete, i s - t a c k l e d through d e c m p o s i t i o n and branch and bound. A p p l i c a t i o n s a r e made t o problems o f b a l a n c e i n s m a l l groups, t r a n s i e n t behaviour o f complex s o c i e t a l systems and s i g n s o l v a b i l i t y of l i n e a r q u a l i t a t i v e systems o f equations.
1.
INTRODUCTION
L e t G = (X,U)(’)
denote a weighted signed g m p h i . e . a graph t o each a r c (xk,xl)eU
o f which b o t h a w e i g h t dkl
E
Rf and a s i g n S k l
E
{+,-I a r e assigned.
s i g n o f a p a t h as t h e p r o d u c t o f t h e s i g n s o f i t s a r c s .
Define t h e
The p r e s e n t paper i s
devoted t o t h e problem o f d e t e r m i n i n g s h o r t e s t signed paths between one v e r t e x and a l l o t h e r s , and i t s a p p l i c a t i o n s .
A l a b e l l i n g a l g o r i t h m f o r t h e g e n e r a l case i s proposed i n S e c t i o n 2; i t i s shown t h a t i t can be implemented so as t o have O(m l o g l o g D/d) t i m e a n d O(n t m
+
D/d) space c o m p l e x i t y , where n =
l a r g e s t and s m a l l e s t w e i g h t of t h e a r c s of
1x1, m
=
( U I , D and d denote t h e
U, r e s p e c t i v e l y .
The more d i f f i c u l t problem r a i s e d by t h e a d d i t i o n a l r e q u i r e m e n t t h a t t h e paths be elementary, i . e . t h a t no path does c o n t a i n more t h a n once t h e same v e r t e x , i s examined i n S e c t i o n 3.
I t i s noted t h a t t h i s problem i s NP-complete and an
a l g o r i t h m u s i n g decomposition and branch-and-bound i s proposed t o s o l v e i t . A p p l i c a t i o n s a r e t a k e n up i n S e c t i o n 4; t o problems of balance i n s m a l l groups, as presented by C a r t w r i g h t and Harary [4],
t o t h e t r a n s i e n t behaviour o f
( l ) We f o l l o w t h e t e r m i n o l o g y and n o t a t i o n s o f Berge [3].
202
P Harisen
s o c i e t a l systems, as modelized by Roberts [24],
and t o the problem o f s i g n -
s o l v a b i l i t y o f l i n e a r q u a l i t a t i v e systems o f equations.
An a l g o r i t h m implement-
i n g t h e necessary and s u f f i c i e n t c o n d i t i o n s f o r s i g n - s o l v a b i l i t y o f t h e Bassett Maybee and Q u i r k theorem [l] i s given f o r t h e l a t t e r case; i t embeds a polynomial a l g o r i t h m f o r t h e r e c o g n i t i o n o f s t r o n g l y s i g n s o l v a b l e systems. Conclusions a r e drawn i n s e c t i o n 5.
2.
THE DOUBLE-LABEL ALGORITHM
I n the f o l l o w i n g a l g o r i t h m , two l a b e l s if
and A - a r e assigned t o each v e r t e x j X, hence i t s name; these l a b e l s a r e i n t e r p r e t e d , a t a c u r r e n t i t e r a t i o n , 3
x. E J as t h e normalized l e n g t h s o f the s h o r t e s t p o s i t i v e and negative paths from
x1 t o x . y e t found. The a l g o r i t h m i s s i m i l a r i n s p i r i t t o t h a t o f D i j k s t r a 116.1 J and makes use o f buckets as i n Denardo and Fox 1151. I t s r u l e s are as f o l l o w s : a ) InitiaZisation a.1) Determine D = max k 9 1 1 (xk , X i d = min S1
dkl
and
\ ( Xk SX1)EU d k l
-
+
+
a . 2 ) Set a1 = 0, A. = = f o r j = 2,3 ,..., n; a = f o r j = i,2 J j + - = 0 f o r j = 1,2 + n; T = T = (1,2 nl. pj = p j
,...,
,...,
b) Selection of vertices
I
b.1) L e t a = Min {LA;] if
= u go t o d ) .
j E T+,
1A;J
I 1=
LXil, k
S- = {k
1
[Ail, k E T - 1
b.2) Set T+:= T+ \ S + ,
j
E
T-);
Otherwise l e t
S+ = {k
=
I
E T'l
T-:= T-
\ S-;
i f T+ u T- = @ go t o d ) .
if A t
,Ak+ + dkl/d
+
++
s e t A1 = ak
dk,/d
and p:
= k+
,..+,n;
203
Shortest paths in signed graphs t
c.2) Ik E S- : V 1 E T i f X1 > X i t d V 1 E T-
1
kl
d)
(x,,x,)
E
/d s e t x1 =
(xk,xl)E
i f X i > h i t dkl/d go t o b ) . Final ZabeZs.
I
t
i-
U,
ski
X i t dkl/d
=
-
+
and p1 = k-;
U, sk, = t set
= X i t dk,/d
and p i = k-;
+
S e t A t := A . x d and A - : = A T x d, j = 1,2 J J j~ t - - + P r i n t t h e xj, hj, p . and p i f o r j = 1,2 n. J
,...,n
,...
The s i g n e d s h o r t e s t paths a r e found through backwards r e c u r s i o n w i t h t h e p o i n t e r s
p i , p: t a k i n g s i g n s i n t o account. Note t h a t t h e i m p l i c i t assumption i s made J t h a t an empty p a t h has p o n ' t i v e s i g n . Note a l s o t h a t 0, computed i n s t e p a ) i s n o t used f u r t h e r i n t h e s e t o f r u l e s j u s t s t a t e d ; i t i s however necessary t o know t h i s v a l u e , and t h a t o f r D / d l , f o r t h e a s y m p t o t i c a l l y most e f f i c i e n t implementation o f the a l g o r i t h m , as d i s c u s s e d below.
An example o f t h e use o f
t h e a1 g o r i thm, on t h e graph o f f i g u r e 1, i s summarized i n t a b l e 1.
204
A; O
P. Hansen
Aj
x; a
a
A;
A; a
r
A; r
- x -2
A1
a
a
a
0:
a
a
A;
A;
A;
a
a
1 1 1 1 1
hi
a
={l} a {21 I31 10/3 0 10/3 E4,53 10/3 I61
0 4 / 3 = a 0 4 / 3 2 = 0 4 / 3 2 = 0 4/3 2 13/3 0 4/3 2 13/3
4 4
16/3
P;
P i
Pi
-
-
-
-
-
p1
p2
p3
p4
p5
p6
0
0
0
0
0
0
0
P;
Pj
0
0
0
1
+
0
1 + 1 + 1+ 1+
4 4 44-
0 0 0
Lengths
0
Pi
a
0:
3 3 3
a
13/3 13/3 13/3
0
0
0
0
0 0 0
2-
0 0 22-
0 0
5+
0
13
12
16
a 9 1 3
0 0
0
0
0 2-
0 0
0 0 1 + 0 0 1 + 3+ 5- I + 3+ 5- I + 3+ 5- 1+
a
7/3 7/3 7/3 7/3
s+
sfl I41 I51 I2361
I31 @
0
0 2 + 0 2+ 3+ 2+ 3+ 2+ 3+
:
4
6
3
7
10
T a b l e 1. R e s o l u t i o n o f t h e example o f f i g u r e 1 by t h e d o u b l e - l a b e l a l g o r i t h m .
L e t us now prove t h e a l g o r i t h m ' s c o r r e c t n e s s : 7;zz JaZueS of the labeZs A t and A T given b y the double label algorithm J m e equcl t o the l u n g ~ h so f the shortest positive and negative paths from x to
Yheorsr-r 2 .
1
x . for, uEi j fro1" 1 t o n. J P r o o f . The p r o o f i s by i n d u c t i o n on t h e number o f i t e r a t i o n s o f s t e p b ) . L e t us f i r s t n o t e t h a t , f r o m t h e r u l e s d e f i n i n g T+ and T-, l a b e l s x t o r :x such t h a t J J lit1 o r l<:l = A a t a g i v e n i t e r a t i o n a r e n o t m o d i f i e d a t any subsequent J + A = 0, which a l l o w s us i t e r a t i o n . Moreover, a t t h e f i r s t i t e r a t i o n o n l y LA L)
A=
1
t o begin t h e induction.
-
L e t us assume t h a t t h e l a b e l s o f v e r t i c e s s e l e c t e d
up t o i t e r a t i o n t - 1 a r e e q u a l t o t h e l e n g t h s o f t h e c o r r e s p o n d i n g s i g n e d s h o r t e s t paths ( a f t e r m u l t i p l i c a t i o n by d ) and c o n s i d e r t h o s e o f i t e r a t i o n t.
205
Shortest paths in signed graphs
Assume, by contradiction, one such l a b e l , say
t x., J
i s l a r g e r than t h e lenoth of
the s h o r t e s t positive path from x , to x . divided by d; l e t Pt denote t h i s path, J
l ( P t ) i t s length, xk t h e predecessor o f x x1 t h e vertex w i t h a label associated j' with Pt previously selected in s t e p b ) c l o s e s t t o x under t h a t condition and xr j
t h e follower of x1 on P t .
t
From s t e p c ) , x1 = xk implies ?, x d g l ( P + ) , so j x1 # xk. Now the subpath of Pt from xr t o x - has length a t l e a s t d and from J s t e p b ) the label of xr on Pt ( h i o r 1;) i s such t h a t LA> (orLAJ) 4 x, hence
1 ( P + ) 2 (1 - + 1 ) x d contradicting again h jt x d by induction.
Remark 1 .
1 (P').
The proof then follows
Beineke and Harary [Z] have introduced t h e concept of marked graphs,
i . e . graphs i n which each vertex xk has a s i g n s k e { + , - I . Let us consider a signed and marked graph and look a t the nroblem of finding signed paths from x 1 to a l l x . , the sign o f a path being the product of t h e signs of both i t s J arcs and i t s v e r t i c e s (including end v e r t i c e s ) . This problem can be reduced to the previous one by the following transformations, which leave the signs of a l l
paths unchanged: I f s 1 = - s e t s 1 = t and reverse t h e signs of a l l a r c s with x 1 as i n i t i a l vertex ; b ) For a l l x . such t h a t s = - s e t s = + and reverse the signs of a l l a r c s with J j j x as t e r n i n a l vertex; j c ) Erase a l l signs of v e r t i c e s . a)
Remark 2. structures.
The double-label algorithm can be implemented w i t h various data For sparse graphs and D/d moderate, asjmptotically the b e s t implement-
?, mod LD/d] and ation seems to obtain by u s i n g a t a b l e o f possible values of chained l i s t s of indices o f v e r t i c e s with Lxfimod LD/d] o r L?,:]mod LD/dl equal
J
J
to such values. A very e f f i c i e n t device of Van Emde Boas, Kaas and Z i j l s t r a €291 [30] and Johnson 0 5 1 allows t o imolewnt t h e usual p r i o r i t y queue operations on such a table in time proportional t o the double algorithm o f , and snace
proportional t o , i t s length. T h i s y i e l d s an O(m log log D/d) O ( n t m t D/d) space imoTementation (see Karlsson a n d Poblete ion of the application o f these data s t r u c t u r e s t o D i j k s t r a ' s some values of n, m, D a n d d, o t h e r s t r u c t u r e s such as binary
time and [16] f o r a discussa1 gori t h m ) . For countinq t r e e s [lo]
o r heaps could y i e l d f a s t e r imnlementations. Remark 3. When a l l arcs have p o s i t i v e signs the problem reduces t o the usual s h o r t e s t path problem with non-negative weights; the normalization of lengths i m p l i c i t i n the algorithm a n d the use o f buckets solves, to some e x t e n t , the problem raised by Gallo and Pallotino [7] of having very long t a b l e s o f possible values f o r the labels when verv p r e c i s e data i s used.
P Hansen
206 Remark 4.
I n the case o f graphs i n R2, as e.g. road networks, the v e r t i c e s
selected i n step b ) belong t o d i s j o i n t ring-shaped regions i n c r e a s i n g l y f a r from
x,;
using buckets allows t o replace t h e sum and comparison o p e r a t i o n s by
less time consuming l o g i c a l t e s t s f o r arcs b o t h e n d v e r t i c e s o f which belong t o the same such r e g i o n . Remark 5.
When a l l d . are equal t o 1, an O(m) implementation i s e a s i l y Jk
obtained [9] . 3.
ELEMENTARY SHORTEST PATHS
A signed s h o r t e s t path between x, and some v e r t e x x . mav c o n t a i n a c i r c u i t , w i t h J negative s i g n . Such a p a t h may c o n t a i n a p o s i t i v e c i r c u i t o r several p o s i t i v e
and negative ones o n l y i f a l l arcs o f a l l these c i r c u i t s , except p o s s i b l y a negative one, have w e i g h t 0; should t h i s be t h e case, d e l e t i o n o f redundant c i r c u i t s y i e l d s a s h o r t e s t path w i t h a s i n g l e and n e g a t i v e c i r c u i t .
I f , however, i t i s r e q u i r e d t h a t t h e p a t h c o n t a i n no c i r c u i t a t a l l t h e Droblem becomes NP-complete.
Indeed, as very r e c e n t l y mentioned by Johnson [16],
LaPaugh and Papadimitriou L21] have shown t h a t t h e e x i s t e n c e nroblem f o r an elementary path w i t h an even number o f arcs j o i n i n g a v e r t e x x 1 t o a v e r t e x xn i n a graph G = (X,U)
i s NP-complete.
Now, i f a l l arcs o f G are given negative
signs, any such path i s a p o s i t i v e elementary p a t h from x, t o xn and conversely; i t s existence would be d e t e c t e d by a signed elementary s h o r t e s t oath a l g o r i t h m . Note t h a t t h e corresponding s h o r t e s t p a t h problem on an u n d i r e c t e d graph i s polynomial and has been solved by Edmnds (see
091)through
an e l e g a n t r e d u c t i o n
t o matching. Problems o f moderate s i z e may be solved by t h e f o l l o w i n g a l g o r i t h m , based upon decomposition and branch-and-bound ( d u r i n g t h e w s o l u t i o n the graph G =
(X,U)
w i l l be m o d i f i e d and may c o n t a i n two arcs from x, t o a v e r t e x x., w i t h signs J + and -; t h e corresponding weights w i l l be noted d+ and d- ) . 1J 1j a)
Initin2izatii.n
G
t h e t e r m i n a l v e r t e x o f which i s x,.
a.1)
Suppress a l l arcs o f
a.2)
Determine by d e p t h - f i r s t search (see T a r j a n [28])
= (X,U)
t h e s t r o n g components
o f G and then t h e b l o c k s (subgraphs w i t h o u t c u t - v e r t i c e s ) o f these s t r o n g components. a.3) Rank the v e r t i c e s o f G and re-index them i n such a way t h a t i ) v e r t ces o f the same s t r o n g b l o c k have consecutive indices;
belong t o d i f f e r e n t s t r o n g blocks (x,,x,)
g
u
=>
k
ii) if >
1.
x k and x
207
Shortest paths in signed graphs
b)
SeZection o f a subgraph
Consider t h e subgraph G' = ( X I , U x , ) o f G where X ' i s composed o f t h e v e r t i c e s o f G i n i n c r e a s i n g o r d e r o f i n d i c e s up t o and i n c l u d i n g those o f a s t r o n g b l o c k w i t h more than one v e r t e x ( o r up t o n i f no such b l o c k remains). c)
S h o r t e s t signed path and recognition of non-elementary ones
Apply t h e d o u b l e - l a b e l a l g o r i t h m t o t h e subgraph G ' .
Determine f r o m t h e p o i n t e r s
p?, p: t h e s i g n e d s h o r t e s t p a t h s f r o m x 1 t o x . f o r j = l , Z , , . . , I X ' l . N o t e which J J J o f these a r e non-elementary. I f t h e r e a r e no p a t h s w i t h c i r c u i t s go t o e ) ; o t h e r w i s e go t o d ) . d)
Branch-and-bound algorithm
Determine t h e s h o r t e s t elementary s i g n e d p a t h s f o r a l l j and s i g n s
t
or
-
such
f
corresponds t o a non-elementary p a t h b y a p p l y i n g f o r each o f them t h a t A . o r :A J J i n t u r n a branch-and-bound method. Such a method c o u l d use as s e p a r a t i o n p r i n c i p l e t h e e x c l u s i o n o f one o f t h e arcs o f t h e n e g a t i v e c i r c u i t w h i c h i s n o t used t w i c e i n t h e p a t h and as bounding r u l e
the f a c t t h a t t h e length o f the s h o r t e s t
s i g n e d p a t h g i v e n by t h e d o u b l e - l a b e l a l g o r i t h m i s , o f course, a l o w e r bound on t h e l e n g t h o f t h e elementary s h o r t e s t s i g n e d path. Update t h e values of 1' and :X the absence o f elementary s i g n e d p a t h b e i n g j J' n o t e d by an i n f i n i t e Val ue. e)
Modification o f G e.1)
Suppress a l l a r c s j o i n i n g v e r t i c e s o f X '
e.2)
I f x. Q X', J
1
I f x. E X', J
1j
If I X ' I
<
+ j
< = add an a r c f r o m x
< = addan a r c f r o m x 1 t o
n go t o b ) .
t
= X . and s . = 1 t o x J. w i t h dti j J 1J
xJ. w i t h dAj = X-j
and s 15. =
Othetwise, end.
The c o r r e c t n e s s o f t h e a l g o r i t h m f o l l o w s f r o m t h e f a c t s t h a t i ) ii)
any elementary p a t h f r o m x t o x . must go through a sequence o f v e r t i c e s 1 J w i t h indices o f increasing value except possibly w i t h i n a strong block; any elementary c i r c u i t i s c o n t a i n e d i n a s t r o n g b l o c k , and thus s t r o n g b l o c k s may be processed s e q u e n t i a l l y .
The r e s o l u t i o n o f t h e example of f i g u r e 1 w i t h t h i s a l g o r i t h m i s summarized i n Table 2 and f i g u r e 2.
+;
-.
208
P. Hansen X' =
~x1,x2,x3,x4,x5~
A;
'1
x;
1;
A;
4
6
13
12
0
Pi 0
P; 1+
-
-
Pi 4-
P; 2-
'2 9
p1 0
p2 3+
-
Pi 2-
-
'1 a
-
-
-
A 3 '4 1 3 3
'5 7
-
-
p3 5-
p4 1+
-
p5 2+
Non-elementary path : . 1+ -r 4- + 3+ + 2- + 4+ 4 '
Result of branch-and-bound
x'
see f i g u r e 2 .
At
A+
= {X1YX2,X3,X4YX5,X61 A;
-
-
'1
'2 9
1;
A;
A;
hi
0
4
6
a 1 2 1 6
=
Pf
P; I+
Pi
Pi
P;
p1
1+
0
1+
1+
1;
P; 5-
=rr:
4
-
0
-
p2 1+
-
'3 1
-
'4 3 3
p3 1+
-
p4 1+
-
-
'5 '6 7 1 0
-
-
p5 1+
p6 3+
Table 2 . Resolution of the example of f i g u r e 1 by the algorithm f o r elementary signed s h o r t e s t paths.
x,
x1
Figure 2. Modified graph, a f t e r f i r s t i t e r a t i o n of elementary signed s h o r t e s t path algorithm.
4. APPLICATIONS 4.1
BALANCE I N SMALL GROUPS
Cartwright and Harary [4] have proposed t o analyse q u a l i t i v e l y the r e l a t i o n s h i p s ( c m u n i c a t i o n , cooperation, love, . . . ) between e n t i t i e s o f a small group (persons c o u n t r i e s , ...) with the help of signed graphs : v e r t i c e s a r e associated with t h e e n t i t i e s , arcs or edges w i t h the r e l a t i o n s h i p s and signs with t h e i r favourable or
Shortest paths in signed graphs
unfavourable character.
209
A m a j o r concern i s t h e n s t a b i l i t y , i . e .
t h e tendency o f
t h e system t o r e m a i n unchanged i n t i m e , and v a r i o u s concepts o f b a l a n c e a r e i n t r e duced i n [12]
0 3 1 i n o r d e r t o s t u d y how s t a b l e i s a group.
A w e l l known r e s u l t
p a r t l y a n t i c i p a t e d by Konig DgJ, i s
Theorem 4.1 a)
.
G = (X,E)
(Harary [l21 ) The following statements are equivalent:
i s a baZanced signed graphs ( i . e . t h e product of t h e s i g n s of t h e
edges o f any cycle is p o s i t i v e ) . b)
G has a p o s i t i v e cycle basis ( i . e . the product of t h e signs o f the edges of each cycle o f a cycZe b a s i s o f G i s p o s i t i v e ) .
c)
The negative edges o f G forn a cocycle ( i - e . t h e v e r t i c e s of G may be p a r t i tionned i n t o subsets X1 and X p such t h a t t h e sign of an edge is negative i f and only i f i t j o i n s v e r t i c e s from d i f f e r e n t s u b s e t s ) .
I n p r a c t i c e long Balance o f a s i g n e d graph can be checked i n O(m) t i m e [9] [12]. n e g a t i v e c y c l e s may, however, n o t u p s e t much s t a b i l i t y ; G i n N-balanced [13] i f and o n l y i f i t c o n t a i n s no n e g a t i v e c y c l e o f l e n g t h 6 N.
A p p l y i n g t h e double-
l a b e l a l g o r i t h m w i t h each v e r t e x o f G as o r i g i n i n t urn a l l o w s t o check N-balance i n O(m) t i m e [9]
(when A; becomes d e f i n i t i v e t h e a l g o r i t h m may b e stopped; N i s
N-balanced i f and o n l y i f 1; > N f o r each o r i g i n ) .
A graph G i s path-balanced ( o r chain-balanced) i f and o n l y i f f o r e v e r y x . , x k € X J a l l elementary chains from x t o x k a r e o f t h e same s i g n . A f t e r r e p l a c i n g each j edge o f G by a p a i r o f o p p o s i t e a r c s w i t h t h e same s i g n , one can u s e t h e G i s l o c a l l y balanced a t x . J i f and o n l y i f a l l elementary c y c l e s passing by x . a r e balanced. A d i r e c t e d J g r a p h G = (X,U) i s c i r c u i t balanced i f and o n l y i f a l l elementary c i r c u i t s a r e
a l g o r i t h m o f s e c t i o n 3 t o t e s t G f o r path-balance.
p o s i t i v e and l o c a l l y c i r c u i t balanced a t x . i f t h i s i s t r u e o f a l l elementary J c i r c u i t s passing by x . . These p r o p e r t i e s can a l s o be checked w i t h t h e a l g o r i t h m J o f s e c t i o n 3.
4.2.
TRANSIENT BEHAVIOUR OF COMPLEX SYSTEMS
Roberts [24]
has proposed t o s t u d y s o c i e t a l i s s u e s (energy, p o l l u t i o n e t c ) i n
canplex systems by m o d e l i z i n g them w i t h signed graphs : v e r t i c e s a r e a s s o c i a t e d t o r e l e v a n t v a r i a b l e s , arcs t o d i r e c t i n t e r a c t i o n s between them and p o s i t i v e o r n e g a t i v e s i g n s t o t h e augmenting o r i n h i b i t i n g e f f e c t o f an i n c r e a s e o f t h e v a l u e o f t h e i n i t i a l v a r i a b l e on t h e t e r m i n a l one.
Then p o s i t i v e c i r c u i t s a r e d e v i a t i o n
amp1 i f y i n g and n e g a t i v e c i r c u i t s d e v i a t i o n - c o u n t e r a c t i n g .
Roberts and Brown [26]
have s t u d i e d t h e s t a b i l i t y o f complex systems under p u l s e processes, i n which
P. Hansen
210
d e v i a t i o n s a r e t r a n s m i t t e d a t equal i n t e r v a l s i n t i m e . Problems o f t r a n s i e n t b e h a v i o u r o f t h e systems a r e a l s o o f i n t e r e s t and c o u l d be L e t us mention, among o t h e r s ,
s t u d i e d w i t h t h e a l g o r i t h m s o f s e c t i o n s 2 and 3.
t h e f o l l o w i n g q u e s t i o n s : i ) what i s t h e f i r s t e f f e c t o f an i n c r e a s e i n v a l u e o f a v a r i a b l e upon i t s e l f ? when does i t o c c u r ( d e l a y s f o r i n t e r a c t i o n b e i n g equal o r not)
i i ) which v a r i a b l e s can most i n f l u e n c e , t h r o u g h a s i n g l e p a t h , a g i v e n
one by augmenting ( i n h i b i t i n g ) i t ?
i i i ) which v a r i a b l e s have o n l y an augmenting
( i n h i b i t i n g ) e f f e c t on a g i v e n one?
4.3.
S I G N SOLVABILITY
OF
SYSTEMS OF QUALITATIVE EQUATIONS
L e t AX = b denote a square system o f l i n e a r e q u a t i o n s and assume t h a t t h e s i g n s , b u t n o t t h e magnitudes, o f a l l c o e f f i c i e n t s o f A and b a r e known.
AX = b i s
s i g n - s o l v a h h i f and o n l y i f t h e e x i s t e n c e o f a s o l u t i o n and t h e s i g n s o f a l l components o f X a r e determined b y t h e s i g n s o f t h e c o e f f i c i e n t s o f A and b
A X = b i s strongly sign-solvable i f and o n l y i f i t i s
( c f . Samuelson [ Z ; ) ;
s i g n - s o l v a b l e and no component of X i s e q u a l o f 0 ( c f . Klee and Ladner [17-1). F o l l o w i n g a d i s c u s s i o n o f Samuelson i n ?oun&tions of Economic A n a l y s i s , Lancaster
IZ . d asked f o r necessary and s u f f i c i e n t c o n d i t i o n s f o r AX . ,
sign solvable.
= b t o be
He a l s o n o t e d t h a t s i g n s o l v a b i l i t y i s n o t a f f e c t e d b y permuta-
t i o n or rows o r columns o f A o r by m u l t i p l i c a t i o n o f a r o w o r v a r i a b l e by -1. Any s i g n s o l v a b l e system can b e m o d i f i e d i n o r d e r t o have aii bi
\<
0 f o r i = l,Z,
... ,n
t o be i n such a form.
<
0, xi
>
0 and
by such t r a n s f o r m a t i o n s ; we assume f r o m now on A X = b
The s i g n s o l v a b i l i t y p r o b l e m was s o l v e d , i n a non-
c o n s t r u c t i v e way by B a s s e t t , Maybeeand Q u i r k [l]. A s s o c i a t i n g w i t h A X = b a graph G = (X,U) and skl
w i t h v e r t i c e s c o r r e s p o n d i n g t o rows and columns, (xk,xj)
= t when akl
0, (xk,xl)
6
u
and skl
can be expressed as f o l l o w s ( c f . Roberts [25],
fienrar: 4.2. ij’ i ) G
..
11 )
( B a s s e t t , laybee, Q u i r k [l])
C O ~ ; C ~ Z < M S ):G
bi . 0 +?-l4ee
Paybee r23]
=
-
when akl
<
EU
0, t h e i r r e s u l t
l a n b e r c21-1).
AX = b
is s i g n soZvabZe if and onZy
e l e r m t u q pos
X.
1
as temninaLT v e r t e x are p o s i t i v e .
c a l l s ~ i i s t i n ~ ~ i s hvertex ed any v e r t e x o f G w h i c h i s t h e t e r m i n a l
vertex o f p o s i t i v e paths only:
he d e f i n e s a siqi-solvable graph
PO]
as a graph
w i t h no p o s i t i v e c i r c u i t and a d i s t i n g u i s h e d v e r t e x i n each s t r o n g component. Manber (213 has shown AX = b i s s t r o n g l y s i g n - s o l v a b l e i f and o n l y i f G i s s i g n s o l v a b l e , bi
<
0 f o r a l l d i s t i n g u i s h e d v e r t i c e s i n t e r m i n a l s t r o n g components o f
G and bi = 0 f o r a l l n o n - d i s t i n g u i s h e d v e r t i c e s .
We now c o n s i d e r t h e a l g o r i t h m i c
aspect o f the p r o b l e m o f d e t e r m i n i n g a l l s i g n - p a t t e r n s AX = b i s s i g n - s o l v a b l e ,
f o r b ( i f any) f o r which
w i t h t h e f o l l o w i n g s i p - s o Z v a b i l i t y algom’thm:
21 1
Shortest paths in signed graphs a)
P o s i t i v e partial! subgraph.
Consider t h e p o s i t i v e p a r t i a l subgraph
o f G o b t a i n e d b y d e l e t i n g from G a l l n e g a t i v e a r c s .
Gt
=
(X,U+)
Check i f G c o n t a i n s a
c i r c u i t ; i f y e s , AX = b i s n o t s i g n - s o l v a b l e . b)
Distinguished v e r t i c e s .
t
Consider i n t u r n a l l subgraphs Gx-Ixkl
o f Gt o b t a i n e d
by d e l e t i n g f r o m Gt a v e r t e x xk which i s t h e i n i t i a l v e r t e x o f a n e g a t i v e a r c
-.
Then (x,,x,) i n G. Label a l l v e r t i c e s x1 f o r which (xk,xl) e U and s k 1 = l a b e l a l l v e r t i c e s xm f o r which t h e r e i s a p a t h f r o m a l a b e l l e d v e r t e x x1 t o xm t
i n GXJXk}.
A l l v e r t i c e s which remain u n l a b e l l e d a r e d i s t i n g u i s h e d . c)
Determine a l l a n c e s t o r s o f a l l d i s t i n q u i s h e d
Undistinguished subgraph.
v e r t i c e s ; l e t Y C X denote t h e s e t o f d i s t i n g u i s h e d v e r t i c e s and t h e i r ancestors. Consider t h e subgraph o f 6 generated b y X
\ Y and check by d e p t h - f i r s t search
whether i t c o n t a i n s an elementary p o s i t i v e c i r c u i t ; i f so, G i s n o t s i g n - s o l v a b l e and o t h e r w i s e i t i s .
I f G c o n t a i n s no elementary p o s i t i v e c i r c u i t i t f o l l o w s f r o m theorem 4.2. t h a t those b f o r w h i c h AX = b i s s i g n - s o l v a b l e a r e g i v e n by bi < 0 + x
i s distinguishi Correctness o f t h e a l g o r i t h m t h e r e f o r e depends on t h e f a c t t h a t e l e m e n t a r y
ed.
p o s i t i v e c i r c u i t s are detected.
I f Y = X, G i s s i g n - s o l v a b l e and t h e r e s u l t s
f o l l o w s f r o m theorems 2 . 2 . and 3.1. o f [ll];o t h e r w i s e i t f o l l o w s f r o m t h e a p p l i c a t i o n o f those theorems t o t h e subgraph generated by Y and o f s t e p c) t o t h e subgraph generated by X \ Y. time.
Note t h a t i f Y = X t h e a l g o r i t h m r e a u i r e s O(nm)
Step c ) i s , o f course, non p o l y n o m i a l .
Several a l t e r n a t i v e s t o d e p t h - f i r s t
search e x i s t : i ) use the a l g o r i t h m o f s e c t i o n 3 f o r a v e r t e x o f X \ Y ; i f no e7ementary p o s i t i v e c i r c u i t i s found d e l e t e i t and i t e r a t e ; i i ) use dynamic programming, e t c . 5.
CONCLUSIONS
Two a l g o r i t h m s have been proposed f o r d e t e r m i n i n g s h o r t e s t p a t h s and e l e m e n t a r y s i g n e d s h o r t e s t paths i n s i g n e d graphs. f o r t h e usual s h o r t e s t p a t h problem. f o r an NP-complete problem.
The f o r m e r one has a low c o m p l e x i t y , as
The l a t t e r one i s e x p o n e n t i a l and designed
Several a p p l i c a t i o n s , among many p o t e n t i a l ones,
have been o u t l i n e d . Acknow 1edgemen t s Thanks a r e due t o Jack Edmonds and t o M a r t i n Grhtschel f o r u s e f u l d i s c u s s i o n s .
P. Hansen
2 12
+ x1
0 0 b =
0
or
0 0 0
d)GX\ y
Figure 3. Illustration o f the use o f the sign solvability algorithm.
Shortest paths in signed graphs
213
REFERENCES: B a s s e t t , L., Maybee, J. and Q u i r k , J., Q u a l i t a t i v e economics and t h e scope o f t h e correspondence p r i n c i p l e , Econometrica, 26 (1968) 554-563. Beineke, L.W. and Harary, F., Consistency i n marked d i g r a p h s , J. Math. Psych. 18 (1978) 260-269. Berge, C.,
Graphes e t hypergraphes, ( P a r i s , Dunod, 1970).
C a r t w r i g h t , D. and Harary, F., S t r u c t u r a l b a l a n c e : a g e n e r a l i z a t i o n o f H e i d e r ' s t h e o r y , Psychol. Review 63 (1956) 277-293. Denardo, E.V. and Fox, B.L., S h o r t e s t paths methods : 1. and b u c k e t s , Oper. Res. 27 (1979) 161-186.
Reaching, p r u n i n g
D l j k s t r a , E.W., A n o t e on two problems i n c o n n e c t i o n w i t h graphs, Numerische Math. 1 (1959) 269-271. G a l l o , G. and P a l l o t i n o , S . A new a l g o r i t h m t o f i n d s h o r t e s t paths, D i s c r e t e A p p l i e d Math. 4 (1982) 23-35. G r o t s c h e l , M. and P u l l e y b l a n k , W.R., Weekly b i p a r t i t e graphs and t h e max-cut problem, Oper. Res. L e t t e r s 1 (1981) 23-27. Hansen, P., L a b e l l i n g a l g o r i t h m s f o r b a l a n c e i n s i g n e d graphs, 215-217, i n : Bermond,J.C. e t a l . ( e d s . ) , Problemes c o m b i n a t o i r e s e t t h 6 o r i e des graphes, ( C o l l . I n t e r du CNRS no 260 P a r i s , E d i t . CNRS, 1978). Hansen, P., An O(m log 0) a l g o r i t h m f o r s h o r t e s t paths, D i s c r e t e A p p l i e d Math. 2 (1980) 151-153. Hansen, P. , Recognizing s i g n s o l v a b l e graphs, D i s c r e t e A p p l i e d Math. 6 (1983) 237-241. Harary, F., On t h e n o t i o n o f balance o f a signed graph, M i c h i g a n Math. J. 2 (1953) 143-146. Harary, F., On l o c a l b a l a n c e and N-balance i n signed graphs, M i c h i g a n Math. J. 3 (1955) 37-41. Harary, F. and K a b e l l , J.A., An e f f i c i e n t a l g o r i t h m t o d e t e c t b a l a n c e i n signed graphs, Mathematical S o c i a l Sciences, 1 (1980) 131-136. Johnson, D.B., A s i m p l e p r i o r i t y queue i n which i n i t i a l i z a t i o n and queue o p e r a t i o n s t a k e 0 ( l o g l o g n) time, Proc. o f t h e 1978 Conf. on I n f o r m a t i o n Sciences and Systems, 66-71. Johnson, D.S., The NP-completeness 3 (1982) 381-395.
column: An ongoing g u i d e , J. A l g o r i t h m s
Karlsson, R.G. and Poblete, P.V., An 0 (m l o g l o g D) a l g o r i t h m f o r s h o r t e s t paths, D i s c r e t e A p p l i e d Math. 6 (1383) 91-93. Klee, V . and Ladner, R., Q u a l i t a t i v e m a t r i c e s : s t r o n g s i g n s o l v a b i l i t y and weak s a t i s f i a b i l i t y , 293-320,in: Greenberg, H.J. and Maybee, J.S. (eds.) Computer-assisted a n a l y s i s and model s i m p l i f i c a t i o n , (New York: Academic Press, 1981).
P. Hanseri Klee, V . , Ladner, R. and Manber, R., S i g n s o l v a b i l i t y r e v i s i t e d , Technical r e p o r t 82-04-04, Department o f Computer Science, U n i v e r s i t y o f Washington, S e a t t l e 98195. Konig, D., (1950).
Theorie d e r e n d l i s c h e und unendlische graphen, r e p r i n t , Chelsea
LaPaugh, A.S. and Paoadimitriou, C.H., digraphs, Networks ( f o r t h c o m i n g ) .
The even p a t h problem f o r graphs and
Lancaster, K., The scope o f q u a l i t a t i v e economics, Rev. Economic Studies, 29 (1962) 99-132. Manber, R., Graph t h e o r e t i c a l approach t o q u a l i t a t i v e s o l v a b i l i t y o f l i n e a r systems, L i n e a r Algebra and i t s A p p l i c a t i o n s 48 (1982) 457-470. Maybee, J . S . , Sign s o l v a b l e graphs, D i s c r e t e A p p l i e d Math. 2 (1980) 57-63. Maybee, J . S . , S i g n s o l v a b i l i t y , 201-257, i n : Greenberg, H.J. and Maybee J.S. ( e d s . ) Computer-assisted a n a l y s i s and model s i m p l i f i c a t i o n , (New York: Academi c Press, 1981 ) . Roberts, F.S., (1976).
D i s c r e t e mathematical models, Englewood C l i f f s : P r e n t i c e H a l l
Roberts, F.S., Graph t h e o r y and i t s a p p l i c a t i o n s t o problems o f s o c i e t y , ( P h i l a d e l p h i a : SIAM, 1978). Roberts, F.S. and Brown, R.A., Signed graphs and Math. Monthly 82 (1975) 577-594.
he energy c r i s i s , American
Samuelson, P., Foundations o f economic a n a l y s i s , New York: Atheneum, 1971, o r i g i n a l l y pub1 ished b y Harvard U n i v e r s i t y Press, 1947). Tarjan, R.E., D e p t h - f i r s t search and l i n e a r graph algorithms, Computing 1 (1972) 146-160.
S I A M 3. on
Van Emde Boas, P., Preserving o r d e r i n a f o r e s t i n l e s s than l o g a r i t h m i c time and l i n e a r space, I n f . Proc. L e t t e r s 6 (1977) 80-82. Van Emde Boas, P., Kaas, R . and Z i j l s t r a , E., Design and implementation o f an e f f i c i e n t p r i o r i t y queue, Math. Systems Theory, 10 (1977) 99-127.
Annals of Discrete Mathematics 19 (1984) 215-228 0 Elsevier Science Publishers B.V. (North-Holland)
A BOOLEAN ALGEBRAIC ANALYSIS
B.L. Hulme and A.W.
Shiver
Sandia N a t i o n a l L a b o r a t o r i e s A1 buquerque, NM 87185 U.S.A.
OF FIRE PROTECTION P.J. S l a t e r U n i v e r s i t y o f Alabama H u n t s v i l l e , AL 35899 U.S.A.
I n a comolex f a c i l i t y , such as a n u c l e a r power p l a n t , t h e d e s t r u c t i o n by f i r e o f c e r t a i n c r i t i c a l combinations o f equipment can have p o t e n t i a l l y s e r i o u s consequences. T h i s paper d e s c r i b e s a computational procedure which can be used t o f i n d minimum c o s t ways t o p r o t e c t t h e c r i t i c a l combinat i o n s o f equipment f r o m a s i n g l e - s o u r c e f i r e by p r o t e c t i n g c e r t a i n areas and s t r e n g t h e n i n g c e r t a i n b a r r i e r s a g a i n s t f i r e . The procedure y i e l d s a complete s e t o f optimum s o l u t i o n s by i t e r a t i v e l y computing upper and l o w e r bounds on t h e minimum c o s t . The f i r e p r o t e c t i o n s e t s e v o l v e f r o m Boolean a l g e b r a i c computations which o b t a i n minimum c o s t b l o c k i n g s e t s a s s o c i a t e d w i t h t h e l o w e r bounds w h i l e t h e upper bounds a r e produced by maxflow-mincut c a l c u l a t i o n s i n a network. The problem can be viewed as one o f m i n i m i z a t i o n o v e r t h e cobases o f an independence system.
1.
THE FIRE PROTECTION PROBLEM
The f i r e spread p o s s i b i l i t i e s i n a f a c i l i t y a r e modeled by a f i r e spread network n (V,A),
a weighted d i r e c t e d graph c o n s i s t i n g o f a s e t V o f v e r t i c e s , a s e t A o f
d i r e c t e d a r c s , d i s j o i n t f r o m V b u t j o i n i n g one v e r t e x o f V t o another, and nonn e g a t i v e w e i g h t s a s s o c i a t e d w i t h t h e a r c s and v e r t i c e s .
The v e r t e x s e t V i s
determined by p a r t i t i o n i n g t h e f a c i l i t y l a y o u t i n t o s u i t a b l e areas and a s s i g n i n g one v e r t e x t o each area.
The a r c s e t A i s determined by f i r s t i d e n t i f y i n g which
f i r e areas a r e p h y s i c a l l y a d j a c e n t and t h e n d o i n g combustion c a l c u l a t i o n s [2]
for
each f i r e area t o see whether t h e f u e l load, v e n t i l a t i o n r a t e , room c o n f i g u r a t i o n , e t c . , can s u p p o r t a f i r e s u f f i c i e n t t o p e n e t r a t e t h e s u r r o u n d i n g w a l l s , c e i l i n g o r f l o o r i n t o an a d j a c e n t area.
The a r c s a r e d e f i n e d i n p a i r s .
I f f i r e can
spread i n one o r b o t h d i r e c t i o n s between t h e v e r t i c e s i and j , t h e n b o t h t h e arcs ( i , j )
and ( j , i )
E
A a l t h o u g h t h e y may have d i f f e r e n t weights.
V e r t i c e s where v i t a l components o f s a f e t y systems a r e l o c a t e d a r e c a l l e d t a r g e t vertices.
A minimal s e t o f t a r g e t v e r t i c e s whose d e s t r u c t i o n c o u l d have unaccep-
t a b l e consequences i s c a l l e d a t a r g e t s e t .
The c o l l e c t i o n T = IT1,T 2,...,Tk3
of
t a r g e t s e t s t o be p r o t e c t e d i s t y p i c a l l y determined by f a u l t t r e e a n a l y s i s [1,9]. Being minimal, no t a r g e t s e t Ti c o m p l e t e l y c o n t a i n s a n o t h e r .
Since a l l v e r t i c e s
6.L. Hulme et al.
216
i n a t a r g e t s e t must be d e s t r o y e d i n o r d e r t o i n i t i a t e an u n a c c e p t a b l e e v e n t , p r o t e c t i n g one t a r g e t v e r t e x i s s u f f i c i e n t t o p r o t e c t any t a r g e t s e t t o which i t belongs. The a r c s and v e r t i c e s o f
11
a r e weighted w i t h f i r e p r o t e c t i o n costs.
The v e r t e x
weight, c ( i ) , i s t h e c o s t o f p r e v e n t i n g f i r e f r o m p r o p a g a t i n g t h r o u g h area i as
For example,
w e l l as p r e v e n t i n g f i r e damage t o any v i t a l equipment l o c a t e d t h e r e . t h i s m i g h t be t h e c o s t o f a n adequate s p r i n k l e r system. i n d i v i d u a l arc weight, c ( i , j ) , through the b a r r i e r ( i , j ) ,
Each a r c has an
w h i c h i s t h e c o s t o f p r e v e n t i n g t h e spread o f f i r e
f o r example by i m p r o v i n g t h e f i r e r a t i n g o f t h e
b a r r i e r o r by r e d u c i n g t h e f u e l l o a d i n a r e a i. The a r c s come i n p a i r s , as mentioned above, even though one a r c m i g h t have a z e r o w e i g h t i n d i c a t i n g t h a t i t i s already protected against f i r e . j o i n t arc weight, b ( i , j ) ,
Each p a i r of a r c s , ( i , j ) and ( j , i )
t h e c o s t o f p r o t e c t i n g b o t h ( i , j ) and ( j , i ) .
has a Since
t h e r e may be economies p o s s i b l e i n p r o t e c t i n g both, we have max { c ( i , j ) ,
A network n ' ( V ' , A ' ) in
17'
c(j,i)l 4 b(i,j) 6 c(i,j)
i s a subnetwork o f n ( V , A )
a r e simply r e s t r i c t i o n s from
sequence vOalvla
11 t
o n'.
+
.
c(j,i)
i f V ' s V, A '
C_
(11
A, and t h e w e i g h t s
A path i n n i s an a l t e r n a t i n g
?... akvk o f d i s t i n c t v e r t i c e s vi and d i s t i n c t a r c s ai such t h a t
t o vi, 1 6 i 4 k. D e l e t i o n o f a v e r t e x v f r o m M means t h e removal ai j o i n s v . 1-1 n o t o n l y o f v b u t a l s o o f a l l a r c s i n c i d e n t t o o r f r o m v, w h i l e d e l e t i o n o f a n a r c means removal o f o n l y t h e a r c . The most l i k e l y a c c i d e n t a l f i r e s a r e s i n g l e - s o u r c e f i r e s .
They may s t a r t a t any
v e r t e x and f l o w a l o n g p a t h s o f p o s i t i v e l y w e i g h t e d a r c s and v e r t i c e s .
I f every
v e r t e x i n some t a r g e t s e t has a common predecessor v e r t e x a l o n g such paths, t h e n t h e f a c i l i t y i s vulnerable t o a f i r e propagating from t h a t vertex.
This motivates
the following definitions.
An s - t n e t f o r (Iz,T)
i s a minimal, p o s i t i v e l y weighted, subnetwork a' i n M con-
s i s t i n g o f a source v e r t e x s, a t a r g e t s e t Ti II
such t h a t f o r e v e r y t a Ti
set f o r
(II,T)
E
T, and o t h e r v e r t i c e s and a r c s o f
t h e r e i s a p a t h i n M' f r o m s t o t .
A f i r e protection
i s a s u b s e t R o f a r c s and v e r t i c e s whose d e l e t i o n ( p r o t e c t i o n )
y i e l d s a subnetwork
11-R
n o t c o n t a i n i n g any s - t n e t .
The c o s t o f a f i r e p r o t e c t -
e c t i o n s e t R i s t h e sum o f t h e w e i g h t s o f i t s elements, u s i n g t h e j o i n t w e i g h t b(i,j)
instead o f c ( i , j )
+
c ( j , i ) when b o t h ( i , j )
and ( j , i ) E R.
The minimum c o s t f i r e p r o t e c t i o n problem i s t o f i n d a l l o f t h e minimum c o s t f i r e protection sets f o r
(M,r).
A boolean algebraic analysis of fire protection
2.
211
A BLOCKING SYSTEM
F o l l o w i n g t h e t e r m i n o l o g y o f Edmonds and F u l k e r s o n [4],
we d e f i n e a c l u t t e r s on a
f i n i t e s e t E t o be a c o l l e c t i o n o f subsets o f E, no one o f which c o n t a i n s a n o t h e r . The c l u t t e r R on E c o n s i s t i n g o f t h e minimal subsets o f E having nonempty i n t e r s e c t i o n w i t h e v e r y member o f S i s c a l l e d t h e b l o c k i n g c l u t t e r , o r t h e b l o c k e r , of S.
We s h a l l c a l l t h e members o f R b l o c k i n g s e t s a n d t h e p a i r (S,R) a b l o c k i n g
sys tern
.
F o r p r e s e n t purposes we l e t E = V
t h e s e t o f a r c s and v e r t i c e s i n n.
An s - t
The c o l l e c t i o n S o f a l l t h e s - t n e t s f o r ( n , T ) i s a c l u t t e r
n e t i s a subset o f E.
on E .
u A,
The b l o c k i n g c l u t t e r R o f S i s a c o l l e c t i o n o f minimal subsets o f a r c s and
v e r t i c e s such t h a t each member o f R c o n t a i n s a t l e a s t one element o f e v e r y s - t net. O b v i o u s l y t h e b l o c k i n g s e t s i n R a r e minimal f i r e p r o t e c t i o n s e t s .
Our t a s k i s t o
f i n d t h e minimum c o s t b l o c k i n g s e t s i n R , and we propose t o do t h i s w i t h o u t constructing either
s
o r R s i n c e t h e s e c o l l e c t i o n s can be q u i t e l a r g e .
We a r e i n d e b t e d t o t h e r e f e r e e who p o i n t e d o u t t h a t t h i s problem can a l s o be p l a c e d i n t h e framework o f o p t i m i z a t i o n o v e r independence systems
P O , p.191.
An
independence system S = ( E , I ) i s a f i n i t e s e t E t o g e t h e r w i t h a c o l l e c t i o n I o f if I
subsets o f E c l o s e d under i n c l u s i o n , i . e . ,
E
I and J
G
I , then J
E
I.
A maximal independent s e t i s a
elements o f I a r e c a l l e d independent s e t s .
The
&.
A subset o f E n o t i n 1 i s c a l l e d a dependent s e t , and a minimal dependent s e t i s
a circuit. E\Bi
The independence system S* = (E,I*)
whose bases a r e t h e complements
o f t h e bases Bi o f S i s c a l l e d t h e dual independence system o f S.
S* i s a cobase o f S, and a c i r c u i t o f S* i s a c o c i r c u i t o f S. system S i s c a l l e d a m a t r o i d i f I and J
E
A base o f
An independence
I w i t h IJI=II 1+1 i m p l i e s t h a t t h e r e
e x i s t s an element j e J \ I such t h a t I IJ { j }e I .
Among t h e many p r o p e r t i e s o f
m a t r o i d s a r e t h e f a c t s t h a t a l l bases have t h e same c a r d i n a l i t y and t h a t t h e greedy algorithm
[lo,
p.3061 w i l l always f i n d a maximum w e i g h t independent s e t
g i v e n any nonnegative w e i g h t f u n c t i o n on E. I t i s known f o r m a t r o i d s [7,
pp.36-371 and more g e n e r a l l y f o r independence
systems t h a t t h e c i r c u i t s and t h e cobases f o r m a b l o c k i n g system. problem, where E =
VU
independence system.
I n the f i r e
A, l e t S, t h e c o l l e c t i o n o f s - t nets, be t h e c i r c u i t s o f an The cobases, which a r e b l o c k i n g s e t s o f S, a r e minimal f i r e
p r o t e c t i o n s e t s , whose complements, t h e maximal p r o t e c t e d subnstworks, a r e bases. A subset (subnetwork)
X
C
E i s independent ( p r o t e c t e d ) i f X does n o t c o n t a i n any
c i r c u i t ( s - t n e t ) , otherwise
X i s dependent ( u n p r o t e c t e d ) . T h i s independence
system i s n o t a m a t r o i d because t h e bases can have d i f f e r e n t c a r d i n a l i t i e s . Therefore, t h e greedy a l g o r i t h m w i l l n o t always s o l v e t h e f i r e problem.
Our
i t e r a t i v e a l g o r i t h m , which uses Boolean a l g e b r a t o produce t h e b l o c k i n g s e t s , i s
B. L. Hulme et al
218
d e s c r i b e d i n t h e language o f t h e f i r e problem t o m i n i m i z e o v e r t h e m i n i m a l f i r e p r o t e c t i o n s e t s o b t a i n e d as b l o c k i n g s e t s o f t h e s - t n e t s .
It i s e a s i l y t r a n s -
l a t e d t o an a l g o r i t h m f o r m i n i m i z i n g o v e r t h e cobases o f any independence system, where t h e cobases a r e o b t a i n e d a l g e b r a i c a l l y as b l o c k i n g s e t s o f t h e c i r c u i t s .
3.
3.1.
A SOLUTION ALGORITHM
The B a s i c Prdcedure
The b a s i s f o r o u r a l g o r i t h m can be e x p l a i n e d by a b r i e f d e s c r i p t i o n o f a t h e o r e t i c a l a l g o r i t h m w h i c h o m i t s a l l t h e r e f i n e m e n t s needed f o r a p r a c t i c a l computation. The t h e o r e t i c a l a l g o r i t h m c o n s i s t s o f g e n e r a t i n g a n e s t e d sequence o f subc l u t t e r s Si
in
s, sl= s* '=... c s i ==... c-s,
(2)
Every b l o c k i n g s e t R e Ri i n t e r s e c t s e v e r y
and t h e i r c o r r e s p o n d i n g b l o c k e r s Ri.
From ( 2 ) i t f o l l o w s t h a t R a l s o i n t e r s e c t s e v e r y member o f
member o f Si.
T h i s means t h a t e v e r y b l o c k i n g s e t R E Ri i s a s u p e r s e t o f Si-,,Si-2,...,S,. some b l o c k i n g s e t f o r e v e r y p r e v i o u s Sj, 1 Q j Q i - 1 , so t h a t min c ( R ) 6 m i n c ( R ) 4 RER R6R
... .c min
R6Ri
c ( R ) .c m i n c ( R )
R e
I n each b l o c k e r Ri t h e l e a s t c o s t b l o c k i n g s e t s Ri,j c ( R ~ , ~=) m i n c ( R ) z Li R=Ri
.
(3)
and t h e i r c o s t ,
,
(4)
Thus, ( 3 ) shows t h a t Li i s a l o w e r bound on t h e minimum c o s t o f
are i d e n t i f i e d . f i r e protection.
Every l e a s t c o s t b l o c k i n g s e t Ri,j
E
Ri
i s t e s t e d t o see i f i t b l o c k s n o t o n l y
e v e r y s - t n e t i n Si b u t a l s o e v e r y s - t n e t i n S.
T h i s means a s k i n g i f M-R
i,j c o n t a i n s any s - t n e t s , a q u e s t i o n answerable by s t a n d a r d p a t h f i n d i n g t e c h n i q u e s . If
11-R.
1 ,j
c o n t a i n s s - t n e t s , some o f them s h o u l d be saved t o u n i t e w i t h Si a t t h e
n e x t stage. Moreover, Ri,j
If
11-R. . c o n t a i n s no s - t n e t s , t h e n Ri,j i s a b l o c k i n g s e t i n R. 1 ,J i s a minimum c o s t member o f R because o f ( 4 ) and ( 3 ) , and hence
R . . i s one o f t h e minimum c o s t f i r e p r o t e c t i o n s e t s which we seek. 1 ,J I f t h e r e a r e any o t h e r minimum c o s t members o f R , t h e y w i l l a l s o b e l o n g t o R i and be i d e n t i f i e d a l o n g w i t h Ri,j.
T h i s can be seen as f o l l o w s .
I f R i s another
member o f R h a v i n g t h e same minimum c o s t , c ( R ) = c ( R ~ , ~ ) ,t h e n R i n t e r s e c t s e v e r y
R i s minimal w i t h respect t o t h e p r o p e r t y o f i n t e r s e c t i n g e v e r y member o f Si. I f t h i s were n o t so, t h e n R would c o n t a i n an element i n a d d i t i o n t o a b l o c k i n g s e t o f Si. From t h e p o s i t i v i t y o f
member o f S and hence e v e r y member o f Si.
A boolean algebraic analysis of fire protection
219
t h e w e i g h t s o f t h e elements i n s - t n e t s , R would have a c o s t c(R) > c ( R ~ , ~ ) a, Being a minimal subset o f E which i n t e r s e c t s every member o f S .
contradiction.
R
E
Ri.
1’
T h e r e f o r e , whenever one optimum s o l u t i o n i n R i s found among t h e l e a s t a l l t h e o t h e r s w i l l be found t h e r e a l s o .
c o s t members o f Ri,
The t h e o r e t i c a l a l g o r i t h m must converge t o a complete s e t o f optimum s o l u t i o n s because S i s f i n i t e and hence t h e number o f i t e r a t i o n s i s f i n i t e . l a r g e t h i s process can r e q u i r e enormous amounts o f t i m e .
When S i s
B e f o r e d i s c u s s i n g ways
t o make t h i s a l g o r i t h m more p r a c t i c a l , we summarize i t s s t e p s as f o l l o w s . Theoretical Algorithm
8, Z
8.
1.
Set i = 0, So =
2.
I f n has s - t n e t s , then save some i n AS
=
0
e l s e stop, h a v i n g shown t h a t n i s
a1 ready p r o t e c t e d .
3.
Set i
4.
C o n s t r u c t t h e b l o c k e r Ri o f Si.
5.
O b t a i n a l l o f t h e l e a s t c o s t b l o c k i n g s e t s R i,j i n R 1. .
6.
F o r e v e r y Ri,j
=
i+l, Si = S i - l u
i f M-R.
1 ,j
ASi-,.
o b t a i n e d i n 5, do has no s - t nets, t h e n save R .
1
,j
i n Z e l s e save some s - t n e t s i n
ASi.
7.
3.2
If Z =
0,
t h e n go t o 3 e l s e stop, having a l l t h e optimum s o l u t i o n s i n Z .
Aspects o f a P r a c t i c a l A l g o r i t h m
The f i r s t p r a c t i c a l m a t t e r t o be c o n s i d e r e d i s t h e presence o f s i n g l e t o n s i n T. Any t a r g e t v e r t e x t which forms a s i n g l e t o n Ti = I t } i n T must be p a r t o f any f i r e p r o t e c t i o n set.
I n p a r t i c u l a r , a l l s i n g l e t o n t a r g e t v e r t i c e s must belong t o
a l l minimum c o s t f i r e p r o t e c t i o n s e t s .
I n o r d e r t o s i m p l i f y t h e computations we
assume t h a t a l l s i n g l e t o n s have been d e l e t e d from T , t h a t t h e c o r r e s p o n d i n g t a r g e t v e r t i c e s have been e i t h e r d e l e t e d f r o m n o r g i v e n z e r o weights, and t h a t these v e r t i c e s and t h e i r c o s t s w i l l be combined w i t h t h e a l g o r i t h m i c r e s u l t s t o f o r m complete s o l u t i o n s t o t h e f i r e p r o t e c t i o n problem. The procedure s t a t e d above i s r e a l l y n o t an a l g o r i t h m u n t i l we s t a t e s p e c i f i c a l l y (a)
how t o s e l e c t t h e new s - t n e t s &‘i-l
(b)
how t o c o n s t r u c t t h e b l o c k e r Ri o f Si.
i n f o r m i n g Si and
These i s s u e s a r e r e l a t e d by t h e f a c t t h a t t h e more s - t n e t s t h a t a r e produced a t On t h e
each stage, t h e l a r g e r i s I S i I and t h e more d i f f i c u l t i t i s t o o b t a i n R i .
B. L. Hulme et al.
220 o t h e r hand, i f ISi/
grows t o o s l o w l y t h e method w i l l be slow t o converge.
Our
approach t o i s s u e ( b ) d e t e r m i n e s o u r approach t o ( a ) , so we s h a l l d e a l f i r s t w i t h (b). T h e o r e t i c a l l y t h e b l o c k e r Ri o f a c l u t t e r Si can be c o n s t r u c t e d a l g e b r a i c a l l y by f o r m i n g t h e Boolean p r o d u c t o v e r t h e members o f Si o f t h e Boolean sum of t h e elements i n t h e members, expanding t h e p r o d u c t by t h e d i s t r i b u t i v e l a w ( a ( b = ab
v ac),
v
c)
and s i m p l i f y i n g t h e sum-of-products by idempotence (aa = a, a V a = a )
and a b s o r p t i o n ( a
v
ab = a ) .
The terms i n t h e r e s u l t i n g d i s j u n c t i v e normal f o r m
correspond t o t h e b l o c k i n g s e t s i n Ri. Both t h e t i m e and space r e q u i r e m e n t s o f t h i s a l g e b r a i c t e c h n i q u e can grow exponent i a l l y w i t h ISi o f Ri,
1.
We have chosen t o combat t h i s g r o w t h by n o t c o n s t r u c t i n g a l l
b u t i n s t e a d o n l y a s u b s e t R i G Ri of b l o c k i n g s e t s h a v i n g a c o s t no g r e a t -
e r t h a n a c u r r e n t upper bound, Ui,
a s s o c i a t e d w i t h some f e a s i b l e s o l u t i o n .
The Boolean a l g e b r a i c m a n i p u l a t i o n language SETS [8] has t h e c a p a b i l i t y n o t o n l y o f expanding and s i m p l i f y i n g Boolean f o r m u l a s b u t a l s o o f t r u n c a t i n g sums-ofp r o d u c t s on any g i v e n n u m e r i c a l t h r e s h o l d f o r t e r m v a l u e s , where t e r m v a l u e s can be computed as sums o f v a r i a b l e v a l u e s .
We a s s o c i a t e f i r e p r o t e c t i o n c o s t s w i t h
t h e Boolean v a r i a b l e s r e p r e s e n t i n g t h e v e r t i c e s and a r c s o f t h e s - t n e t s .
I n any
Boolean sum c o n t a i n i n g t h e v a r i a b l e I - J f o r a r c ( i , j ) we i n c l u d e 1 - 4 r e p r e s e n t i n g t h e j o i n t a r c s ( i , j ) and ( j , i ) , ensures t h a t , i f ( i , j ) and ( j , i )
provided t h a t b ( i , j )
c c(i,j)
c(j3i).
t
This
a r e a t t r a c t i v e choices f o r a b l o c k i n g set, then
t h e c o r r e s p o n d i n g Boolean t e r m w i l l c o n t a i n t h e v a r i a b l e 1 - 4 h a v i n g c o s t b ( i , j ) rather than the product ( I - J ) ( J - I )
having c o s t c ( i , j )
t
c(j,i).
Also, i t i s
e f f i c i e n t t o o m i t f r o m a Boolean sum t h e v a r i a b l e I - J f o r an a r c ( i , j ) , l e a v i n g 1 - 4 f o r t h e a r c p a i r , whenever c ( i , j )
= b(i,j).
a r c s can be p r o t e c t e d f o r t h e same c o s t as t h e s i n g l e a r c . c(j,i) c(j,i),
while
T h i s i s because b o t h
O f course, i f
= 0 so t h a t I - - J has been o m i t t e d f r o m t h e sum because b ( i , j )
= c(i,j)
t
t h e n I - J i s l e f t i n t h e sum.
Thus, we need an upper bound Ui on which t o t r u n c a t e terms so t h a t t h e r e s u l t i n g Boolean f o r m u l a F i , whose terms a r e t h e b l o c k i n g s e t s i n R t , m i g h t be more manageable i n s i z e .
N o t i c e t h a t t h e s e t o f s - t n e t s Si does n o t need t o be saved
f o r t h e n e x t stage.
I t i s enough t o save t h e t r u n c a t e d f o r m u l a Fi.
A t the
i + l - s t stage a Boolean p r o d u c t o v e r t h e new s - t n e t s ASi can be m u l t i p l i e d by Fi, expanded, t r u n c a t e d on Uitl
Q
Ui,
and s i m p l i f i e d t o f o r m Ryt1.
In t h i s way o n l y
b l o c k i n g s e t s w i t h c o s t s l o w enough t o p o s s i b l y produce minimum c o s t s o l u t i o n s a r e c o n s t r u c t e d and saved a t each stage. I t i s i m p o r t a n t i n p r a c t i c e t o n o t i c e t h a t t h e c o m p u t a t i o n o f RY+,
i s much f a s t e r
A boolean algebraic analysis of fire protection
i f , b e f o r e t h e m u l t i p l i c a t i o n by Fi, expanded, t r u n c a t e d on Uitl,
221
t h e Boolean product-of-sums c v e r aSi i s
and s i m p l i f i e d w h i l e repeated use i s made o f t h e
d i s t r i b u t i v e law ( a v b ) ( a V c ) = a V bc. T h i s r e p e a t e d l y saves p r o d u c i n g t h e c r o s s p r o d u c t s ac
v ab and t h e n
d e l e t i n g them
by e i t h e r t r u n c a t i o n o r a b s o r p t i o n . Our need d u r i n g t h e Boolean computation f o r an upper bound c o s t Ui a s s o c i a t e d w i t h a f e a s i b l e s o l u t i o n determines how we deal w i t h i t e m ( a ) .
Constructing s - t
n e t s s i m p l y by p a t h f i n d i n g i n M-R. . does n o t produce a f e a s i b l e s o l u t i o n . We use 1 ,J a s u b r o u t i n e FIRE [5] which computes b o t h f e a s i b l e s o l u t i o n s and s - t n e t s . By u s i n g D i n i c ' s a l g o r i t h m [3] t o f i n d a minimum c u t Q . . s e p a r a t i n g two b l o c k i n g 133
s e t s o f T i n a network r e l a t e d t o n-Ri,j
and weighted w i t h v e r t e x and j o i n t a r c
u
f o r (M-R. . , T ) . Hence, R . . 1 ,J 133 h a v i n g c o s t c(R. .) + c ( Q ~ , ~ ) A . lthough
weights, FIRE produces a f i r e p r o t e c t i o n s e t Qi,j
(M,J)
Qi,j
i s a f i r e protection set f o r
Qi,j
i s m i n i m a l , i t may n o t be a minimum c o s t supplement t o RiYj.
c(Ri , j )
+
1 ,J
However,
c(Qi , j ) i s an upper bound which may be used t o o b t a i n Ui+l.
See [6]
f o r t h e d e t a i l s o f t h e minimum c u t c a l c u l a t i o n s . FIRE produces t h e supplementary s - t n e t s
. 1 ,J
Q.
f o r minimality.
as an immediate byproduct o f t e s t i n g
I n t h e network G = M - R ~ , -~ Qi,j
-
{ z e r o weighted a r c s and
v e r t i c e s } , which c o n t a i n s no s - t nets, each element e e Q . . i s p u t back i n , one 1 ,J a t a time, and a p a t h f i n d e r t e s t s G + e f o r t h e e x i s t e n c e o f an s - t n e t . I f G + e has no s - t n e t , e i s n o t e s s e n t i a l t o Qi,j,
so e i s d e l e t e d f r o m QiYj.
Otherwise,
e i s essential t o Q
because t h e s - t n e t j u s t f o u n d c o n t a i n s e and w i l l be uni, j p r o t e c t e d i f e i s n o t p r o t e c t e d . Thus, FIRE produces o n l y one s - t n e t f o r each
a r c and v e r t e x i n Q. ., h e l p i n g t o keep I A S ~ fI r o m b e i n g t o o l a r g e , and each s - t 1 ,J n e t c o n t a i n s i t s corresponding a r c o r v e r t e x . Consequently, p r o t e c t i n g Q . . p r o 1,J
t e c t s a l l t h e new s - t n e t s , ASi,
and p r o t e c t i n g Ri,j
i n Si,
so t h a t Ri,jU Qi,j
cost.
The Boolean computation o f Ri+l
c o s t p r o t e c t i o n s e t s Ritl
p r o t e c t s Si+l
,j
= Si
u ASi,
protects a l l o f the s - t nets b u t perhaps n o t w i t h minimum
a t t h e n e x t stage w i l l produce minimum
f o r Sitl.
The f i n a l p r a c t i c a l m a t t e r concerns Step 6 o f t h e t h e o r e t i c a l a l g o r i t h m . sometimes i n e f f i c i e n t t o t e s t
fl o f
3.3
., p a r t i c u 7 1J I t i s always s u f f i c i e n t t o t e s t o n l y
t h e l e a s t c o s t blocking sets R.
l a r l y when t h e r e i s a l a r g e number o f them. t h e f i r s t one, Ri,,,
It i s
u n t i l t h e f i n a l s t a g e m, and o n l y t h e n t o t e s t a l l Rm,j.
THE ALGORITHM
With t h e above d e t a i l s i n mind we can now s t a t e a much more p r a c t i c a l a l g o r i t h m .
22 2
1. 2.
B. L. Hulme et al,
Set i = 0 , S 0 = 0, Z = 0, L0 = 0 , Fo = 1 (Boolean). Compute Qo = a minimal f i r e protection s e t f o r ( n , T ) , U1 = c ( Q 0 ) = c o s t of Qo, aS0 = an associated c o l l e c t i o n of s - t nets i n by
3.
If U, = 0 , then go t o 16.
4.
Set i = i + l .
5.
Form the Boolean products =
11
t h a t would be protected
Q,, one s-t net f o r each a r c a n d vertex in Qo.
A Ve SenSi-, eeS
,
Fi-1 A Pi-1
Fi
si
( i n e f f e c t constructing
=
si-l U AS^-^).
6.
Expand F i , truncating terms on t h e value U i , and simplify t o form R Y , the family of blocking s e t s R e R i having c(R) 6 U i . Call the truncated equation Fi.
7.
Find the l e a s t c o s t blocking s e t s R i , l , R i , 2 L i = c(R
8. 9.
i,1 ) and U i + l
=
,... , R i , k .
in Rf and set 1
Ui.
For j = 1 , 2 ,... , k i do compute Q i , j = a minimal f i r e protection set for (n-R.
.,T),
1 ,J
c ( Q . , ) = c o s t of Q i , j , 1
,J
nSi = an associated c o l l e c t i o n of s - t n e t s in M-R. t h a t would i, j be protected by Q . ., one s - t net f o r each a r c and vertex 1 ,J i n Q. : 1 ,J’
+
10.
set Ui+l
11.
if c ( Q ~ , =~ 0, ) t h e n set Z =
12.
i f Z # 0, then go t o 14;
13.
i f Ui+l
= rnin(Ui+l,Li
=
c(Q.1 , J. ) ) ;
Zu R.1 ,J.,
and go t o 14;
L i , then go t o 14, e l s e go t o 15.
14. Continue. 1 5 . I f Z = 0, then go t o 4.
16. Stop. Z i s the c o l l e c t i o n of minimum c o s t f i r e protection s e t s , and L i = U . 1+1 i s t h e minimum c o s t .
A boolean algebraic analysis of fire protection
223
4. AN EXAMPLE A f i r e spread network n i s shown i n F i g u r e 1 w i t h t h e j o i n t a r c w e i g h t s enclosed i n parentheses.
F i g u r e 1.
A F i r e Spread Network n
L e t t h e c o l l e c t i o n o f t a r g e t s e t s be
T = IT1,Tzl T1 = I2,31
T2 = {3,81.
There a r e many f i r e p r o t e c t i o n s e t s R f o r ( n , ~ ) ,t h e f o l l o w i n g ones b e i n g easy t o f i n d by i n s p e c t i o n
R
43
{2,3 ,83 {31
50.0
I ( 2 , 1 ) ;(3,4),(4,3); (5,6),(6,5) ;(7,8),(8,7)l { ( I 1 3 ) $4 );(3,4) ,(433);(3,5),(5,3)1
27.0
26.0
.
F i n d i n g an optimum s o l u t i o n by i n s p e c t i o n , however, i s d i f f i c u l t even f o r such a small problem as t h i s . Our Boolean a l g e b r a i c a l g o r i t h m f o u n d 11 s - t n e t s b e f o r e c o n v e r g i n g i n t h r e e i t e r a t i o n s t o t h e s e t o f two optimum s o l u t i o n s h a v i n g c o s t 2 2 . 0 .
We summarize t h e
i n t e r m e d i a t e and f i n a l r e s u l t s by d i s p l a y i n g t h e l e a s t c o s t b l o c k i n g s e t s Ri,j, t h e c u t s Qi,j,
t h e bounds Li and Ui,
p r o d u c t o v e r t h e new s - t n e t s ASi-l,
as w e l l as Boolean f o r m u l a s f o r Pi-l, and Fiy
the
t h e t r u n c a t e d d i s j u n c t i v e normal f o r m
Si which have c o s t s 6 Ui, 1 6 i 6 3. As Boolean v a r i a b l e s we use t h e v e r t e x numbers I , t h e i n d i v i d u a l a r c names I-J, and t h e j o i n t
whose terms a r e b l o c k i n g s e t s o f a r c names I--J,
1 6 I , J 6 8.
z = g Fo = 1
(Boolean)
Lo = 0.0
Qo
= I ( 2 , 1 ) ;( 2 A ),(4,2);(3,5),(5,3);(4,6),(6,4)1
U1 = c ( Q 0 ) = 23.0
224
B. L. Hulme et a[. 7 v 8 v 3 - 5 v 5-7 v 7-8)A v 3 v 2-1 v 1-3)A ( 5 v 6 v 4 v 3 v 7 V 8 v 5--6 v 6-4 v V 5-7 v 7-8) A (5v 3 v 7 v 8 v 5-3 v 5-7 v 7-8)A ( 2 v 4 v 3 v 2--4 v 3 - - 4 ) A (3 v 4 v 2 v 3--4 v 2--4)
Po = ( 3
v
5 (2v 1
F,
= Fo
A A
Po ( t r u n c a t e d on U1 )
3--4
term value
A 2--4v
13.0
5 - 7 A 2-1
A
2--4V
14.0
A
A
2--4v
14.0
7-8 A 2-1 A 2--4V
15.0
5-7 7-8 5-7
A
1-3 1-3 1-3
A A
3--4V
18.0
5 - 7 A 2-1 3--4V 7-8A 1 - 3 A 3--4V
19.0
7-8A 2-1 A 3 - - 4 v 3-5 A 1-3 A 5--6 A 5-3 A 2 - - 4 V 3-5 A 1-3 A 3--4 A 5-3 V
20.0
A 3-5 A 1-3 A 3-5 A 2-1 A 3-5 A 2-1 A
22.0
3 - 5 A 2-1
A 5-3 A 2--4V 6-4 A 5-3 A 2--4 v 3 - 4 A 5-.3V 6-4 A 5-3 A 2--4 5-6
19.0 21 .o
22.0 22.0
23.0 23.0
term value
18.0 19.0 19.0 20.0 21 .o
A boolean algebraic analysis of fire protection
225 22.0 22.0 22 .o
22.0 22.0
22.0 23.0 23.0 23.0 23.0 23.0 23.0 23.0 23.0
term value 22.0 22.0 23.0
23.0 23.0
B. L. Hulme et al.
226
Thus, Z i s a complete s e t o f optimum s o l u t i o n s having c o s t 22.0.
5.
CODE PERFORMANCE
A procedure f i l e , PROTECT, w r i t t e n i n t h e Cyber Control Language, implements o u r a l g o r i t h m on the CDC 7600 [ti]. Although FIRE f i n d s minimum c u t s i n polynomial time w i t h D i n i c ' s O(n4 ) algorithm, where n i s t h e number o f v e r t i c e s , t h e Boolean expansion o f products i n t o d i s j u n c t i v e normal form can r e q u i r e an amount o f time t h a t i s exponential i n t h e number o f s - t n e t s found, even w i t h t r u n c a t i o n . Table I shows t h a t r e a l i s t i c problems have been completely solved i n 60 t o 108 seconds.
We a l s o stopped t h e i t e r a t i o n a f t e r 890 seconds on a r e a l i s t i c problem
having v e r t e x weights which were l a r g e r e l a t i v e t o t h e a r c weights. we accepted t h e one f e a s i b l e s o l u t i o n , Ri ,1
U Qi,l,
having c o s t Ui+l
In t h i s case as t h e best
s o l u t i o n o b t a i n a b l e i n t h i s amount o f time.
No. No. oof f No. oof f No. No. oof f No. oof f Minimum Minimum CDC7600 CDC7600 No. No. No. oof f Target Target Target Target No. No. oof f Cost Run Time Time No. oof f No. Prob. No. ss- -t t nneet st s Cost Run Prob. Arcs VVeer tritci ceess Sets Sets I It teer raat ti oi onnss Found Found Solutions S o l u t i o n s (Seconds) (Seconds) No. VVertices e r t i c e s Arcs No.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 3 5 4 4 8 8 9 9 20 20 20 20 20 20 20 20 a5 85 85
2 6 12 10 10 20 20 32 32 62 62
62 60 60
60 60 60 512 512 512
2 3 2 4 3 3 4 5 5 5 4 4 4 6 4 5 8 15 1 5. 20
1 2 1 2 1
2 2 4 3 2 3 3 2 3 4 6 7 ia ia 22
2 2 2 3 4 3 4 5 3 9 3
6 5 5 3 3 3 3 6 16
2 6 7 5 4 11 9 10 6 46 12 22 5
24 12 24 12 12 22 20
1 13 10
13.5 13.5 14.8 17.4 20.7 21.2 23.1 35.6 24.9
85.6 1 1 1 1 2 56 15 4
20.9 43.1 25.0 38.0 19.9 24.1 21.8 75.5 60.1 107.6
227
A boolean algebraic analysis of'fire protection
6.
CONCLUSION
As i n d i c a t e d i n S e c t i o n 2, t h e f i r e a l g o r i t h m o f S e c t i o n 3.3 g e n e r a l i z e s t o an a l g o r i t h m f o r f i n d i n g a l l o f t h e minimum w e i g h t cobases o f any independence system whose c i r c u i t s a r e p o s i t i v e l y weighted.
I n a d d i t i o n t o t h e t r a n s l a t i o n o f s - t nets
t o c i r c u i t s and minimal f i r e p r o t e c t i o n s e t s t o cobases, one o t h e r correspondence i s needed.
We assume t h e e x i s t e n c e o f an independence a l g o r i t h m A which, g i v e n
any subset XE E , decides whether o r n o t X i s independent.
We f u r t h e r assume
t h a t , i f X i s dependent, t h e n A produces one c i r c u i t f r o m X.
The a l g o r i t h m A i s
needed i n t h e greedy a l g o r i t h m , which r e p l a c e s t h e FIRE s u b r o u t i n e a t steps 2 and 9.
u Qi,j
T h i s greedy a l g o r i t h m f i n d s n o t o n l y a s u p e r s e t Ri,j
a l s o new c i r c u i t s aSi a t each stage.
o f a cobase b u t
The p r o o f o f c o r r e c t n e s s f o r t h i s general
a l g o r i t h m i s s i m p l y a t r a n s l a t i o n o f t h e argument g i v e n i n S e c t i o n 3.1.
REFERENCES Barlow, R.E., F u s s e l l , J.B. and S i n g p u r w a l l a , N.D., e d i t o r s , R e l i a b i l i t y and F a u l t Tree A n a l y s i s , Proceedings o f t h e Conference on R e l i a b i l i t y and F a u l t Tree A n a l y s i s , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , September 3-7, 1974, SIAM, P h i l a d e l p h i a , 1975.
-
B e r r y , O.L., Nuclear power p l a n t f i r e p r o t e c t i o n p h i l o s o p h y and a n a l y s i s , SAND80-0334, Sandia N a t i o n a l L a b o r a t o r i e s , Albuquerque, NM, May 1980. O i n i c , E.A., A l g o r i t h m f o r s o l u t i o n o f a problem o f maximum f l o w i n a network w i t h power e s t i m a t i o n , S o v i e t Math. Dokl., 11 (1970), 1277-1280. Edmonds, J . , and Fulkerson, D.R., 8 (1970) 299-306.
B o t t l e n e c k extrema, J. C o m b i n a t o r i a l Theory,
Hulme, B.L., S h i v e r , A.W. and S l a t e r , P.J., FIRE: a s u b r o u t i n e f o r f i r e p r o t e c t i o n network a n a l y s i s , SAND81-1261, Sandia N a t i o n a l L a b o r a t o r i e s , Albuquerque, NM, December 1981. Hulme, B.L., Shiver, A.W. and S l a t e r , P.J., Computing minimum c o s t f i r e p r o t e c t i o n , SAND82-0809, Sandia N a t i o n a l Laboratoires, Albuquerque, NM, June 1 9 8 2 Welsh, O.J.A.,
M a t r o i d Theory (Academic Press, London, 1976).
W o r r e l l , R.B., S e t e q u a t i o n t r a n s f o r m a t i o n system (SETS), SLA-73-0028A, N a t i o n a l L a b o r a t o r i e s , Albuquerque, NM, January 1975.
Sandia
W o r r e l l , R.B. and Stack, D.W., A SETS u s e r ' s manual f o r t h e f a u l t t r e e analyst, SANO77-2051, Sandia N a t i o n a l L a b o r a t o r i e s , A1 buquerque, NM, November 1978. Zimmermann, U., L i n e a r and C o m b i n a t o r i a l O p t i m i z a t i o n i n Ordered A l g e b r a i c S t r u c t u r e s , v o l . 10, Annals of D i s c r e t e Mathematics (North-Holland, Amsterdam, 1981).
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Annals of Discrete Mathematics 19 (1984) 229-256 0 Elsevier Science Publishers B.V. (North-Holland)
229
ITERATION AND SUMMABILITY I N SEMIRINGS
B.
Mahr*
Fachbereich I n f o r m a t i k Technische U n i v e r s i t a t B e r l i n F r a n k l i n s t r . 28/29 1000 B e r l i n 10
An a l g e b r a i c t r e a t m e n t o f v a r i o u s q u e s t i o n s i n l i n e a r a l g e b r a , automata and f o r m a l language t h e o r y , combinat o r i a l o p t i m i z a t i o n , d a t a f l o w a n a l y s i s , graph t h e o r y , semantics and o t h e r s has e x h i b i t e d t h e importance o f t r a n s i t i v e c l o s u r e o v e r s e m i r i n g s and t h e broad a p p l i c a b i l i t y o f Gauss-Jordan e l i m i n a t i o n method. I n view o f these a p p l i c a t i o n s we s t u d y t h e s o l v a b i l i t y o f e q u a t i o n s o f t h e f o r m x=l+ax, c a l l e d i t e r a t i o n , and i n t r o d u c t a concept o f s u m m a b i l i t y i n p o s i t i v e s e m i r i n g s .
INTRODUCTION I t &as e a r l y d i s c o v e r e d i n /Ca 71/ t h a t t h e r e i s a c l o s e r e l a t i o n between methods
f o r s o l v i n g p a t h problems and methods f o r s o l v i n g systems of l i n e a r e q u a t i o n s o r i n v e r t i n g r e a l matrices.
Gauss-Jordan e l i m i n a t i o n which i s among t h o s e common
methods, was l a t e r seen / E i 74/, /BC 75/ t o be used b y Kleene f o r t r a n s f o r m i n g f i n i t e automata i n t o e q u i v a l e n t r e g u l a r expressions, see a l s o /Co 71/.
Since then
f u r t h e r areas o f a p p l i c a t i o n o f Gauss-Jordan e l i m i n a t i o n have been found, such as d a t a f l o w a n a l y s i s I T a 81a/,
graph t h e o r y /Mar 761 and semantics / E l 77/.
So i t
was n a t u r a l t o ask f o r general s p e c i f i c a t i o n and s o l u t i o n schemes a p p r o p r i a t e t o t r e a t u n i f o r m l y a l l t h e s e " s i m i l a r " problems.
Based on e a r l y a t t e m p t s t o genera+
i z e Boolean m a t r i x t h e o r y /Yo 61/ and t o d i s c u s s s p e c i f i c p a t h problems i n an a l g e b r a i c frame /Mo 61/, /CH 65/ and /HR 68/ f o r example, i t was i n /Ca 71/ where a f i r s t g e n e r a l a l g e b r a i c s e t t i n g o f p a t h problems was g i v e n . T h i s s e t t i n g i n v o l v e s i n f i n i t e sums and c e r t a i n s e m i r i n g s , and was l a t e r g e n e r a l i z e d i n / E r 741, /AHU 741, /BC 75/, /Le 77/ and /Wo 79/.
A u n i f i e d approach t o
p a t h problems which was based on Salomaa's a x i o m a t i z a t i o n o f t h e a l g e b r a o f regul a r expressions, and which d i f f e r s s l i g h t l y f r o m t h e o t h e r approaches i s g i v e n i n /Ta 81b/.
*
To m e n t i o n i s a l s o t h e e a r l y work of G i f f l e r .
T h i s work was p a r t l y supported by t h e MINERVA Foundation f o r German I s r a e l i collaboration.
230
B. Mahr
Path problems and t r a n s i t i v e closure are c l o s e l y r e l a t e d and u n i f o r m l y t r e a t e d as the problem o f f i n d i n g a s o l u t i o n t o the m a t r i x equation X=l+AX over a p p r o p r i a t e semirings /Le 77/.
I n /Ma 82/ these r e l a t i o n s a r e e x t e n s i v e l y studied, based on
the n o t i o n o f summability i n semirings.
D e t a i l e d d e s c r i p t i o n s o f path problems
and t r a n s i t i v e c l o s u r e over general semirings can a l s o be found i n the books
/CG 79/. /Ca 79/ and / Z i 81/.
The l a t t e r has a l s o t h e most complete b i b l i o g r a p h y
on t h i s s u b j e c t . This paper i s concerned w i t h the a b s t r a c t study o f s o l v a b i l i t y o f equations x=l+ax and existence o f i n f i n i t e sums
Z ai i n semirings. it1
Both questions are
c r u c i a l i n a general a l g e b r a i c s e t t i n g o f p a t h problems and t r a n s i t i v e c l o s u r e . S u m a b i l i t y i n such general s t r u c t u r e s as semirings and i n p u r e l y a l g e b r a i c terms
i s f o r the f i r s t time s t u d i e d i n /Ma 82/ and i n t h i s paper We s t a r t by reviewing t h e b a s i c n o t i o n s and p r o p e r t i e s o f semirings i n s e c t i o n 1 and b r i e f l y discuss a d d i t i o n a l axioms.
I n s e c t i o n 2 we i n v e s t i g a t e s o l v a b i l i t y
o f equations x=l+ax i n general semirings, and i n t h e semiring o f matrices over some semiring.
I n s e c t i o n 3 summability i n p o s i t i v e semirings i s i n t r o d u c e d and
studied, i n p a r t i c u l a r w i t h reference t o a p p l i c a t i o n s f o r t r a n s i t i v e c l o s u r e . Section 4 i s devoted t o t r a n s i t i v e closure, and s u f f i c i e n t c o n d i t i o n s f o r i t s existence. Most o f the m a t e r i a l presented i n t h i s paper can a l s o be found i n /Ma 82/, where p r o o f s a r e given f o r a l l those r e s u l t s which here are o n l y s t a t e d .
A fully
worked o u t p r e s e n t a t i o n o f t h e c o n t e n t o f t h i s paper would by f a r break the
l i m i t s o f space a v a i l a b l e . We have t h e r e f o r e decided t o omit p r o o f s i t they are n o t o f some i n t e r e s t i n t h e i r own r i g h t .
ACKNOW LEOGEMENT
I thank Clemens Lautemann, Daniel Lehmann, Michel Minoux, Michael Y o e l i and Uwe Zimmermann f o r h e l p f u l remarks and discussions.
Special thanks go t o Gunter Rote
f o r p o i n t i n g o u t t o me some i n c o n s i s t e n c i e s i n a previous v e r s i o n o f t h i s paper.
1.
SEMIRINGS:
B A S I C DEFINITIONS
I n t h i s s e c t i o n we g i v e t h e b a s i c n o t i o n o f a semiring and b r i e f l y discuss
a d d i t i o n a l axioms.
23 1
Iteration and summabilitv in semirings
1.1
DEFINITION
A s e m i r i n g S=(S,+,.,O,l)
i s a s e t S, c a l l e d domain, t o g e t h e r w i t h two, n o t
n e c e s s a r i l y d i f f e r e n t , s e l e c t e d elements 0 and 1, and two b i n a r y o p e r a t i o n s
.
t
and
such t h a t t h e f o l l o w i n g axioms h o l d :
Al)
a+(b+c)=(a+b)+c
A2)
a+b = b+a
A3)
a+O = a
A4)
a. ( b . c ) = ( a . b ) . c
A5)
a.1 = 1.a = a
A6)
a.(b+c)=a.b+a.c
A7)
(a+b)-c=a.c+b.c
A8)
a - 0 = 0.a = 0
We c a l l
+
addition,
.
multiplication, 0
a and 1 unit.
Axiom ( 8 ) i s c a l l e d t h e
zero r u l e . Homomorphisms between s e m i r i n g s a r e n a t u r a l l y d e f i n e d .
We c a l l i n j e c t i v e homo-
morphisms embeddings and b i j e c t i v e homomorphisms isomorphisms To improve r e a d a b i l i t y we o f t e n o m i t t h e s i g n
.
f o r m u l t i p l i c a t i o n and make no
f o r m a l d i s t i n c t i o n between o p e r a t i o n s i n d i f f e r e n t s e m i r i n g s as l o n g as t h e r e i s n no c o n f u s i o n t o expect. We a l s o use c ai f o r al+ ...+an. i=1 Our n o t i o n o f s e m i r i n g c o i n c i d e s w i t h t h e one g i v e n i n / E i 74/ and seems t o be w i d e l y accepted now.
One o f t h e c h a r a c t e r i s t i c s o f s e m i r i n g s i s , t h a t t h e y cover
so d i f f e r e n t s t r u c t u r e s l i k e u n i t a r y r i n g s , f i e l d s , d i s t r i b u t i v e l a t t i c e s w i t h We l i s t a number o f examples
g r e a t e s t and s m a l l e s t element o r Boolean a l g e b r a s . t o i l l u s t r a t e t h e wide range o f a p p l i c a t i o n s .
1.2
EXAMPLES
.
i s t h e so c a l l e d t r i v i a l w i t h t r i v i a l o p e r a t i o n s + and (1) l = ( { l } , t , . , l , l ) 1 i s t h e o n l y s e m i r i n g where z e r o and u n i t a r e equal. semiring. (2)
lB=({O,l},v,A,O,l)
Boolean semi r i ng
.
t h e two element Boolean a l g e b r a i s a s e m i r i n g and c a l l e d
232 (3)
B. Malir
GF(2) = ({O,l},+,.,O,l)
the f i e l d o f two elements i s a semiring and beside
IB
t h e o n l y o t h e r semiring on a two element s e t . the n a t u r a l numbers w i t h usual a d d i t i o n and m u l t i p l i c a t i o n ( 4 ) No = (NO,+,.,O,l) form a semiring. No i s the i n i t i a l semiring i n the category o f semirings, i . e . f o r each semiring S t h e r e i s e x a c t l y one homomorphism h: No
-f
S.
(5)
Q+ and W+, t h e p o s i t i v e r a t i o n a l and r e a l numbers form a semiring.
(6)
Min = (Ru{m},min,+,m,O)
the r e a l numbers w i t h g r e a t e s t element
adjoint,
minimum-operation as a d d i t i o n and r e a l a d d i t i o n as m u l t i p l i c a t i o n form a semiring. This semiring u n d e r l i e s the well-known s h o r t e s t p a t h problem.
I t remains a semi-
r i n g i f i t s domain i s r e s t r i c t e d t o t h e p o s i t i v e r e a l numbers P+. ( 7 ) Max = (Ru{-ml,max,+,-m,O)
d e f i n e d i n analogy t o M i n i s a semiring.
t h e p o s i t i v e r e a l s w i t h maximum o p e r a t i o n and r e a l m u l t i ( 8 ) S = (R+, max,.,0,1) p l i c a t i o n form a semiring /Mo 61/.
( 9 ) b(E) = (2E*,u,o,$,IA))
the powerset o f t h e f r e e monoid generated by E w i t h
union as a d d i t i o n and complex product as m u l t i p l i c a t i o n , I$ as zero and u n i t where X denotes the empty word i n E*,
{A1 as
forms a semiring, c a l l e d t h e language
semiring over E. (10)
A P ( A ) = ( 2 ,u,n,@;A)
t h e powerset o f a s e t A w i t h union and i n t e r s e c t i o n
from a semiring. More examples can be found i n /Br 74/, /Ca 79/ o r
/GM 82/, where a l s o correspond-
i n g a p p l i c a t i o n s a r e discussed.
A s p e c i a l c l a s s o f semirings i s i n t r o d u c e d i n / E i 74/ and c a l l e d p o s i t i v e semirings.
These semirings s a t i s f y the f o l l o w i n g axioms
A9)
a+b=O=> a=O and b=O
A10)
a.b=O*a=O
or
b=O
We w i l l c a l l a semiring p o s i t i v e , i f i t s a t i s f i e s axiom A9.
W e c a l l a semiring 5 ordered, i f t h e r e i s a p a r t i a l o r d e r 6 d e f i n e d on S such t h a t t h e f o l l o w i n g axiom i s s a t i s f i e d : All)
If a 4 b and c 6 d i n S, then a + c 6 b + d.
S i s t o t a l l y ordered, i f 6 i s t o t a l o r d e r on S .
Iteration and summabilit>3in semirings
233
Often a n a t u r a l o r d e r i n g on a s e m i r i n g S i s g i v e n by t h e f o l l o w i n g d i f f e r e n c e r e l a t i on: A12)
a 6 B <=> t h e r e i s x e S w i t h a t x = b
We c a l l S o r d e r e d b y t h e d i f f e r e n c e r e l a t i o n i f t h e o r d e r i n g on S s a t i s f i e s axiom A12).
Then S i s a l r e a d y p o s i t i v e .
The d i f f e r e n c e r e l a t i o n i s a r e f l e x i v e and t r a n s i t i v e r e l a t i o n on any s e m i r i n g , and s a t i s f i e s axiom A l l ) . An i m p o r t a n t c l a s s o f s e m i r i n g s which can be o r d e r e d b y t h e d i f f e r e n c e r e l a t i o n i s t h e c l a s s o f idempotent s e m i r i n g s .
We c a l l a s e m i r i n g S idempotent, i f i t s a t i s -
f i e s t h e f o l l o w i n g axiom A13)
ata=a
Idempotent s e m i r i n g s b e a r w i t h r e s p e c t t o a d d i t i o n a s e m i - l a t t i c e s t r u c t u r e and have a+b as s m a l l e s t upper bound o f a and b . Idempotent semi r i n g s a r e f r e q u e n t l y used i n t h e 1it e r a t u r e and c a l l e d " p a t h a1 geI,
b r a i n /Ca 79/.
An i m p o r t a n t s u b c l a s s o f idempotent s e m i r i n g s i s t h e c l a s s o f
s i m p l e s e m i r i n g s , where a s e m i r i n g S i s c a l l e d s i m p l e i f i t s a t i s f i e s t h e f o l l o w i n g axiom A14)
lta=l
Simple s e m i r i n g s were f i r s t s t u d i e d i n /Yo 61/ under t h e name Q - s e m i r i n g s .
A f u r t h e r n a t u r a l subclass o f idempotent s e m i r i n g s i s formed by t h e e x t r e m a l semir i n g s , where a s e m i r i n g S i s extremal, i f i t s a t i s f i e s t h e axiom A15)
atb
E
{a,bl
A s e m i r i n g S i s extremal i f and o n l y if S i s idempotent and t o t a l l y o r d e r e d ( b y the difference relation). Many examples o f extremal semirings, l i k e t h e s e m i r i n g s [Min and Max a r e d e r i v e d from t o t a l l y o r d e r e d monoids M = (M,t,O,c). = (Mu{z), 6
min,t,z,O)
Converting
o r Max(M) = (Mu{z),max,+,z,O)
M i n t o a s t r u c t u r e Min(M)
by d e f i n i n g min and max t h r o u g h
on M, and a d j o i n i n g z, one o b t a i n s e x t r e m a l s e m i r i n g s which, i n a d d i t i o n , a r e
positive. I f i t i s f u r t h e r assumed t h a t 0 i s s m a l l e s t ( o r l a r g e s t ) element i n M, then Min(M),
and Max(M) r e s p e c t i v e l y , a r e simple. c a l l e d Oijkstra-semirings
i n /Le 77/.
Semirings, which a r e e x t r e m a l and s i m p l e a r e They have been g i v e n t h i s name, because
B. Mahr
234
they a r e r i c h enough t o y i e l d correctness o f D i j k s t r a ' s a l g o r i t h m f o r p a t h problems over such semirings. F i n a l l y i n t h i s s e c t i o n we d e f i n e t h e semiring o f m a t r i c e s o v e r some semiring. An nxn-matrix over a semiring S i s a mapping M: [n]
with [n]:={l
,...,n l .
can d e f i n e on M(n,S) L e t f o r %M(n,S) i , j - e n t r y o f A.
x
[n]
-+
S
L e t M(n,S) denote t h e s e t o f nxn-matrices over S, then we a semiring s t r u c t u r e i n the well-known way:
and i , j 6 n t h e element A ( i , j )
be denoted by A . 1j
We then d e f i n e f o r nxn m a t r i c e s A and B
and c a l l e d t h e
a d d i t i o n C=A+B by C..:=A..+B. IJ
multiplication
C=A.B
1~ I j by C..:=Ail.B 1.J
zero unit -
.+...+ Ain-Bnj
13
0 by O..:=O 1J
1 by 1..:=1
i f i = jand lij:=O
otherwise.
1J
It i s easily verified that
(M(n,S),+,.,O,l)
i s a semiring.
I n /Le 77/ a semi-
r i n g i s defined l i k e i n d e f i n i t i o n 1.1, except t h a t t h e zero r u l e A8 i s n o t assSince t h e zero r u l e i s needed t o show t h a t I i s u n i t i n
umed t o be s a t i s f i e d .
the semiring M ( n , S ) , we have i n c l u d e d t h i s axiom i n o u r d e f i n i t i o n i n c o n t r a s t t o t h e a x i o m a t i z a t i o n i n /Le 77/. F o r n 3 2, M(n,S)
can never s a t i s f y A10.
F o r n = 1 we have M(n,S) isomorphic t o S .
Obviously, ifS i s idempotent, then so i s M(n,S). It also i s true, t h a t i f
i s Pl(n,S).
S is ordered b y t h e d i f f e r e n c e r e l a t i o n (axiom A I Z ) , so
To show t h i s , one d e f i n e s an o r d e r i n g on M(n,S), which i n a n a t u r a l
way i s induced from S: A g B <=> A.. 6 Bij 1J
f o r i,j 6 n
we c a l l t h i s o r d e r i n g the p o i n t w i s e o r d e r on matrices. o r d e r on matrices can never b F t o t a l be simple i f
n B 2.
on M(n,S).
For n
>2
the p o i n t w i s e
And a l s o no semiring M(n,S)
can
Consequently m a t r i c e s over a semiring cannot form an extremal
o r D i j ks tra-semi r i n g .
2.
ITERATION IN SEMIRINGS
I n t h i s s e c t i o n we study s o l v a b i l i t y o f f i x e d p o i n t equations o f t h e form x=b+ax, We w i l l discuss i t e r a t i o n i n t h e semiring o f m a t r i c e s over some semiring. We c a l l a
c a l l e d i t e r a t i o n , f o r a r b i t r a r y elements a and b o f a g i v e n semiring.
23 5
Iteration and summabi1it.v irr semirings
s e m i r i n g 5 c l o s e d , i f f o r a l l a E S t h e r e i s a s o l u t i o n o f x = l t a x i n S. phisms p r e s e r v e s o l v a b i l i t y o f i t e r a t i o n , i . e .
h:S
+
Homomor-
i f z s o l v e s x=b+ax i n S, and
S ' i s a homomorphism, then h ( z ) s o l v e s x=h(b)+h(a)x i n S ' .
2.1 P r o p o s i t i o n L e t S be a s e m i r i n g .
The f o l l o w i n g statements a r e e q u i v a l e n t
(1)
S i s closed, i . e .
(2)
x=a+ax i s s o l v a b l e f o r a l l a c S x=b+ax i s s o l v a b l e f o r a l l a,b E S.
(3)
x=l+ax i s s o l v a b l e f o r a l l a E S
The s i m p l e p r o o f o f t h i s propos t i o n shows more: t h e r e a r e c e r t a i n r e l a t i o n s between s o l u t i o n s . 2.2 P r o p o s i t i o n L e t S be a s e m i r i n g , then (1)
i f z s o l v e s x=l+ax, t h e n z.b s o l v e s x=btax
(2)
i f z s o l v e s x=a+ax, t h e n l + z s o l v e s x = l t a x
(3)
i f z s o l v e s x=b+ax, and t s o l v e s x=O+ax, t h e n z+t.c s o l v e s x=b+ax f o r a l l
C E
s.
I t i s n o t t h e case t h a t a l l s o l u t i o n s o f x=b+ax a r e o f t h e f o r m z.b f o r some s o l u t i o n z o f x=l+ax. T h i s i s seen i n an example t o 2.6. One cannot e x p e c t much t o know about s o l v a b i l i t y o f i t e r a t i o n i n g e n e r a l s e m i r i n g s More i n t e r e s t i n g r e s u l t s a r e o b t a i n e d i n p a r t i c u l a r s e m i r i n g s o r i n s e m i r i n g s which s a t i s f y a d d i t i o n a l axioms.
2.3 Examples the f o l l o w i n g i s true: ( 1 ) I n t h e semiringINo = (lNo,+,.,O,l) x=b+ax i s s o l v a b l e i n No i f and o n l y i f b = 0. I n t h i s case 0 i s t h e o n l y solution. the following i s true: ( 2 ) I n t h e s e m i r i n g IR = (lR,+,.,O,l) x=b+ax i s s o l v a b l e i n IR f o r a l l b and a l l a f 1. I n t h i s case
i s the only
solution. T h i s i s a l s o t r u e i n any f i e l d .
I f a s e m i r i n g i s ordered, we may ask f o r a minimum s o l u t i o n o f i t e r a t i o n . However, i t seems t o be d i f f i c u l t t o answer t h i s q u e s t i o n i n any g e n e r a l i t y (see
236
8. Mahr
a l s o the l a s t paragraph i n t h i s s e c t i o n ) . Notation: L e t S be a semiring and a and
an:=an-’.a
E
S , then
for n > 0
(1)
aO:=l
(2)
acn>:=ao+. . .an f o r n >, 0.
Using t h i s n o t a t i o n we can s t a t e
2.4 P r o p o s i t i o n Let S be a semiring which i s ordered by t h e d i f f e r e n c e r e l a t i o n , then
(1)
i f z solves x=b+ax, then f o r a l l n >, 0.a
n
.b -s z and a<”>.b c: z
( 2 ) i f a<”>.b solves x=b+ax f o r some n t 0, then a
f o r a l l r >/ n,
Idempotent semi r i n g s a r e ordered by the d i f f e r e n c e r e l a t i o n , which then takes the form a 6 b <=> a+b=b.
I n t h i s case we have
2.5 P r o p o s i t i o n L e t S be an idempotent semiring, then
+ z2
(1)
i f z1 and z2 s o l v e x=b+ax, then so does z1
(2)
i f f o r some n
(3)
a 4 1 i f and o n l y i f 1 i s minimal s o l u t i o n o f x=l+ax.
>
0 an=an+’,
then a
A simple semiring i s idempotent and has 1 as l a r g e s t element.
I n t h i s case we
have
2.6 P r o p o s i t i o n L e t S be a simple semiring, then (1)
S i s closed
(2) (3)
b i s the s m a l l e s t s o l u t i o n o f x=b+ax 1 i s the unique s o l u t i o n o f x=l+ax.
I n a simple semiring b i s n o t n e c e s s a r i l y the o n l y s o l u t i o n o f x=b+ax.
This i s
seen from t h e f o l l o w i n g simple semiring: L e t S = ( [O,l],max,min,O,l) i f f 0.5 6 x <,
then x solves the i t e r a t i o n w i t h a = 0.7 and b = 0.5
0.7.
Extremal semirings form another subclass o f idempotent semirings.
They a r e
Iteration and surnrnabifity in semirings
237
e x a c t l y t h e t o t a l l y o r d e r e d idempotent semi r i n g s .
2.7 P r o p o s i t i o n L e t S be an e x t r e m a l s e m i r i n g , S # 1, and 0 t h e o n l y s o l u t i o n o f x=O+ax, t h e n : x=l+ax i s s o l v a b l e i f and o n l y i f a 6 1, and i n t h i s case 1 i s t h e unique solution. The assumption t h a t 0 i s t h e o n l y s o l u t i o n o f x=O+ax i s s a t i s f i e d i n any s e m i r i n g w i t h m u l t i p l i c a t i v e c a n c e l l a t i o n r u l e (a.b=a.c *b=c).
2.8 Examples (1)
I n t h e s e m i r i n g N i n = ( Ru{ml,min,+,m,O)
for a z 1
x=min(O,a+x)
the following i s true:
i s s o l v a b l e i f and o n l y i f aaRO
which i s an immediate consequence o f 2.7 and t h e above remark ( n o t e t h e o r d e r i n N i n i s t h e r e v e r s e d orded o f
R).
I n t h e s e m i r i g n lL(E)=(ZE*,u,o,@,CX1)t h e f o l l o w i n g i s t r u e : m i and A x={X}uAox i s s o l v a b l e f o r any ALE*, i s minimal s o l u t i o n .
(2)
ivo
T h i s s o l u t i o n i n v o l v e s i n f i n i t e union, which w i l l be s t u d i e d as i n f i n i t e “sum“ i n the n e x t section. Any p o s i t i v e s e m i r i n g S w h i c h i s n o n t r i v i a l can be embedded i n t o a c l o s e d semir i n g , s i m p l y b y a d j o i n i n g a new element:
2.9 P r o p o s i t i o n be a p o s i t i v e s e m i r i n g , S # 1 and z $ S, t h e n
L e t S = (S,+,.,O,l) clos(S)=(Su{z},+,.,O,l)
d e f i n e d b:’ a+b
i f a,bcS
a+b= otherwise 1i.b
i f a,baS i f a=z and b t O
a. b=
i f b=z and a t 0 i f a=O o r b=O
i s a c l o s e d s e m i r i n g , c a l l e d c l o s u r e o f S, and t h e r e i s an embedding o f S i n t o clos(S). The assumption t h a t S i s p o s i t i v e , i s needed i n t h e v a l i d i t a t i o n o f t h e two d i s p t r i b u t i v e laws.
Any i t e r a t i o n x=b+ax i n c l o s ( S ) i s s o l v a b l e by t h e new element z.
R. Mahr
238
I n /Le 77/ a s i m i l a r c l o s u r e operator i s used t o c l o s e t h e s e m i r i n g R=(R,+,.,O,l) Since (p i s n o t p o s i t i v e , c l o s ( R ) i s n o t d e f i n e d i n our frame-
o f r e a l numbers. work.
I n /Le 77/ t h e zero r u l e f o r semirings i s n o t assumed.
Since h i s d e f i n i -
t i o n a.z=z.a=z f o r a l l a e S i s c o n s i s t e n t w i t h axioms A1 t o A 7 , b u t n o t w i t h the zero r u l e A8, h i s c l o s u r e o f the r e a l numbers i s c o r r e c t . Closed semirings a r e o f t e n s t u d i e d i n the l i t e r a t u r e as semirings w i t h an addit i o n a l unary o p e r a t i o n A16)
*
d e f i n e d such t h a t f o r a e S the axiom
a*=l+a.a*
i s satisfied.
We c a l l such semirings S = ( S , t , .
d i s t i n g u i s h them from closed semirings.
,*,0,1)
i t e r a t i v e semirings t o
Obviously any closed s e m i r i n g can be
expanded t o an i t e r a t i v e semiring by an a p p r o p r i a t e choice f u n c t i o n * : S which f o r any a
E
-f
S,
S chooses an element a* E. S such t h a t A16 i s s a t i s f i e d .
D i f f e r e n t axioms on the o p e r a t i o n
*
are discussed by many authors, see e.g.
/Co 71/, /Ca 79/, / T a 81/ and / Z i 81/.
I n the c o n t e x t o f t h i s paper i t e r a t i v e
semirings p l a y a minor r o l e . We now want t o study i t e r a t i o n i n the semiring o f m a t r i c e s over a semiring.
An
a l g e b r a i c form o f Gauss-Jordan e l i m i n a t i o n i s g i v e n i n t h e f o l l o w i n g r e c u r s i v e d e f i n i t i o n o f a m a t r i x A*. 2.10 D e f i n i t i o n L e t S=(S,t,.,*,O,l) be an i t e r a t i v e semiring. are defined f o r k = 0,
..., n
A!?) 1J
=
Then from a m a t r i x A matrices A ( k )
by
A.. 1J
The m a t r i x A*:=l+A(n) i s c a l l e d t h e Kleene-matrix o f A over S . 2.11 P r o p o s i t i o n L e t S be a semiring and n a r b i t r a r y , then: S i s closed ifand o n l y i f M(n,S) i s closed.
Proof:
According t o the above remark we can change S i n t o an i t e r a t i v e semiring.
We f i r s t show t h a t A("),
d e f i n e d i n 2.10,
solves X=A+AX.
Kleene-matrix o f A, i s then shown t o solve X = l + A X . A(')=A+A.A(")
h o l d s t r u e i f and o n l y if f o r a l l i , j 6 n
By 2.2 (2) A*,
the
Iteration and surnrnability in sernirings
239
To d e r i v e ( l ) , we show by i n d u c t i o n on k f o r a l l i , j 6 n k A(k) A!k) = A . . + c A. 1J 1J r = l ir rj
.
I f k=o,then A!?) 1J
= Aij
0 + c Air r =1
. A‘o), rJ
proving (2).
u s i n q a*=l+aa* i n t h e r i g h t b r a c k e t s , we o b t a i n
u s i n q t h e i n d u c t i o n h y p o t h e s i s , we o b t a i n
Thus ( 2 ) h o l d s t r u e f o r a l l k=O,
...,n,
which shows A(”) =A+A.A(”).
F o r t h e converse d i r e c t i o n c o n s i d e r t h e m a t r i x AEM(n,S) (a
A.. = 1J f o r a r o i t r a r y a&. we have
Since M(n,S)
d e f i n e d by
i f i=j=1
{ LO
otherwise
i s closed, t h e r e i s a s o l u t i o n
o f X=l+AX and
B. Mahr
240
Since A l k = 0 f o r k = l ,
..., n,
i t follows t h a t A
11
= l+a.ill.
T h u s x=l+ax i s
solvable in S. Proposition 2.11 i s a s l i g h t improvement of the main r e s u l t in /Le 77/, where a*=l+aa*=l+a*a i s assumed i n S. As i n our proof, the zero r u l e i n S i s not used i n /Le 77/. B u t our proof i s much simpler.
In general, i t e r a t i o n in M(n,S) has not n e c e s s a r i l y only one s o l u t i o n , even in those cases, where x = l t a x i s uniquely solvable in S.
3 SUMMABILITY IN POSITIVE SEMIRINGS In t h i s section we introduce the notion of s u n a b i l i t y in semirings. This allows t o t a l k about existence o f i n f i n i t e sums which i n many a p p l i c a t i o n s i s very much t o be desired.
S u m a b i l i t y i n semirings i s based on t h e notion of complete semirings as given i n /Ei 74/ and /AHU 74/, and covers convergence i n R+ a s well as suprema of countable s e t s i n d i s t r i b u t i v e l a t t i c e s with l a r g e s t and smallest
element. Throughout t h i s and t h e next s e c t i o n we assume t h a t semirings a r e positive. 3 . 1 Definition A p a r t i a l complete semiring S = ( S , e , . l ) i s a domain S together with one s e l e c t e d
element 1 , c a l l e d unit, a binary operation., c a l l e d m u l t i p l i c a t i o n , and f o r each countable set I a p a r t i a l l y defined operation 1
1:s I
-->s
c a l l e d generalized a d d i t i o n , such t h a t the followinq axioms a r e s a t i s f i e d
D1)
1 i s t o t a l l y defined f o r a l l f i n i t e I and I=@ I
D2) :a u = a1. :iEJ i \
D3)
L ai i s defined by
CL
i f f for all k S
ie1 7 ( b a i ) i s defined by ba
ie I
and
24 1
Iteration and summability in semirings C
(sib)
i s d e f i n e d by ab
icI 04)
Given
{ai
C ai
I
i a I 1 and decomposition I =
i s d e f i n e d by
CI I jeJ j’
i f f for a l l j e J
a
then ai
C
is
i€1
icI
j
d e f i n e d by, say, aj
,
and
i s d e f i n e d by a
C a jeJ
D5 )
a. (b. c ) = (a. b ) c
D6)
a.l=l-a=a
I We c a l l an element 66s an I - i n d e x e d f a m i l y and denote i t by
6={ailicI}.
I
is
c a l l e d an i n d e x s e t .
A p a r t i a l complete s e m i r i n q i s c a l l e d complete, i f f o r a l l
countable index sets
I the generalized a d d i t i o n
i s a t o t a l l y defined
operation.
A p a r t i a l complete s e m i r i n g i s an a l g e b r a w i t h p a r t i a l l y defined i n f i n i t a r y o p e r a t i o n s f o r each i n d e x s e t I. While t h i s l o o k s l i k e a monstrous o b j e c t , i t i s v e r y c o m f o r t a b l e t o work w i t h .
E s s e n t i a l l y , t h e r e i s o n l y one i n f i n i t a r y p a r -
t i a l l y defined operation:
L e t k:J
+
I be b i j e c t i v e , t h e n f o r a l l indexed f a m i l i e s { a i l i zai= id
E
I]
z a jrJ k(j)’
Thus g e n e r a l i z e d a d d i t i o n i s i n v a r i a n t under renaming and rearrangement o f i n d e x s e t s , which i s a consequence o f axioms D2 and D4.
F u r t h e r i d e n t i t i e s can b e
o b t a i n e d which a r e f r e q u e n t l y needed:
3.3
Fact
I n a p a r t i a l complete semi-ring
(1) (2) (3)
( C ai) i aI
.
( 1 b.) jeJ
=
C .a..bj ( i ,j )~IXJ
’
iE1
jcJ ”
z ( c a..) j e J ioI
c
( , c aij)
=
ieI
J ~ J
c
(
c a. . )
=
the following i d e n t i t i e s are true
’’
c
aij
( i ,j k I x J
and one s i d e o f these i d e n t i t i e s i s d e f i n e d whenever t h e o t h e r s i d e i s d e f i n e d .
B. Mahr
242
We say t h a t a p a r t i a l complete semiring S=(S,c,.,l) p a r t i a l ordering
Q
i s ordered, i f t h e r e i s a
d e f i n e d on S such t h a t the f o l l o w i n g axiom i s s a t i s f i e d :
07) f o r a l l indexed f a m i l i e s { a i l i c I l and t b i l i c I l for a l l i t 1
ai-< bi
=)
C a . c c bi irI
provided
C ai
and
'
C bi
ipI
a r e defined.
icI
ipI
And we c a l l a p a r t i a l complete semiring idempotent, i f f o r a l l acS the sum i s d e f i n e d and equals a.
3.4
C a
i cI
Proposition
I f S i s an idempotent p a r t i a l complete semiring, then S i s ordered by
and t h e f o l l o w i n g i s t r u e
(1)
L a . i s supremum of I a i l i c I i
iLI
c
(2)
1
a
i
i f i t i s defined
i s smallest s o l u t i o n o f x=l+ax i f i t i s d e f i n e d . n
i e No
Obviously, t h e r e i s a c l o s e r e l a t i o n between semirings and p a r t i a l complete semir i n g s which we w i l l use subsequently t o d e f i n e summability i n semirings. L e t S=(S,c,.,l)
be a p a r t i a l complete semiring, then the semiring r e s t r i c t i o n o f
- i s defined t o be t h e unique s e m i r i n g S=(S,+,.O,l) 0:= al+a2:=.
with
c ai ia$
c 1E{l ,Z}
ai
Note t h a t the semiring r e s t r i c t i o n o f a p a r t i a l complete s e m i r i n g i s p o s i t i v e
[GR83].
We a l s o c a l l
determined by S.
5
a p a r t i a l completion of S , which o f course i s n o t uniquely
However, t h e r e i s a unique'lninimal" p a r t i a l completion o f a
p o s i t i v e semiring S which i s obtained by d e f i n i n g f o r each f i n i t e index s e t I the sum .C ai as i t e r a t e d b i n a r y a d d i t i o n f o r an a r b i t r a r y f i x e d o r d e r i n g o f I , and 161 l e a v i n g C ai f o r i n f i n i t e I undefined. We c a l l t h i s p a r t i a l completion o f S the id g e n e r a l i z a t i o n o f S.
243
Iteration and summability in semirings
3.5
Definition
Given a s e m i r i n g S=(S,+,-,O,l) and an indexed f a m i l y 6 ={ai l i e 1 1 i n S.
We c a l l 6
C ai e x i s t s i n S, i f t h e r e i s a p a r t i a l i eI i s defined.
a d d i t i v e and e q u i v a l e n t l y say t h a t c o m p l e t i o n o f S, i n which
C ai
icI Given a s e t F= {6.ljaJ} o f indexed f a m i l i e s 6 J j' We say F i s a d d i t i v e and e q u i v a e x i s t s f o r a l l & = { a . m . ) , i f t h e r e i s a p a r t i a l complel e n t l y say t h a t ai J-1 Jie1 j t i o n o f S i n which a l l 6 . a r e a d d i t i v e . J Semirinqs may have d i f f e r e n t p a r t i a l completions, so t h a t t h e mere e x i s t e n c e o f C ai irI
does n o t guarantee t h a t a unique s e m i r i n g element i s t h e r e b y denoted.
We
w i l l t h e r e f o r e i n a d d i t i o n t o e x i s t e n c e say how C ai can be d e f i n e d , and c a l l ieI
2 ai=b c o n s i s t e n t i n S, i f t h e r e i s a p a r t i a l c o m p l e t i o n o f S i n which C ai i s it1 icI d e f i n e d by b.
3.6
Examples
(1) L e t INo be t h e s e m i r i n g o f n a t u r a l numbers, t h e n t h e f o l l o w i n g i s t r u e : c ai=b i s c o n s i s t e n t i n INo i f and o n l y i f t h e r e i s a f i n i t e s e t F d such t h a t id ai=O f o r i r I \ F and b= C ai iEF
( 2 ) L e t d b e t h e s e m i r i n g o f p o s i t i v e r e a l numbers, then t h e f o l l o w i n g i s t r u e : A p a r t i a l completion o f p i s o b t a i n e d by d e f i n i n g f o r each c o u n t a b l e index s e t I C ai:=b icI
i f t h e r e i s an o r d e r i n g o f v : I N o > I such t h a t
TR'with
C a i s convergent i n jENo v ( j )
v a l u e b.
F u r t h e r examples a r e d e r i v e d f r o m t h e f o l l o w i n g more g e n e r a l s i t u a t i o n s , which i n c o n n e c t i o n w i t h t r a n s i t i v e c l o s u r e a r e most i m p o r t a n t : 3.7
Definition
L e t S be a s e m i r i n g .
An indexed f a m i l y {ai l i e 1 1 i n S i s c a l l e d s t a t i o n a r y , if
f o r any KcI and t h e induced s u b f a m i l y { a i l i e K } such t h a t f o r a l l f i n i t e R w i t h F(K)GRsK
t h e r e i s a f i n i t e s e t F(K)cK
B. Mahr
244
z
a
~a~ ieR
=
ieF(K) holds.
3.8 Let
We c a l l F(K) a c h a r a c t e r i s t i c index s e t o f K.
Fact {ai l i
E. I } be a s t a t i o n a r y indexed f a m i l y w i t h c h a r a c t e r i s t i c index sets
F o ( I ) and F 1 ( I ) , then ai =
X irFo( I )
1
ai
ieFl ( I )
3.9 P r o p o s i t i o n L e t S be a p o s i t i v e semiring and I a i l i E I } a s t a t i o n a r y indexed f a m i l y i n S, then f o r any c h a r a c t e r i s t i c index s e t F ( 1 ) o f I
Z ai ieI
=
a.
1
'
iaF(1)
i s c o n s i s t e n t i n S. Proof: 7: ai
5 of
We d e f i n e a p a r t i a l completion
S as f o l l o w s :
i s defined i f f i a i ) i o D i s s t a t i o n a r y , and i n t h i s case we d e f i n e
ieI
Z ai:= ieI
Z
i r F ( 1 ) ai
f o r c h a r a c t e r i s t i c index s e t F ( I ) o f I .
By f a c t 3.8 t h i s y i e l d s a w e l l - d e f i n e d
expansion o f S which we have t o show t o be a p a r t i a l completion o f S. the semiring r e s t r i c t i o n o f ?iequals S.
(D2) and (03) are s t r a i g h t forward.
Obviously
Definedness c o n d i t i o n s i n axioms ( D l ) ,
Definednesr c o n d i t i o n s i n (04) a r e d e r i v e d
as f o l l o w s : Let K
I and K
=
U jcJ
K . be d i s j o i n t , then by d e f i n i t i o n the f a m i l y { a i ( i E K:) J
s t a t i o n a r y f o r any j EJ.
I t i s l e f t t o show t h a t a l s o { C a i I j iaK j
s t a t i o n a r y : Given J '
We then consider the f a m i l y
c_
J.
6 = {aili
r
U j c J ' Kj}
6 i s a subfamily o f l a i l i r I}, and thus s t a t i o n a r y . L e t X be a c h a r a c t e r i s t i c index s e t of 6.
We then show t h a t
F ( J ' ) = { j E J ' I K j n X # $I}
E
J} i s
is
Iteration and summability in semirings i s a c h a r a c t e r i s t i c index s e t o f the family {
a . [ j c J ' } which completes t h e ieK. J
proof. __ Fact:
245
Given f o r a l l j o J ' f i n i t e s e t s Q . such t h a t J F( K j
€9 .sK i
RsJ'
then f o r a l l f i n i t e
c
j
we have
c
a.
'
itF(K.) J
jrR
C
C
jcR
ieQ.
=
a 1.
J
We have t o v e r i f y t h a t f o r a r b i t r a r y f i n i t e R w i t h F ( J ' ) r R sJ' the equation
c
C
jeF(J')
a.=
'
ieF(Kj)
C
1
a.
i c F ( KJ. )
jeR
F o r t h e l e f t s i d e of (1) we d e r i v e from t h e f a c t by choosing
holds t r u e .
R=F(J' ) and Q . = F ( K . ) u ( K.nX) J J J
C
C
jGF(J')
icF(K.) J
a. =
'
C ai icX
For t h e r i g h t s i d e o f (1) we d e r i v e f r o m t h e f a c t by choosing R a r b i t r a r y and Qj=F(Kj)u(K.nX) J
c
C
i r F ( KJ. )
jeR
Since X =
U
jeF (J ' )
a . = C jcR
C a i
(3)
iaQi
(KjnX) by d e f i n i t i o n o f F ( J ' ) , and
and c o n s e q u e n t l y a l s o
we o b t a i n by d i s j o i n t n e s s o f t h e Q j X a i = X a i ieX id). J
C joR
(1) then i s deduced from ( 2 ) and t h e combination o f z a 1. l j r J } i s s t a t i o n a r y . ieK
Thus a l s o {
j
(3)
and
(4).
B. Mahr
246
I t i s now s t r a i g h t forward t o complete t h e p r o o f .
3.10 Examples
( I ) Let 6 = { a i \ i
f o l 1owing
E I ) be an indexed f a m i l y i n a semiring S w i t h the
property : t h e r e i s a f i n i t e s e t F(K) s I such t h a t ai
= 0 for a l l
E
I V ( 1 ) then
i s called finitary. Any f i n i t a r y f a m i l y i s s t a t i o n a r y w i t h c h a r a c t e r i s t i c index s e t F ( I ) , and thus additive.
(2)
Let 8 =
I1 be an indexed family i n an idempotent semiring S w i t h
E
the f o l l o w i n g property: there i s a f i n i t e s e t F(1) c_ I such t h a t f o r a l l i E I t h e r e i s J E F ( 1 ) w i t h ai = a
j
then 6 i s c a l l e d weakly f i n i t a r y . Any weakly f i n i t a r y f a m i l y i n an idempotent semiring i s s t a t i o n a r y with characteri s t i c index s e t F ( I ) , and thus a d d i t i v e . ( 3 ) L e t 6 = t a i l i e I } be an indexed f a m i l y i n an idempotent semiring S w i t h t h e fallowing property: for all K
G
I t h e r e i s a f i n i t e s e t F(K) such t h a t f o r a l l i
E
K there
i s j E F(K) w i t h ai L a .
J
then 6 i s c a l l e d f i n i t e l y bounded.
Any f i n i t e l y bounded f a m i l y i n an idempotent semiring i s s t a t i o n a r y w i t h character i s t i c index s e t F ( I ) , and thus a d d i t i v e . I n s p e c i f i c semirings, such as f r e e semirings w i t h elements represented by formal power series, one can f i n d f u r t h e r examples o f a d d i t i v e indexed f a m i l i e s (see /Ma 821). For t h e semiring o f m a t r i c e s over some semiring S we o b t a i n t h e f o l l o w i n g r e s u l t : 3.11 _Proposition L e t S be a semiring and n a r b i t r a r y .
Then an indexed f a m i l y { A k l k E I } i n M(n,S)
i s a d d i t i v e i f and o n l y i f f o r a l l i,j c n t h e s e t o f f a m i l i e s { A k ( i , j ) l k additive i n S.
E
I} i s
Iteration and summability in semirings
4.
247
SEMIRINGS
TRANSITIVE CLOSURE I N POSITIVE
I n t h i s s e c t i o n we apply t h e theory developed i n the previous sections t o t r a n s i t i v e c l o s u r e o f matrices over semirings. t r a n s i t i v e closure
There are several ways t o d e f i n e
o f matrices over general semirings.
We t h e r e f o r e i n v e s t i g a t e
t h e i r r e l a t i o n f i r s t , and then study existence. I f S i s a p o s i t i v e semiring and A e M(n,S),
then we might simply ask f o r a solu-
t i o n T(A) o f t h e i t e r a t i o n X=l +Ax T(A), however, i s n o t uniquely determined and we do n o t
i n t h e semiring M(n,S).
know much about the minimal s o l u t i o n s , i f S i s ordered.
Another drawback o f t h i s
s p e c i f i c a t i o n o f t r a n s i t i v e c l o s u r e i s t h a t i t t e l l s l i t t l e about u n d e r l y i n g a p p l i c a t i o n s , a t l e a s t i n the c o n t e x t o f path problems. We g i v e t h r e e ways of d e f i n i n g matrices T(A) from a m a t r i x A and show t h a t i n a c e r t a i n way they are a l l equivalent, and, whenever they e x i s t , solve t h e above iteration: L e t S be a p o s i t i v e semiring and A
E
M(n,S).
Then matrices A',
A* and A
T
are
f o r m a l l y d e f i n e d as f o l l o w s :
A':
A'
. = C Aijk f o r i,j 4 n
ij-
kaO
A*:
A* : = 1
+
A(n) where A(n) i s d e f i n e d i n 2.10
w i t h a* denoting the formal element
1 ai
irNo AT: ATj:
=
C
v a l A ( p ) f o r i,j 6 n
pep..
1J
Here P.. denotes the s e t o f a l l paths from i t o j i n the complete S - l a b e l l e d graph 1.I
which has A as adjacency m a t r i x .
valA(p) denotes the l a b e l l i n g o f path p which
i s d e f i n e d by valA(p) = A(el). f o r p = el. ..er
and ei
Q
...
.A(er)
[n]x[n].
These d e f i n i t i o n s do n o t i m p l y existence o f the matrices A', f o r e i n t r o d u c e t h e f o l l o w i n g convention: Let
5 denote a p a r t i a l completion o f
S.
Then we say:
A* o r AT.
We there-
B. Mahr
248
A'
i s d e f i n e d i n M(nS) i f f o r a l l i , j r n t h e sum k A..
Z kc INo
A* i s d e f i n e d i n rl(n,S)
i s defined i n
s
lJ
i f f o r a l l k h n t h e sum
Z
(ALt-l))i
i s defined i n
i e No
AT i s d e f i n e d i n M(m,S) i f f o r . a l l i , j a the sum valA(p)
C
i s defined i n
5
pep..
1J
The f o l l o w i n g r e s u l t shows t h a t a l l t h r e e concepts o f t r a n s i t i v e c l o s u r e are e q u i valent: 4.1 P r o p o s i t i o n Let S be a p o s i t i v e semiring and
be a p a r t i a l completion o f S.
Then are
equivalent: (1)
A'
(2)
A* i s d e f i n e d i n M(n,s)
(3)
AT i s d e f i n e d i n M(n,S)
i s d e f i n e d i n M(n,S)
and i n case a l l t h r e e matrices are d e f i n e d i n M(n,S),
then A'
= A*
=
A'.
The p r o o f o f t h i s p r o p o s i t i o n i s based on two lemnata which g i v e w e l l known i n t e r p r e t a t i o n s o f t h e m a t r i c e s A'
and A(r) which d e f i n e A
and A* r e s p e c t i v e l y .
4.2 Using the n o t a t i o n
P
= I p E Pijl
l e n g t h ( p ) = r } , then
I n f o r m a l l y speaking, we o b t a i n the i , j - e n t r y -alued o f paths o f l e n g t h
Proof:
o f t h e m a t r i x Ar i f we sum up a l l
r.
0 1J
I f r = 0 , then A . . = o i f i # j and A P j = I i f i = j .
i # j and P
Since PI:'
= @ if
249
Iteration and summability in semirings 0
A.. =
1
lJ
u s i n g t h e f a c t t h a t v a l A ( X ) = 1.
<1>
If r =1, t h e n A!. = A . . . 1J 1J
Since P . . = { ( i , j ) } , 1J
1
A.. =
we have
1
lJ
Assume t h e e q u a t i o n i n lemma 4.2 i s v a l i d f o r r.
Any p a t h o f l e n g t h r+l f r o m i
t o j can be decomposed u n i q u e l y i n t o a p a t h o f l e n g t h r f r o m i t o k, some p r e decessors o f j , and an edge ( k , j )
.
So we have
Note, a l l sums i n t h i s p r o o f a r e a c t u a l l y f i n i t e
‘0
B e f o r e we s t a t e t h e n e x t lemma, we d i s c u s s some f a c t s about paths i n graphs: By d e f i n i t i o n , P . . denotes t h e s e t o f paths from i t o j . 1J
P . .oP IJ
jk
: = {Pq(P€Pij,qaPjkl
and f u r t h e r f o r O , l , (P..)O: = 11
(Pii)? (Pii)
We d e f i n e
...,n,.. .
IX}
= P 11 ..
n
For r = o , . ..,n
: = (Pii)
n-1
0
P..
11
we d e f i n e
p t ha1 P ! r ) : = I p ~ P ~ ~ l and 1J By i n d u c t i o n on
r
i n n e r p o i n t s of p a r e i n Crll
one e a s i l y v e r f i e s
FACT For r 4 n and i , j 6 n
B. Mahr
250
So a p a t h p i s i n P!r)
i f and o n l y i f i t i s e i t h e r i n
P(r-'),
r n o t as i n n e r p o i n t , o r t h e r e i s a decomposition
1J
1J
i . e . contains
p = plp 2...pn
4.3 For
f o r r=2,. pncP ( r.- 1 ) and piePrr(r-l) rJ
such t h a t pleP\:-'),
. . ,n-l.
Lemma = 1,.
r,i,j,
.. ,n
we have I .
'- ( pap.
r) 1J
valA(p)
(r =
0)
i f and o n l y i f
exists
and i n case o f e x i s t e n c e f o r r = O , l ,
Proof:
... r
..., n
we have t o show ( 2 ) :
.By d e f i n i t i o n o f A ( r ) we have
( A.0) = A. 1J
.,
On t h e o t h e r hand
1J
Since Val ( i , j ) = A . ., we o b t a i n A 13
(*I ( n o t e , t h i s sum i s f i n i t e , and t h e r e f o r e e x i s t s ) ( r = 1) we have t o show (1) and ( 2 ) :
Z
Assume
(Al1( 0 )) i e x i s t s , t h e n f o r i,j-.n
i s INo
exists i n vJe o b t a i n
5.
S i n c e A!o) 1J
= Aij
and valA(i,j)=Aij
P .( O ) 1J
=
C(i,j)l
25 1
Iteration and summability in semirings
Using t h e above f a c t ,
and t h e p r o p e r t y Val ( p q ) = Val ( p ) A A w i t h t h e h e l p of axioms D3 and D4 t h a t
A!!)
=
c
. valA(q),
we o b t a i n
valA(p)
pep..
lJ
1J
exists i n
3.
T h i s proves one d i r e c t i o n o f (1) and ( 2 ) .
Conversely, assume
valA(p)
C
e x i s t s , then since
pep.
1j
U ic
5 Pi:)
by axiom D4 we have t h a t
INo
i s consistent i n
5.
By i n d u c t i o n on
i one e a s i l y shows w i t h t h e h e l p o f 3.3(1)
Again from axiom D4 we deduce
Using e q u a t i o n ( * ) above, we deduce c o n s i s t e n c y i n 5 f o r =
C i e INo
which completes t h e p r o o f of ( 1 ) and ( 2 ) i n t h e case r = 1. shown by i n d u c t i o n i n analogy r o t = 1 ) . We a r e now ready t o g i v e t h e p r o o f f o r p r o p o s i t i o n 4.1.
The case ( r =
rn) i s
B. Mahr
252 PROVING PROPOSITION 4 . 1
-
( 1 ) Assume A“ i s d e f i n e d i n M(n,S),
ks
5.
i s defined i n
Since P . . =
INo
lJ
By lemma 4.2 we o b t a i n
k> P..
L
”
k A..
c
t h a t i s t o say t h a t f o r a l l i , j s n
kaNo
i s a decomposition o f P . .
lJ
i t follows
1.J’
from axiom D4 t h a t
5.
i s consistent i n
Consequently
Since f o r a l l r = o ,..., n axiom 04 t h a t f o r a l l r =
z
P.:)(
A
EP..
1J 0,
T e x i s t s and AT and
1J
..., n
P (. n ) = 1J
= A’,
P..\{k},
we deduce f r o m
1J
valA(p)
pep!:) 1J
exists i n
and t h e r e f o r e by lemma 4 . 3 ( 1 ) f o r a l l k = 1, ... ,n
5, i it
N
0
5.
i s defined i n A?.
=
Thus a l s o A* e x i s t s .
Usino lemma 4 . 3 ( 2 ) we o b t a i n
1 . . + A (. 1 )
1J
=
1J
1.1
5
valA(P)
pep!!) 1J
-- -
valA(p) = AT.
~
PEP.
1J
’
1J
thus
A
*
I-
= A-.
So we h a v e shown:
(2)
A‘
e x i s t s = A* e x i s t s and A T e x i s t s and A‘
Assume k ’ e x i s t s i n M(n,S),
is defined i n 5 .
Since Pij
=
*
= A
T
= A
then f o r a l l i , j c n
u
#ij 7 i s
r40
a decomposition o f P . . we o b t a i n f r o m 1J
Iteration and summability in semirings
253
axiom D4 and lemma 4.2 t h a t f o r i,j 6 n
c re
AYj
No
5.
i s defined i n (3)
Assume A
*
i s defined i n
exists i n
5.
Thus a l s o A
c
exists.
e x i s t s i n M(n,S),
then by d e f i n i t i o n f o r a l l k s n
Using lemma 4 . 3 ( 2 ) , we o b t a i n t h a t f o r a l l r = 1, ..., n
5.
Since P!n)
1 j
=
P..\thl 1J
we deduce e x i s t e n c e o f A
T
and i , j s n
i n N(n,S).
T h i s completes t h e p r o o f o f p r o p o s i t i o n 4.1. We may f u r t h e r ask, how t h e m a t r i c e s AT,
C
A',
A' and A* a r e r e l a t e d w i t h t h e m a t r i x
which i s expressed i n terms o f an i n f i n i t e sum i n t h e s e m i r i n q Fl(n,S)
i e No r a t h e r t h a n S.
b l i t h o u t p r o o f we s t a t e t h e f o l l o w i n g r e s u l t which i s c l o s e l y
r e l a t e d t o p r o p o s i t i o n 3.11:
4.4 P r o p o s i t i o n L e t S be a s e m i r i n g and AEM(n,S).
Then
Z
A' e x i s t s i n M(n,S)
i f and o n l y i f
i a No
there i s a p a r t i a l completion
C
5
o f S such t h a t A'
i s d e f i n e d i n M(n,S) and
Ai = A'
i e No
i s consistent i n M(n,Sl F i n a l l y we o b t a i n
4.5 C o r o l l a r y L e t S be a s e m i r i n g and Ae M(n,S). s o l v e X = 1 + AX i n M(n,S),
C
Then any of t h e t h r e e m a t r i c e s A ,. A
*
and A
T
p r o v i d e d t h e r e i s a p a r t i a l c o m p l e t i o n o f S i n which
they a r e defined.
We now t u r n t o t h e q u e s t i o n o f e x i s t e n c e of t r a n s i t i v e c l o s u r e and s t a t e a number
B. Mahr
254
o f r e s u l t s , which are well-known i n the l i t e r a t u r e (see e.g. never been s t a t e d i n t h i s g e n e r a l i t y .
/ Z i 81/), b u t have
Also these r e s u l t s c o n f i r m our concept o f
sumnabi lity. I f S i s an idempotent semiring and A e M(n,S).
Then A i s c a l l e d semipositive, i f
f o r a l l cycles q i n A valA(q) 4 1
A i s c a l l e d p o s i t i v e , i f Aij
6 1 f o r a l l i , j 4 n.
I n t h e semiring M = (Ru(-l,min,t,m,O), i s semipositive,
which i s idempotent, a m a t r i c A E M(nJMin)
i f a l l i t s cycles q have non-negative value, i . e . v a l A ( q ) bRO
w i t h respect t o the n a t u r a l o r d e r i n g
on
IR. A i s
positive i f a l l i t s entries
are non-negative. 4.6 P r o p o s i t i o n L e t S be an idempotent semiring and A
E
M(n,S) semipositive.
Then we conclude:
n-1 . A = C A ’ i=O i s consistent, and AT i s s m a l l e s t s o l u t i o n o f X=ltAX ( a c c o r d i n g t o the p o i n t w i s e order on m a t r i c e s ) .
T Moreover, A =A*,
t h e Kleene-matrix w i t h r e s p e c t t o ( A i k - ’ ) ) *
=
1 f o r k = 1,
...,n.
Since a m a t r i x over a simple semiring i s always p o s i t i v e , and thus a l s o semip o s i t i v e , we have
4.7 Corol 1a r y The conclusion i n 4.6 i s t r u e i f S i s simple and A a r b i t r a r y . For extremal ( t o t a l l y ordered idempotent) semi r i n g s we o b t a i n 4.8 P r o p o s i t i o n I f S i s extremal and A t M(n,S)
semipositive, then the conclusion i n 4.6 i s t r u e .
Moreover, f o r a l l i,j 4 n t h e r e i s a simple p a t h po
B
P . . such t h a t 1J
ATj = valA(po).
For m a t r i c e s over D i j k s t r a - s e m i r i n g s we o b t a i n the conclusion o f 4.8 g e n e r a l l y . T /Le 77/ has shown t h a t f o r t h e computation o f A over a D i j k s t r a - s e m i r i n g the
Iteration and summability in semirings
255
well-known D i j k s t r a - a l g o r i t h m i s c o r r e c t . 4.9 Remark Summability i n a r b i t r a r y and n o t j u s t p o s i t i v e semirings can be s t u d i e d by a d i f f e r e n t a x i o m a t i z a t i o n o f p a r t i a l complete semirings, which i s obtained by chang i n g t h e " i f f " i n axiom D4 t o " o n l y i f " . This change a f f e c t s f a c t 3.3 and propos i t i o n 4.1: Existence o f AT i m p l i e s e x i s t e n c e o f A c and A*, b u t n o t v i c e versa. ( T h i s was p o i n t e d o u t t o t h e author by G. Rote /GR 83/).
REFERENCES /AHU 74/
Aho, A.V., Hopcroft, J.E. and Ullman, J.D., The Design and A n a l y s i s o f Computer Algorithms (Addison-Wesley, 1974).
/BC 75/
Backhouse, R.C. and Carr6, B.A., Regular algebra a p p l i e d t o p a t h f i n d i n g problems, J . I n s t . Math. A p p l i c s . 15 (1975).
/Br 74/
Brucker, P., Theory o f m a t r i x algorithms, Mathematical Systems i n Economics 13 (Anton Hain, 1974).
/Ca 71/
Carre, B.A., An a l g e b r a f o r network r o u t i n g problems, J . I n s t . Maths. A p p l i c s . 7 (1971).
/Ca 79/
Carr6, B.A.,
/CG 791
Cuninghame-Green,
/CH 65/
Cruon, R. and HervC, P., Reone fr. Rech. oper. No. 34 (1965) 3-19.
/Co 71/
Conway, J.H., 1971).
/ E i 74/
Eilenberg, S.,
/ E l 77/
Elgot, C.C., F i n i t e Automaton from a Flowchart Scheme P o i n t o f View, (IBM Research Report RC 6519, 1977).
/GM 82/
Gondran, M., and Minoux, M., unpub 1ished paper.
/HR 68/
Hammer, P.L. and Rudeanu, S., (Springer, 1968).
/Le 77/
Lehmann, D., A l g e b r a i c s t r u c t u r e s f o r t r a n s i t i v e closure, Theoretic. Computer Science 4 (1977).
/Ma 82/
Mahr, B., Semirings and T r a n s i t i v e Closure, TU B e r l i n , FB 20, B e r i c h t NO. 82-5 (1982).
/Mar 76/
M a r t e l l i , A Gaussian e l i m i n a t i o n a l g o r i t h m f o r t h e enumeration o f c u t sets i n a graph, J . ACM 23 (1976).
/Mo 611
M o i s i l , G.C.,
Graphs and Networks (Clarendon Press, 1979).
R.A.,
Minimax algebra, (LNEMS 166, Springer, 1979).
Regular Algebra and F i n i t e Machines (Chapman and H a l l , Automata, Languages and Machines (Academic Press, 1974).
D i o i d Theory and i t s A p p l i c a t i o n s , Boolean Methods i n Operations Research
Comnunicarile Acad. R e p u b l i c i i Populare Romine, 10, 647-652.
256
B. Mahr
/Ta 81a/
Tarjan, R . E . , (1981).
A unified approach t o path problems, J . ACM, 28 No. 3 ,
/Ta 81b/
Tarjan, R . E . , 3, (1981).
Fast algorithms f o r solving path problems, J . ACM, 28, No.
/Wo 79/
Wongseelashote, A . , Semirings and path spaces, Discrete Mathematics 26 (1979).
/Yo 61/
Yoeli, M . , A note on a generalization of Boolean matrix theory, Americ. Math. Monthly 68, 1961.
/Zi 81/
Zimnermann, U., Linear and combinatorial optimization i n ordered algeb r a i c s t r u c t u r e s , book t o appear, 1981.
/GR 83/
Rote, G . , Personal communication, (1983).
Annals of Discrete Mathematics 19 (1984) 257-356 0 Elsevier Science Publishers B.V. (North-Holland)
251
SUBSTITUTION DECOMPOSITION FOR DISCRETE STRUCTURES AND CONNECTIONS WTTH COMB1NATOR I A L OPTIMIZATION* R.H. MtJhring
T e c h n i c a l U n i v e r s i t y o f Aachen Lehrstuhl f u r Informatik I V 51 Aachen West Germany
F. J
. Radermacher
U n i v e r s i t y o f Passau L e h r s t u h l f u r I n f o r m a t i k und Operations Research 839 Passau West Germany
I n t h i s paper we deal w i t h t h e s u b s t i t u t i o n decomposition as known f o r Boolean f u n c t i o n s , s e t systems and r e l a t i o n s . I t i s shown how c o m b i n a t o r i a l o p t i m i z a t i o n problems, p a r t i c u l a r l y on graphs, p a r t i a l o r d e r s , p r o j e c t networks, ( i n - ) dependence systems and c l u t t e r s , n a t u r a l l y l e a d , under weak assumptions,to t h i s k i n d o f decomposition v i a c e r t a i n uniqueness r e s u l t s . We t h e n g e n e r a l i z e t h e comnon f e a t u r e s o f t h i s t y p e o f decomposition from t h e s p e c i a l cases t o a q u i t e g e n e r a l a l g e b r a i c l e v e l , c o v e r i n g i n f i n i t e cases, and deduce a g e n e r a l Jordan-Holder theorem as w e l l as uniqeness r e s u l t s f o r t h e associated composition t r e e i n t h i s general s e t t i n g . These r e s u l t s a r e t h e n r e i n t e r p r e t e d f o r t h e s p e c i a l cases, and t h e c o m p u t a t i o n a l aspects o f t h e s u b s t i t u t i o n decomposition a r e d i s c u s s e d . Throughout, a p p l i c a t i o n s t o many problems concerning d i s c r e t e s t r u c t u r e s a r e i n c l u d e d , as a r e connections w i t h o t h e r approaches i n these f i e l d s , i n p a r t i c u l a r t h e s p l i t decomposition. CONTENl INTRODUCTION SUBSTITUTION DECOMPOSITION FOR BOOLEAN FUNCTIONS, SET SYSTEMS AND RELATIONS:
I
BASIC RESULTS AND COMMON PROPERTIES 1. 2. 3. 4. 5.
6.
I1
Aspects o f common b e h a v i o u r S u b s t i t u t i o n decomposition f o r Boolean f u n c t i o n s S u b s t i t u t i o n decomposition f o r s e t systems S u b s t i t u t i o n decomposition f o r r e l a t i o n s - A p p l i c a t i o n s o f t h e s u b s t i t u t i o n decomposition t o graphs - A p p l i c a t i o n s o f t h e s u b s t i t u t i o n decomposition t o p a r t i a l o r d e r s Interfaces - Coherent systems: Boolean f u n c t i o n s vs. c l u t t e r s - Conformal c l u t t e r s vs. graphs - P a r t i a l o r d e r s vs. c o m p a r a b i l i t y graphs Connections w i t h t h e s p l i t decomposition
UNIQUE CHARACTERIZATION OF THE SUBSTITUTION DECOMPOSITION FOR RELATIONS AND SET SYSTEMS FROM THE VIEW-POINT OF COMBINATORIAL OPTIMIZATION
1.
*
C o m b i n a t o r i a l o p t i m i z a t i o n o v e r graphs
Work was supported by t h e M i n i s t e r f u r Wissenschaft und Forschung des Landes N o r d r h e i n - N e s t f a l e n , West Germany.
R.H. Mohring and EJ. Radermacher
258
Combinatorial o p t i m i z a t i o n over p a r t i a l orders - d e t e r m i n i s t i c p r o j e c t networks time-cost t r a d e - o f f i n p r o j e c t networks - one machi ne scheduling s t o c h a s t i c p r o j e c t networks 3. Combinatorial o p t i m i z a t i o n over (in-)dependence systems and c l u t t e r s 4. Summary and h i n t s on some o t h e r approaches t o decomposition o f c e r t a i n combinatorial o p t i m i z a t i o n problems
2.
-
111 AN ALGEBRAIC MODEL OF DECOMPOSITION
1.
2. 3. 4. 5. 6. IV
The a l g e b r a i c model The system o f congruence p a r t i t i o n s The Jordan-Holder theorem f o r composition s e r i e s The composition t r e e Connections w i t h t h e s p l i t decomposition On t h e a l g o r i t h m i c complexity o f decomposition
ALGEBRAIC AND ALGORITHMIC ASPECTS OF THE SUBSTITUTION DECOMPOSITION FOR RELATIONS, SET SYSTEMS AND BOOLEAN FUNCTIONS 1. 2. 3.
Relations Set systems Boolean f u n c t i o n s
ACKNOWLEDGEMENT REFERENCES
INTRODUCTION The decomposition o f s t r u c t u r e s i s i m p o r t a n t f o r t h e development of many mathematical theories.
Extension o f fundamental r e s u l t s from group t h e o r y and r e l a t e d
f i e l d s has l e d t o q u i t e general concepts i n u n i v e r s a l algebra.
Some o f t h e
s t r o n g e s t r e s u l t s a r e those o f t h e Jordan-Holder type d e a l i n g w i t h t h e unique f a c t o r i z a t i o n o f s t r u c t u r e s i n t o prime (simple, i r r e d u c i b l e ) s t r u c t u r e s and being r e l a t e d t o t h e m o d u l a r i t y o f t h e congruence p a r t i t i o n l a t t i c e s . Most s t r u c t u r e s i n u n i v e r s a l algebra a r e q u i t e s t r o n g l y i n t e r n a l l y r e l a t e d ; f o r instance, a r e s t r i c t i o n t o a subset o f t h e u n d e r l y i n g base s e t does n o t i n general l e a d t o a subalgebra.
However, f o r many s t r u c t u r e s i n d i s c r e t e mathematics,
operations research and computer science, such behaviour is o f t e n the case; e.g. Boolean f u n c t i o n s , s e t systems (e.g.
(in-)dependence systems, c l u t t e r s and mat-
r o i d s ) and r e l a t i o n s (e.g. graphs, p a r t i a l orders and p r o j e c t networks) a r e o f t h i s type, as are submodular f u n c t i o n s and matrices.
The decomposition o f such
d i s c r e t e s t r u c t u r e s i s o f i n t e r e s t from t h e o r e t i c a l , computational as w e l l as p r a c t i c a l p o i n t o f view.
However, t h e e x i s t i n g t h e o r e t i c a l framework i s n o t
259
Substitution decomposition for discrete structures r e a l l y adequate f o r h a n d l i n g a l l t h e s e d i s c r e t e cases.
I n particular, the
u n i v e r s a l a l g e b r a r e s u l t s a r e n o t e x t e n d a b l e t o t h i s s i t u a t i o n w h i l e many ad-hoc concepts a r e r e s t r i c t e d t o s p e c i a l d i s c r e t e s t r u c t u r e s and o f t e n do n o t y i e l d deeper s t r u c t u r a l i n s i g h t . T h i s paper aims t o overcome t h i s s i t u a t i o n by p r e s e n t i n g a g e n e r a l i z e d v e r s i o n o f an i n t e r e s t i n g approach known f o r Boolean f u n c t i o n s , s e t systems and r e l a t i o n s (and o t h e r s t r u c t u r e s such as u t i l i t y f u n c t i o n s ) and t h e i r a p p l i c a t i o n s .
It
shows a s i m i l a r b e h a v i o u r i n a l l t h r e e c l a s s e s and t h e n a t u r a l l y i n v o l v e d subs t i t u t i o n o p e r a t i o n appears t o be one o f t h e e s s e n t i a l common f e a t u r e s . I n S e c t i o n I we g i v e a s h o r t survey o f e x i s t i n g r e s u l t s f o r t h e s u b s t i t u t i o n decomposition t o g e t h e r w i t h h i n t s on a number o f a p p l i c a t i o n s .
I n t h i s context,
we a l s o d i s c u s s some c o n n e c t i o n s w i t h t h e s p l i t decomposition.
While more general
and f a r - r e a c h i n g i n i t s r e s u l t s , t h i s decomposition l a c k s t h e s i m p l e a p p l i c a b i l i t y o f t h e s u b s t i t u t i o n decomposition.
Such aspects become p a r t i c u l a r l y c l e a r i n
S e c t i o n 11, where i t i s shown how a g e n e r a l f a c t o r i z a t i o n problem i n c o m b i n a t o r i a l o p t i m i z a t i o n o v e r graphs, p a r t i a l o r d e r s , p r o j e c t networks, (in-)dependence systems and c l u t t e r s n a t u r a l l y l e a d s e x a c t l y t o t h e s u b s t i t u t i o n decomposition f o r a1 1 t h e s e c l a s s e s . I n S e c t i o n I 1 1 we p r e s e n t a g e n e r a l and r a t h e r a b s t r a c t a l g e b r a i c s t r u c t u r e t h e o r y f o r such weakly i n t e r n a l l y r e l a t e d d i s c r e t e s t r u c t u r e s as Boolean f u n c t i o n s , s e t systems and r e l a t i o n s . c o u n t e r p a r t s here.
It t u r n s o u t , t h a t many u n i v e r s a l a l g e b r a r e s u l t s have
T h i s i s even t r u e f o r theorems o f t h e Jordan-Holder t y p e ,
a l t h o u g h t h e congruence p a r t i t i o n l a t t i c e s a r e h e r e g e n e r a l l y o n l y upper semimodular ( i n t h e f i n i t e case), i . e . t h e y have weaker p r o p e r t i e s t h a n t h o s e f a m i l i a r from u n i v e r s a l algebra.
I n a d d i t i o n , t h e s i t u a t i o n h e r e t u r n s o u t t o be
n i c e f o r a p p l i c a t i o n s , due t o t h e presence o f c o m p o s i t i o n t r e e s .
Some h i n t s on
t h e use o f t h i s i n s t r u m e n t f o r a l g o r i t h m i c approaches t o decomposition, t o g e t h e r w i t h some c o m p l e x i t y a n a l y s i s f o r t h e s p e c i a l s t r u c t u r e s considered, f o l l o w i n Section I V . I n view o f t h e g e n e r a l i t y o f t h e d i f f e r e n t concepts presented, we hope t h a t t h e paper w i l l s e r v e as a s t e p on t h e way t o an a l g e b r a i c decomposition t h e o r y i n d i s c r e t e m a t h m a t i cs.
I SUBSTITUTION DECOMPOSITION FOR BOOLEAN FUNCTIONS, SET SYSTEMS AND RELATIONS: BASIC RESULTS AND COMMON PROPERTIES I n t h i s p a r t we w i l l g i v e an i n t r o d u c t i o n i n t o a v a i l a b l e i n s i g h t i n t o t h e s u b s t i These
t u t i o n decomposition f o r Boolean f u n c t i o n s , s e t systems and r e l a t i o n s .
R.H. Mohring and F.J. Radermacher
260
t h r e e classes o f d i s c r e t e s t r u c t u r e s have been q u i t e thoroughly s t u d i e d by d i f f e r e n t authors i n various, seemingly u n r e l a t e d contexts, and p r e s e n t l y c o n s t i t u t e
p5],
( a p a r t f r o m u t i l i t y theory
which w i l l n o t be covered here) t h e main f i e l d s
o f a p p l i c a t i o n s o f t h e s u b s t i t u t i o n decomposition.
[39]
There i s some evidence [3g,
p o i n t i n g towards p o s s i b l e f u t u r e i n c l u s i o n o f o t h e r s t r u c t u r e s , e.g.
modular f u n c t i o n s and systems o f l i n e a r equations. r e s u l t s w i l l be done w . r . t .
sub-
The p r e s e n t a t i o n of t h e
the intended g e n e r a l i z a t i o n , and t h e terminology,
d i f f e r e n t i n a l l t h r e e f i e l d s , w i l l be u n i f i e d .
The r e s u l t s are o f t e n reformula-
t i o n s , m o d i f i c a t i o n s , o r extensions o f e a r l i e r , versions, b u t some cdses, i n part i c u l a r Tor i n f i n i t e s e t systems, appear here f o r t h e f i r s t time. I n 1.1 we w i l l l i s t some common features, w i t h examples, which a c t as a guide f o r the treatment o f t h e s u b s t i t u t i o n decomposition i n a l l t h r e e classes o f s t r u c t u r e s (1.2
-
1.4) t o g e t h e r w i t h a d i s c u s s i o n o f t h e major a p p l i c a t i o n s .
We c o n t i n u e
i n 1.5 w i t h h i n t s on i n t e r f a c e s between these classes o f s t r u c t u r e s , which f a c i l i t a t e understanding o f t h e analogous behaviour observed. 1.6 some remarks on s i m i l a r i t i e s and d i f f e r e n c e s w . r . t .
F i n a l l y , we i n c l u d e i n t h e more general, b u t
c l o s e l y r e l a t e d s p l i t decomposition [3g.
1.1
ASPECTS OF COMMON BEHAVIOUR
We g i v e a l i s t o f s i x groups o f p r o p e r t i e s ( P l )
-
(P6), which d e s c r i b e i n t e r e s t i n g
aspects o f the s u b s t i t u t i o n decomposition, comnon t o a l l t h r e e classes o f s t r u c t u r e s considered, and j o i n t l y demonstrated i n Example 1.1 .l. Subsequently, ( P l ) (P5) w i l l be r e f e r r e d t o i n 1.2
-
-
1.4, where h i n t s on t h e p r o o f s f o r t h e respec-
t i v e cases are given, treatment o f (P6) being postponed t o S e c t i o n I V .
We mention
here t h a t any s t r u c t u r e S o f t h e t h r e e types i s d e f i n e d on an u n d e r l y i n g s e t
A = As ( t h e base s e t o f A ) . i.e.
a f u n c t i o n F: (0,1ln
variables o f
F.
-f
For example, if t h e s t r u c t u r e i s a Boolean f u n c t i o n , {O,ll,
t h e base s e t AF i s t h e s e t
I f i t i s a s e t system T, i.e.
{X
l,...,~n}
of
a c o l l e c t i o n o f subsets o f some
s e t A, then t h i s s e t A i s t h e base set. F i n a l l y , i f R i s a k-ary r e l a t i o n on A , k i.e. R G A = A x .. x A ( k - t i m e s ) , then again AR = A. Note t h a t f o r s e t systems
.
and r e l a t i o n s , t h e base s e t may be i n f i n i t e . of these s t r u c t u r e s , each subset
B
Due t o t h e weak i n t e r n a l coherence
o f t h e base s e t induces a new s t r u c t u r e S I B by
simply r e s t r i c t i n g t h e g i v e n s t r u c t u r e t o t h i s subset (sanething n o t t r u e f o r a l g e b r a i c s t r u c t u r e s such as groups e t c . ) .
(Pl)
SUBSTITUTION OPERATION
I n each of the t h r e e cases, g i v e n a s t r u c t u r e S ' on a base s e t A ' and f o r each B c A ' a s t r u c t u r e S,
on a base s e t A
6
with
ABnA 8 '
=
0
f o r 8 # B ' , there i s a
26 1
Substitution decomposition for discrete structures
u n i q u e l y d e f i n e d s t r u c t u r e S on t h e base s e t A = the structures S
into S' for B E A'.
B s a i d t o be o b t a i n e d by s u b s t i t u t i o n .
u
AB, o b t a i n e d by s u b s t i t u t i n g $€A I S i s denoted b y S = S'[S,, 8 e A ' ] and i s
T h i s b a s i c o p e r a t i o n , which i s usuaSly m o t i v a t e d f r o m h i g h e r l e v e l problems, and which g i v e s t h e s u b s t i t u t i o n decomposition i t s name, means i n t h e s p e c i a l cases e i t h e r s u b s t i t u t i o n o f Boolean f u n c t i o n s i n t o a n o t h e r i n t h e sense o f e.g. (41, [40], o r s u b s t i t u t i o n o f s e t systems i n t h e sense o f e.g. [lo], [38], [13q, o r e.g.
t h e s o - c a l l e d X - j o i n o f graphs [132]
o r t h e o r d i n a l sum o f p a r t i a l o r d e r s
P4I. S t r u c t u r e s a r e c a l l e d decomposable, i f t h e y have a r e p r e s e n t a t i o n S = S ' [S B €A']
w i t h I A S a / > 1 and
lABl
1 f o r a t l e a s t one 6
>
E
A ' , and
prime
B' otherwise.
T h i s s t r u c t u r a l decomposition occurs n a t u r a l l y i n a p p l i e d problems such as d e t e r m i n a t i o n o f maximal c l i q u e w e i g h t s o r m i n i m a l c o l o u r i n g s i n graphs (compare t h e uniqueness r e s u l t s i n c o m b i n a t o r i a l o p t i m i z a t i o n g i v e n i n S e c t i o n 1 1 ) .
The a p p l i -
c a t i o n s s t i m u l a t e i n t e r e s t i n t h o s e p a r t i t i o n s IT = { A 6 I B a A ' } o f t h e base s e t 8 e A'], where S a r e A o f a s t r u c t u r e S which a l l o w a r e p r e s e n t a t i o n S = S'[S 6' 6 structures over A B Q A ' and S ' i s t h e q u o t i e n t s t r u c t u r e S ' = S/IT on A ' . These B'
p a r t i t i o n s a r e c a l l e d congruence p a r t i t i o n s o f S.
They can e q u i v a l e n t l y be i n t e r -
p r e t e d by means o f t h e s u r j e c t i v e homomorphism 0,: A
+
A ' d e f i n e d by nIT(a)= B
a aA f r o m S o n t o S/n. So qIT maps elements o n t o t h e same element i f t h e y 5 a r e i n t h e same c l a s s o f IT, which i s h e n c e f o r t h denoted by CY II B o r a E
:<=>
where
[BIT i s t h e c l a s s o f
[BIT,
IT
c o n t a i n i n g B.
Congruence p a r t i t i o n s a r e c l o s e l y r e l a t e d w i t h t h e i r c l a s s e s ( c f . ( P 3 ) below) These s e t s can e s s e n t i a l l y b e i d e n t i f i e d w i t h A (a) = CY f o r a l l a E. AB, i n j e c t i v e homomorphisms i n c f AB A, d e f i n e d by i n c A 6' 8 from t h e u n i q u e l y determined autonomous s u b s t r u c t u r e S ( = S I A ) o f S i n t o S. B B
which a r e c a l l e d autonomous s e t s . 9
-f
Note t h a t i n t h e s p e c i a l cases c o n s i d e r e 4 a s t r u c t u r e S i s u n i q u e l y determined b y s p e c i f y i n g a congruence p a r t i t i o n IT,a q u o t i e n t s t r u c t u r e S ' on A ' , and a u t o n m u s substructures SB, B E A ' .
However, i n t h e g e n e r a l a l g e b r a i c Jpproach i n P a r t 111,
t h i s uniqueness o f S ( w h i c h may be viewed as t h e p o s s i b i l i t y t o r e c o n s t r u c t S f r o m t h e d a t a s p e c i f i e d above b y means o f a " g e n e r a l " s u b s t i t u t i o n o p e r a t i o n ) i s n o t required.
In f a c t , i t t u r n s o u t t h a t t h e weak assumptions o f t h e g e n e r a l We w i l l , however, i n c l u d e h i n t s on those
model p e r m i t such a non-uniqueness.
properties o f a general s u b s t i t u t i o n operation t h a t are s u f f i c i e n t f o r i t s i n t e g r a t i o n i n t o t h e g e n e r a l model; t h i s a p p l i e s p a r t i c u l a r l y t o t h e s u b s t i t u t i o n o f Boolean f u n c t i o n s , s e t systems and r e l a t i o n s .
R.H. Mohring and EJ. Radermacher
262 (P2)
AUTONOMOUS SETS
Autonanous sets a r e t h e classes o f congruence p a r t i t i o n s and are r e f e r r e d t o i n t h e l i t e r a t u r e as e.g. bound sets [40], committees [144l,
[lo],
[41],
closed s e t s [58],
11381, e x t e r n a l l y r e l a t e d s e t s [27l,
p a r t i t i v e sets [64],
[153]
[68],
and s t a b l e sets "1411.
[20],
clumps [3],
modules [14],
[9q,
I n t h e f i n i t e case,
autonomous s e t s p r o v i d e a very easy and n a t u r a l way o f v i z u a l i z i n g t h e s u b s t i t u t i o n decanposition, s i n c e they already determine t h e congruence p a r t i t i o n s ( c f .
I n t h e i n f i n i t e case, however, t h i s may no longer be
(S2) and (S2)* below).
t r u e , and congruence p a r t i t i o n s f o r m t h e n a t u r a l n o t i o n f o r v i z u a l i z i n g t h e decanposi t i o n . Given a s t r u c t u r e S we denote t h e system o f autonanous sets by A ( S ) .
While f o r
s e t systems and r e l a t i o n s t h e autonomous s e t s are q u i t e n i c e l y i n t e r n a l l y described, t h i s i s u n f o r t u n a t e l y n o t t h e case f o r Boolean f u n c t i o n s .
However, A ( S )
w i l l f u l f i l some o f t h e f o l l o w i n g p r o p e r t i e s , where t h e standard p r o p e r t i e s ( A l ) , (A2) and (A3) h o l d i n a l l t h r e e cases. (Al)
ia}
E
A(S) for a l l
(A3) L(A4)
CL
a AS, AS
Q
Bn
c
Bn c + I
( ~ 2 ) B,CC A(s), [(A2)* Bi E A ( S ) f o r i
E.
B,C e A ( S ) , B\C f B,C E A ( S ) , 6%
+
I,
cI =>
Bi # fl
A ( S ) ( s o c a l l e d t r i v i a l autonomous s e t s )
-n E:A(S) iEI
and Bi
0, B n C # 0, C\B # 0 0, Bn C 6 , c\B # k3
+
B Uc E A ( S )
E
A(S),
gI Bi
E
A(S)]
B\C E A ( S ) and C\B
E
A(S)
B A C : = ( B \ C ) U (C\B) E A ( S ) ]
For r e l a t i o n s , (A2)* i s a l s o t r u e , which i s n o t g e n e r a l l y t h e case f o r Boolean I f (A4) holds f o r A ( S ) , then A ( S ) i s s a i d t o be
f u n c t i o n s o r set systems. symmetrically closed.
This i s t r u e e.g. f o r Boolean f u n c t i o n s , s e t systems and
symnetric r e l a t i o n s , b u t n o t i n general f o r p a r t i a l orders ( c f . Theorem 4.1.1). Note t h a t ( A 3 ) w i l l n o t be r e q u i r e d f o r the general model discussed i n Section
111; i t w i l l , however, become e s s e n t i a l when i n t r o d u c i n g t h e composition t r e e i n 111.4. For a l l classes o f s t r u c t u r e s considered, autonomy i s a t r a n s i t i v e property. Even stronger, t h e autonanous s e t s o f a s u b s t r u c t u r e SIB o f S are:
(Sl)
A ( S 1 B ) = !C E A ( S )
I
CSB)
f o r each B
EA(S).
Note t h a t w . r . t. the a l g e b r a i c i n t e r p r e t a t i o n o f autonany v i a i n j e c t i v e hanomorA A phisms i n c g , (S1) means t h a t i n c B induces a b i j e c t i o n between t h e autonomous s e t s o f t h e s u b s t r u c t u r e S I B and t h e autonomous sets o f S contained i n B .
(P3)
CONGRUENCE PARTITIONS
We have already h i n t e d a t t h e f a c t t h a t i n u n i v e r s a l algebra congruence p a r t i t i o n s
Substitution decomposition for discrete structures
determine t h e decanpositions.
263
The s e t of a l l such p a r t i t i o n s i s denoted by V ( S )
and i s a subset o f t h e p a r t i t i o n l a t t i c e Z(A) o f AS ordered by f i n e r than -
or
IT'
IT'
c o a r s e r than
o f IT'.I n p a r t i c u l a r ,
c o a r s e s t p a r t i t i o n o f A.
i f each c l a s s o f
IT)
I T :=' {{a)
I
a
EA}
IT
4 IT'
(read
TI
i s c o n t a i n e d i n some c l a s s
IT
i s t h e f i n e s t and n 1 := {A1 i s t h e
I n f a c t , f o r A f i n i t e , i t t u r n s o u t t h a t V ( S ) i s an
upper semimodular s u b l a t t i c e o f Z(A), and t h u s i n p a r t i c u l a r f u l f i l s t h e JordanDedekind c h a i n c o n d i t i o n [13],
[154];
compare Theorem 3.2.5.
A v e r y t y p i c a l f e a t u r e o f t h e s u b s t i t u t i o n decomposition, w i t h no c o u n t e r p a r t i n e.g.
g r o u p t h e o r y , i s t h e s p e c i a l c o n n e c t i o n between congruence p a r t i t i o n s and t h e
autonomous s e t s . out t h a t
IT
I n f a c t , i n t h e f i n i t e case o r f o r a r b i t r a r y r e l a t i o n s i t t u r n s
i s a congruence p a r t i t i o n i f f a l l c l a s s e s of
c o n d i t i o n ( S 2 ) * below).
IT
I
= iLi
i E I}E V ( S )
'TI
( ~ 3 )
(S3):
Li E. A(S) f o r a l l i
=>
6
I
= {Li
I
i
r L i n L . = I f o r i # j= > n = t L 1 ,...,L r , i a l ( r r a A \ U Ljl€V(S) J j=l I} E V ( S ) <=> Li 6 A(S) f o r a l l i E. I]
= iLi
I ~I
i
I}EV(S),
L1 ,... , L rE A ( S ) , [(S2)*
a r e autonomous ( s e e
I n t h e i n f i n i t e case, t h e c o n n e c t i o n i s , i n g e n e r a l ,
somewhat weaker and i s d e s c r i b e d by ( S Z ) , (S2)
TI
[ ( ~ 3 ) *71 = { L
G
:=
:=
U
I T * E V ( S ~ L)~ --3
ioEI,
I
IT*UIL~
0
i e I}
E.
v ( s ) , TI^
G
E v ( s I L ~ )=>
E
IT
icI\{ioIIEV(S)
v(s)]
ie1 i Thus t h e c l a s s e s o f congruence p a r t i t i o n s a r e autonomous s e t s (S1) and f i n i t e l y many d i s j o i n t autonomous s e t s can be extended t o a congruence p a r t i t i o n by means o f singletons (S2).
Also, a l o c a l r e f i n e m e n t o f a congruence p a r t i t i o n by a con-
gruence p a r t i t i o n a s s o c i a t e d w i t h one o f i t s c l a s s e s r e s u l t s i n a new congruence The i n f i n i t e c o u n t e r p a r t s ( S 2 ) * and ( S 3 ) * a r e g e n e r a l l y t r u e f o r
p a r t i t i o n (S3).
O f course, ( S 2 ) g e n e r a l l y means t h a t
r e l a t i o n s , b u t n o t i n t h e o t h e r cases. q u e s t i o n s concerning autonomous s e t s B
A(S) can be t r a c e d back t o q u e s t i o n s con-
E
c e r n i n g congruence p a r t i t i o n s o f t h e form
IT^
:=
\
IB,Ia)
a E A\BI E V ( S ) .
The importance o f congruence p a r t i t i o n s i n a l g e b r a i c t h e o r i e s l i e s i n t h e f a c t t h a t t h e y d e s c r i b e t h e decompositions o f a s t r u c t u r e S, which may be i d e n t i f i e d w i t h t h e q u o t i e n t s t r u c t u r e s S/TI on A ' := A/IT = IBi c l a s s e s o f t h e p a r t i t i o n n = {Bi
I
mentioned s u r j e c t i v e homanorphisms
I
i E I}( i . e . on t h e s e t o f
i e I)E V ( S ) ) o r e q u i v a l e n t l y w i t h t h e above qTI:
A
+
A'.
I n a l l three classes o f structures
we have t h e f o l l o w i n g c o u n t e r p a r t t o ( S l ) : (S4) ( i ) (ii)
IT,U
E
01e
V(S),
IT 4
v ( s / ~ => )
n I T ( u ) := InIT(B) n-'(u'):=rn~'(B')lB't u
=>
I
B u11
6 E
V(S/n)
v(s),
71
4
nIT-1 ( u s )
The f i r s t p a r t o f (54) i s sometimes r e f e r r e d t o as t h e "Theorem o f Induced Homomorphisms".
I t t e l l s us t h a t , g i v e n
IT,U E
V(S) with
TI
<
U,
there i s a s u r j e c t i v e
R. H. M6hring and F.J. Rademzacher
264
homomorphism from 8/71 onto 810.
As a consequence, t h e system o f congruence p a r t i -
t i o n s V(S/a) o f S / n can be (order-isomorphically) i d e n t i f i e d w i t h t h e dual i d e a l
I
IU E V(S)
u 5
TI)
o f V(S)
induced by
T.
This o b s e r v a t i o n i s a s t r o n g instrument
f o r the p r o o f o f the'orems o f t h e Jordan-Holder type ( c f . Theorem 3.3.2). Combining ( S l ) , (S4) and (S5) below, we o b t a i n the c l o s e l y r e l a t e d c o n d i t i o n C
(54) ' ( i )
E
A(S)
n,(C)
=>
E A(S/n)
C ' E A ( s / ~ )=> ~ , ' ( c ' I
(ii)
=A(s)
I n the f i n i t e case ( 8 4 ) ' i s e q u i v a l e n t t o (S4), even w i t h o u t t h e f o l l o w i n g property (S5), which i s needed i n t h e i n f i n i t e case f o r t h i s equivalence and holds f o r the s t r u c t u r e s considered here. C
(85)
E
A(S),
TI
EV(S)
-
[C]a
E A ( S ) , where [C]T
u
:=
[a]a denotes t h e
a d n-completion o f C .
F i n a l l y , we mention two o t h e r p r o p e r t i e s which h o l d ( t r i v i a l l y ) i n the t h r e e classes.
The f i r s t deals w i t h t h e r e s t r i c t i o n T ) B o f a congruence p a r t i t i o n t o an
autonomous s e t B, w h i l e the second i s t h e analogue o f t h e " F i r s t Isomorphism Theorem" [30],
[66]
i n u n i v e r s a l algebra. = i Li
I
i a I } E V ( S ) ==, n l B := i m L i
(S6)
BeA(S),
(S7)
( S I B ) / ( r l B ) i s isomorphic t o ( S ( [B]T)/(T(
(P4)
PRINCIPLES OF INVARIANCE
TI
I
i E I,BnLi#
PIeV(S1B)
NT).
I t turns o u t t h a t t h e systems o f autonanous s e t s remain i n v a r i a n t under c e r t a i n
operations, e.g.
d u a l i s a t i o n , b l o c k i n g o r complementation w i t h i n t h e r e s p e c t i v e
classes o f s t r u c t u r e s . (P5)
ALMOST ALL STRUCTURES ARE PRIME
Contrary t o the s i t u a t i o n f o r important a l g e b r a i c s t r u c t u r e s (e.g. groups), i t t u r n s o u t t h a t the m a j o r i t y of d i s c r e t e s t r u c t u r e s are prime (indecanposable), i.e.
although t h e number o f decomposable s t r u c t u r e s on A = 11 ,.. .,nl may grow
e x p o n e n t i a l l y w i t h n, t h e i r r e l a t i v e frequency w . r . t .
{l, ..., n) tends t o zero when n goes t o i n f i n i t y .
a l l s t r u c t u r e s on A =
This behaviour was observed i n
[99] f o r many d i f f e r e n t classes, such as k-ary r e l a t i o n s , parametric r e l a t i o n s ( i n c l u d i n g graphs, tournaments), c l u t t e r s and p a r t i a l orders, and a l s o f o r Boolean functions, cf.
[136].
Furthermore, i n most o f these cases t h i s behaviour holds
f o r both the l a b e l e d ( d i f f e r e n t s t r u c t u r e s are d i s t i n g u i s h e d ) and unlabeled case (isomorphic s t r u c t u r e s are i d e n t i f i e d ) .
So i f one randomly p i c k s o r generates a
265
Substitution decomposition for discrete structures
s t r u c t u r e on A = 11 ,.
. . ,n}
w.r. t. t h e u n i f o r m d i s t r i b u t i o n on t h e s e t o f s t r u c -
t u r e s on A, t h e n i t w i l l almost c e r t a i n l y b e a prime s t r u c t u r e f o r l a r g e n.
This
b e h a v i o u r t h a t " n i c e " p r o p e r t i e s have an a s y m p t o t i c a l l y v a n i s h i n g f r e q u e n c y i s q u i t e canmon f o r d i s c r e t e s t r u c t u r e s , c f . f o r example t h e i n v e s t i g a t i o n s on prop e r t i e s o f almost a l l graphs [21].
I t does, however, n o t n e c e s s a r i l y mean much
f o r t h e p r a c t i c a l a p p l i c a b i l i t y o f t h e s u b s t i t u t i o n decomposition ( n o t e e.g. almost a l l p a r t i a l o r d e r s ( n e t w o r k s ) have a l e n g t h o f a t most t h r e e [86]). f a c t , e x p e r i e n c e w i t h a p p l i c a t i o n s (e.g.
that In
i n s w i t c h i n g t h e o r y o r network t h e o r y )
i n d i c a t e s t h a t t h e u n i f o r m d i s t r i b u t i o n i s an u n r e a l i s t i c measure, s i n c e t h e The reason may be t h a t
s t r u c t u r e s encountered a r e v e r y f r e q u e n t l y decomposable.
i n s w i t c h i n g t h e o r y , Boolean f u n c t i o n s o c c u r r i n g a r e o f t e n generated b y symmetric i n t e r n a l c o m p o s i t i o n laws which f a v o u r t h e e x i s t e n c e o f autonomous s e t s [40)
,
w h i l e i n a p p l i c a t i o n s o f e.g. p a r t i a l o r d e r s ( p r o j e c t n e t w o r k s ) , h i e r a r c h i c a l p l a n n i n g techniques, proceeding from one l e v e l t o another, n a t u r a l l y i n v o l v e substitution.
(P6)
UNIQUE FACTORIZATION RESULTS
The s t r o n g e s t r e s u l t s f o r t h e s u b s t i t u t i o n decomposition (which i n most aspects a l s o e x t e n d t o t h e more g e n e r a l s p l i t decomposition, c f . 1.6 and 111.5) have been "uniqueness" r e s u l t s o f c e r t a i n f a c t o r i z a t i o n s , which i n c l u d e those o f t h e JordanH o l d e r t y p e , and i m p l y e.g. t h e independence o f m u l t i - s t e p d e c a n p o s i t i o n f r o m t h e o r d e r and s t a r t i n g p o i n t o f t h e s t e p s .
These r e s u l t s a r e n o t o n l y t h e h i g h l i g h t s
o f the t h e o r e t i c a l treatment b u t are e q u a l l y important f o r p r a c t i c a l applications.
A t y p i c a l s i t u a t i o n ( l a t e r t h e b a s i s f o r t h e c o m p o s i t i o n t r e e B(S)), i s : e i t h e r t h e r e i s a c o a r s e s t n o n - t r i v i a l congruence p a r t i t i o n , i . e . V(S)\{vl}
a g r e a t e s t element i n
meaning t h e r e i s a u n i q u e way o f silmultaneously c o n t r a c t i n g a l l maximal
n o n - t r i v i a l autonomous s e t s , o r t h e s t r u c t u r e S has q u o t i e n t s which a r e e x t r e m e l y s p e c i a l , i . e . "degenerate" o r " l i n e a r " i n t h e f o l l o w i n g sense:
D e f i n i t i o n : A s t r u c t u r e S on A i s c a l l e d degenerate, i f each non-empty subset o f i f A ( S ) = P(A)\{P)}, where P(A) denotes t h e power s e t o f A.
A i s S-autonomous, i . e .
S i s c a l l e d l i n e a r i f t h e r e e x i s t s a l i n e a r o r d e r < on A such t h a t A ( S ) i s t h e
s e t A(,<)
o f convex subsets ( o r e q u i v a l e n t l y o f 4-autonomous s e t s o f (A,s)).
Here B i s convex, i f
a
r
a~ B and
CY
6 y 6
B i m p l i e s Y e 6.
For t h e t h r e e t y p e s o f s t r u c t u r e s we g i v e c a n p l e t e c h a r a c t e r i z a t i o n s o f degenerate and l i n e a r s t r u c t u r e s ( c f . Theorem 4.1.2,
4.1.3,
4.2.3,
4.2.4,
and 4.3.1).
For
i n s t a n c ? , degenerate r e l a t i o n s a r e ( u p t o r e f l e x i v e t u p l e s ) empty o r complete,
266
R.H. Mohring and F.J. Radermacher
degenerate c l u t t e r s c o n s i s t e i t h e r o f s i n g l e t o n s o r t h e base set, and degenerate Boolean f u n c t i o n s a r e ( u p t o complementation o f v a r i a b l e s ) o f t h e type F(x l,...,xn)
= x1
* ... *
x,,
where
product ( - ) o r r i n g sum a d d i t i o n
*
stands f o r e i t h e r Boolean sum (+), Boolean
(0).
The f a c t t h a t s i m i l a r r e s u l t s h o l d f o r Boolean f u n c t i o n s , s e t systems and r e l a t i o n s [40],
[138],
[26],
[24],
[38],
was an i n d i c a t i o n t o t h e e x i s t e n c e o f a more
general framework i n t e g r a t i n g a l l these cases as developed i n S e c t i o n 111.
As the
f o r m u l a t i o n i n t h i s p a r t i s r a t h e r involved, however, we postpone d i s c u s s i o n o f (P6) f o r the special cases t o Section IV.
Example 1.1.1:
Consider the
b i l i t y graph [64],
i.e.
G i n F i g u r e 1.1, which i s i n f a c t a compara-
a graph whose edges can be o r i e n t e d i n such a way t h a t the
r e s u l t i n g r e l a t i o n on the vertex s e t i s a p a r t i a l order.
( E q u i v a l e n t l y , two
v e r t i c e s are j o i n e d by an edge i f f they are comparable i n t h e associated p a r t i a l order).
Such a t r a n s i t i v e o r i e n t a t i o n i s a l s o g i v e n i n Figure 1.1 together w i t h
an a c t i v i t y - o n - a r c diagram o f t h e associated p a r t i a l order o. This i s a t y p i c a l form of r e p r e s e n t a t i o n o f p a r t i a l orders i n p r o j e c t scheduling [48],
[77],
which
i s very u s e f u l f o r i d e n t i f i c a t i o n o f autonomous sets, c f . t h e h i n t s below and a t the end o f 1.4.
In t h i s representation, edges correspond t o t h e elements
( a c t i v i t i e s ) o f t h e p a r t i a l l y ordered s e t and d i r e c t e d path from a t o 6.
CI
&eBmeans t h e existence o f a
[In most cases such a r e p r e s e n t a t i o n r e q u i r e s t h e
i n t r o d u c t i o n o f d m y edges ( c f . t h e dashed edge i n Fig. 1 . 1 1
G
l ) $ ) 0 9 10
b’)o
10
S
5
F i g u r e 1.1:
10
12
11
7
A c o m p a r a b i l i t y graph G, a t r a n s i t i v e o r i e n t a t i o n and t h e associated p a r t i a l order o
Consider f u r t h e r the (conformal) c l u t t e r C ( G ) o f c-maximal c l i q u e s o f G, here given by C = i il,2,51 ,[1,3,5} {lO,ll,lZ}j,
,11 ,4,5) ,{6,7,91,{6,7 ,lZ} ,{6,8,9)
,{6,8,12},
a s w e l l as t h e (monotonic) Boolean f u n c t i o n F: 10,1}12
+
I0,lI
267
Substitution decomposition for discrete stntctiires
defined by F(x 1y...,x12)
= 1
i f f t h e r e i s C €C(G) w i t h xi = 1 f o r a l l i E C .
We f i r s t ask f o r t h e autonomous s e t s f o r a l l f o u r s t r u c t u r e s G,o,c,F,
where
autonany means "uniform behaviour t o t h e o u t s i d e " and i s p r e c i s e l y d e f i n e d f o r each case i n 1.2-1.4.
Due t o t h e i n t e r f a c e s d e a l t w i t h i n 1.5, i t i s A(G) = A(C)
= A(F) and t h i s i d e n t i t y extends ( b y t h e p r i n c i p l e s o f i n v a r i a n c e
(P4)) t o t h e
canplementary graph o f G, t h e b l o c k e r and t h e a n t i b l o c k e r of C ( c f . Theorems 1.3.7 t h e d u a l f u n c t i o n of
and 1.3.8),
F, and so f o r t h .
Also, A(G) e s s e n t i a l l y equals
A(o), except f o r sane symnetric d i f f e r e n c e s , c f . Theorem 15.1.
A(o) i s e a s i l y
determined v i a t h e g i v e n a c t i v i t y - o n - a r c diagram, s i n c e B a A(@) i f f t h e diagram r e s t r i c t e d t o B has e x a c t l y one source and e x a c t l y one s i n k and i s connected t o t h e r e s t o f t h e diagram by these two v e r t i c e s o n l y c77], {1,2,3,4}
A(o), b u t n o t {4,5}.
6
[80].
The whole system A(G) = A(C) = A(F), ordered by
i n c l u s i o n , i s g i v e n below i n F i g u r e 1.2 by i t s Hasse-diagram. A(G) = A(@) U {1,51,
F i g u r e 1.2:
where {1,5} = {1,2,3,4}~{2,3,4,5)
TI
TI
Q'
and F ' accordingly.
Further, l e t G,,
TI,
i = 1 ,...,
61
C = C'[Ci,
(cf. Theorem 1.5.1).
= {11 , 2 , 3 , 4 ~ ; ~ 5 1 ; { 6 , 7 , 8 } ~ ~ 9 ~ ; ~ 1 0 , 1 1 ~ ; ~o1f 2 ~ ~
We w i l l see t h a t
The q u o t i e n t s G ' and classes o f
Note t h a t
The system A(G) = A(C) = A(F), ordered by i n c l u s i o n
Now consider the p a r t i t i o n
A = {l, ...,121.
Therefore
i s a congruence p a r t i t i o n i n a l l f o u r cases.
are g i v e n i n F i g u r e 1.3, and d e f i n e ( t h e q u o t i e n t s ) C '
and l e t oi,Ci
...,G6
be the r e s t r i c t i o n s o f G t o t h e Then G = G'[Giy
and Fi be s i m i l a r l y defined.
i n t h e sense o f t h e X - j o i n , Q = o ' [ o .1 ' i
i = 1 ,...,61 and f i n a l l y F = F'[Fi,
=
1
,...,61,
i = 1,..., 61, i . e . a l l these f o u r
s t r u c t u r e s are represented n o n - t r i v i a l l y by means o f s u b s t i t u t i o n .
R.H. Mohring and F.J. Radermacher
268
G'
( a ' ) = G/n
I. a
e
f
c
d
e
F i g u r e 1.3: The q u o t i e n t s G/n and o / n
- Note t h a t f o r A(S) the properties (Al) - (A3) a r e f u l f i l l e d .
In a d d i t i o n (A4)
i s t r u e except f o r 0.
- Also note the t r a n s i t i v i t y of being autonomous ( S l ) , e.g. A(G\{6,...y12}) [C ( 6 , ...,1'2)IC E A(G)). - As was demonstrated above, n i s a congruence p a r t i t i o n . Consequently the
-
-
=
c l a s s e s of TI a r e autonomous. On the o t h e r h a n d , every non-trivial p a r t i t i o n i n t o autonomous sets w i l l allow a non-trivial s u b s t i t u t i o n representation, i . e . the f i n i t e version of ( S 2 ) i s valid. Obviously, TI* = {{1);{2,31;{4)1 i s a congruence p a r t i t i o n o f G1{1,2,3,4}. This implies t h a t the refinement = ~~1~;~2y3~;~4~;~5~;~6,7,8~;~9~;~1 o f TI i s i t s e l f a congruence p a r t i t i o n ; t h i s i s exactly the statement o f (S3). The canonical mapping ,TI from G onto G/n i s as follows: u e A
5
1 2 3 4 I
n,(.)
a
6 7 8
b '
9 I
1
c
I
' d l
1012 e
12 I
f
S t a r t i n g from a = ~11,2,3,4,51;16,7,8,9y10y11y1211 3 TI, we have u ' c V(G/a) as w e l l as n,,-'(u')
{c,d,e,fll:=
A l s o , nn-'(
-
= u
r V(G),
~ ~ ( =0 {{a,bl; )
which i s s t a t e d i n (S4).
i a , b l ) E A(G), r e f l e c t i n g t h e c o n t e n t o f ( S 4 ) ' .
F i n a l l y , we have [{2,3,4,51]~
= I1,2,3,4,5l
and t h e f a c t t h a t (GlI2,3,4,51)/I{2,3,41;
((1,2,3,41;(5)1
c A ( G ) , i . e . t h e statement o f (S5), {511 i s i s a n o r p h i c t o (G({1,2,3,4,51)/
(which i n any case i s isomorphic t o G'I{a,b)),
demonstrating
t h e c o n t e n t o f (56) and (S7). (Of course, t h e l a s t t h r e e o b s e r v a t i o n s a l s o h o l d f o r t h e o t h e r s t r u c t u r e s i n s t e a d o f 6).
269
Substitution decomposition for discrete structures
1.2
SUBSTITUTION DECOMPOSITION FOR BOOLEAN FUNCTIONS
F o r h i s t o r i c a l reasons, we s t a r t w i t h t h i s c l a s s o f s t r u c t u r e s though t h e r e a r e
no p r e v i o u s r e s u l t s on t h e congruence p a r t i t i o n s o r on Jordan-Holder t y p e theorems f o r t h i s case.
T h i s may be due p a r t l y t o t h e a p p l i c a t i o n s i n t e r e s t i n g h e r e and
p a r t l y t o a m i s s i n g i n t e r n a l d e s c r i p t i o n o f autonomy. r e s t r i c t e d t o t h e f i n i t e case o f f u n c t i o n s F: I 0 , 1 l n
+
A l l considerations are I 0 , l l . (We would,
however, l i k e t o m e n t i o n t h a t sane o f t h e n a s t y e f f e c t s a r i s i n g f o r i n f i n i t e s e t systems (compare Example 4.2.2)
s t r a i g h t f o r w a r d l y c a r r y over t o Boolean f u n c t i o n s
w i t h i n f i n i t e l y many v a r i a b l e s ) .
Furthermore, we r e s t r i c t o u r s e l v e s t o f u n c t i o n s
w i t h o u t i n e s s e n t i a l v a r i a b l e s (known t o i m p l y no loss o f g e n e r a l i t y w . r . t
[4q ) .
decanposi t i o n
I n d e t a i l , f r o m now on we assume t h a t A = 11
,...,n l
denote t h i s by x A = (xB,xc) and F ( x A ) = F(xB,xc). and any 0
-
and i d e n t i f y i w i t h t h e we w i l l
Furthermore, where we s p l i t up A i n t o d i s j o i n t s e t s B,C,
v a r i a b l e xi.
1 v e c t o r chB
Furthermore, g i v e n any B S A
o f constants, we denote by F I i A \ B t h e s u b f u n c t i o n
FBCA\B(xB) := F ( X ~ , C ~ \o~f )F on B, induced b y B and t h e c o n s t a n t s c * \ ~ .
The
b a s i c d e f i n i t i o n i s t h e n as f o l l o w s :
Definition: A' = 11
m
A = .U Bi. 1=1
i = 1, ...,m, be Boolean f u n c t i o n s , where
L e t F ( x A , ) and F(xgi),
,...,m }
and t h e s e t s Bi,
i = 1 ,...,m a r e m u t u a l l y d i s j o i n t .
The Boolean f u n c t i o n d e f i n e d by
i s c a l l e d t h e c a n p o s i t i o n o f t h e f u n c t i o n s F ' and Fi, b y F = FIEFi, yi,
Put
i E. A'].
i = 1 ,... ,m,
i = l,...,m
and i s denoted
F i s said t o be obtained b y s u b s t i t u t i o n o f the variables
i n F ' by t h e f u n c t i o n s Fi,
The c o m p o s i t i o n i s proper i f
IA'I
>
i = 1 ,...,m.
1 and I B i I
>
1 f o r some i E A ' .
A Boolean
f u n c t i o n i s s a i d t o be decomposable i f i t has a r e p r e s e n t a t i o n as a proper composition.
Otherwise, i t i s s a i d t o be indecomposable o r
Definition:
prime.
L e t F be a Boolean f u n c t i o n .
a) A p a r t i t i o n T = {Bi I i E: I 1 o f A i s c a l l e d a congruence p a r t i t i o n o f F if ) such t h a t t h e r e e x i s t Boolean f u n c t i o n s F i ( ~ ~ i ) , and a Boolean f u n c t i o n F ' ( y A/. F ' i s c a l l e d t h e q u o t i e n t o f F modulo 71 and i s denoted by F / r . F = F ' [Fi, i E I]. V(F) denotes t h e system o f congruence p a r t i t i o n s o f F. b)
A subset B o f A i s c a l l e d an (F-)autoncmous s e t i f t h e r e is
T 6
V(F) w i t h
R. H. Mehring and F. J. Rau'crmacker
270 B
A(F) denotes t h e system o f a l l F-autonanous s e t s .
0 7.
We discuss how r e s u l t s a v a i l a b l e f r a n l i t e r a t u r e imply, w . r . t . g i v e n above, a l l t h e aspects ( P l )
-
(P5) mentioned i n 1.1.
the definitions
The d e f i n i t i o n s
already r e f l e c t t h e p r a c t i c a l m o t i v a t i o n f o r decomposition and t h e n o t i o n o f subO f course, t h e p r a c t i c a l i n t e r e s t i n decompo-
s t i t u t i o n f o r t h i s case ( ( P l ) ) .
s i t i o n u s u a l l y goes the other way round, and c o n s i s t s i n r e p r e s e n t i n g a g i v e n Boolean f u n c t i o n as a composition o f simDler Boolean f u n c t i o n s , a b a s i c problem i n SWITCHING THEORY.
Many approaches, and n o t o n l y t h e above d i s j u n c t i v e approach,
have heen t r i e d (cf. e.g. [4q, studied by e.g. Ashenhurst [4]
[41],
[8g ). I n our context, the d i s j u n c t i v e approach
and C u r t i s [40],
e x a c t l y t o the s u b s t i t u t i o n deconposition. instance ( c f . [40, Cor. 3.1 A]) t h e f u n c t i o n s Fi,
i s c h a r a c t e r i s t i c and leads
I n t h e i r work they showed f o r
i E A'],
t h a t g i v e n a r e p r e s e n t a t i o n F = F'[Fi,
i a A ' , are determined up t o complementation ( o f t h e f u n c t i o n s )
and t h a t F ' i s determined up t o complementation o f v a r i a b l e s . choice o f F' determines t h e Fi,
i E A',
However, each
uniquely, and vice-versa.
I n order t o
o b t a i n the uniqueness o f the q u o t i e n t f u n c t i o n s we s h a l l i n t h e f o l l o w i n g (as i s usual i n t h e decomposition o f s w i t c h i n g f u n c t i o n s ) regard Boolean f u n c t i o n s as
equal
i f one can be obtained from t h e o t h e r by complementation o f v a r i a b l e s .
I t i s easy t o see ( c f . a l s o [40])
t h a t t h e autonomous s e t s o f a Boolean f u n c t i o n
F are t h e bwnd s e t s o f F considered i n s w i t c h i n g theory (which are d e f i n e d as sets B occurring i n a s o - c a l l e d simple d i s j u n c t i v e decomposition F =
c f - [40])-
G(H(XB).XA\B),
The p r o p e r t i e s o f bound s e t s then y i e l d ( A l )
-
(A3) f o r A(F) and show f u r t h e r m o r e
t h a t A(F) f u l f i l l s (A4), i . e . i s symmetrically closed ( c f . 140, Th. 4.4-4.53, [41,Th.
11.131).
( S l ) i s r a t h e r o b v i w s and f o l l o w s , f o r instance, from the r e p r e s e n t a t i o n o f a simple d i s j u n c t i v e decomposition o f a Boolean f u n c t i o n and i t s complement by means o f the Ashenhurst decanposition c h a r t , c f . [4]. Because o f ( S l ) and t h e f i n i t e n e s s , (S2) and (S3) reduce t o t h e p r o p e r t y : = {Bi
1
i ri I 1 E V(F)
B d A(F) f o r a l l i € 1 . This i s e x a c t l y t h e i a s s e r t i o n t h a t m u l t i p l e d i s j u n c t i v e decompositions can be i t e r a t i v e l y obtained by II
<=>
simple d i s j u n c t i v e decomposition, c f . [40,Th.
4.7-4.8).
I n t h e f i n i t e case, (S5) f o l l o w s frcm (A2) and (S2), because of ( 5 2 ) .
d e c m p o s i t i o n o f F and ( i . e . qil(Ci})
=
and (S4) reduces t o ( S 4 ) '
I n order t o show ( S 4 ) ' , l e t C a A ( F ) , F = F'[FiJ
TI^
be t h e n a t u r a l mapping associated w i t h
Ai f o r a l l i E A ' ) .
Because o f ( S 5 ) ,
i TI
E
A']
= {Ai
be a
I
i r A')
Substitution deconiposition for discrete structures
B : = [C]T
and A\C =
=
u
A. = q-'(QB))
Ai'lCfP)
Ak+,U
1
L e t w.1.o.g.
B =
AIU ...
TI
... U.,,A,
Boolean f u n c t i o n s
i s F-autonanous.
27 1
u Ak
A p p l y i n g well-known r e s u l t s on t h e decomposition o f 4.7-4.81) one o b t a i n s t h e f o l l o w i n g r e p r e s e n t a t i o n
(141, [40,Th.
of F w i t h s u i t a b l e Boolean f u n c t i o n s G, H,Fi:
w i t h H(xB) = F ( x6 ' c Ak+ly*'''c$,
),
=: H ' ( F ( X
..., F
) f o r suitable 0
P u t t i n g yi := F . ( x
'
1 cA.. J
)).
(x
A1
-
Ak
one o b t a i n s
. ) y
A,
F ' (Y1 1.
which shows t h a t I$C)
* *
%Ykdk+l
9 -
= n,(B)
*
* Y Y ~=) G(H'(Y19..
- dk)dk+i
I . .
. sYm)
9
= {l, ...,k l i s F'-autonomous.
The second a s s e r t i o n of ( S 4 ) ' f o l l o w s i m m e d i a t e l y f r o m c40, p. 292).
F i n a l l y , (S6)
and ( 5 7 ) h o l d t r i v i a l l y . Concerning (P4), t h e i n v a r i a n c e s of t h e s u b s t i t u t i o n decomposition a l r e a d y d i s cussed g i v e t h e i n v a r i a n c e o f A(F) w.r.t. plementation o f the function. i s t h e d u a l f u n c t i o n o f F, i . e .
complementation of v a r i a b l e s and
corn-
Thus, i n p a r t i c u l a r , we have A(F) = A(F*), where F* F*(x
,,...,xn)
=
F(?,
,...,xn),
where o f course
F = (F*)*. F i n a l l y , concerning (P5), Shannon [136]
proved, t h a t ( i n t h e sense e x p l a i n e d i n
(P5) above) "almost a l l " l a b e l e d Boolean f u n c t i o n s a r e prime. whether t h i s remains t r u e f o r t h e u n l a b e l e d
I t i s n o t y e t known
case, b u t t h i s seems t o be so.
We
f u r t h e r m e n t i o n t h a t "most" Boolean f u n c t i o n s o c c u r r i n g i n a p p l i c a t i o n s o f s w i t c h i n g t h e o r y seem t o be decomposable [440). We w i l l c l o s e 1.2 w i t h some more h i n t s on a p p l i c a t i o n s . t o switching theory i s self-evident.
O f course t h e a p p l i c a t i o n
S u r p r i s i n g l y , however, connections w i t h
o t h e r f i e l d s have a l s o been found, due t o t h e c o i n c i d e n c e o f monotone ( i n c r e a s i n g ) Boo1 ean f u n c t i o n s and c l u t t e r s ( o r m o n o t o n i c a l l y c l o s e d dependence systems) ; c f . e.g.
t h e connection between C and F i n Example 1.1.1.
We w i l l d i s c u s s t h i s b a s i c
i n t e r f a c e i n 1.5 b u t h e r e m e n t i o n t h a t monotone Boolean f u n c t i o n s a r e c l o s e d under substitution.
Even s t r o n g e r , g i v e n a r e p r e s e n t a t i o n F = F'[Fi,
i €A']
of a
monotone Boolean f u n c t i o n F, t h e r e e x i s t monotone Boolean f u n c t i o n s G ' , Gi, d e f i n e d on t h e same s e t s o f v a r i a b l e s , such t h a t F = G ' [Gi,
i e A'].
i I n other
words, decomposition o f monotone Boolean f u n c t i o n s w i t h i n t h e c l a s s o f monotone
A',
R.H. Mohring and F.J. Radennacher
272
Boolean f u n c t i o n s i s e s s e n t i a l l y the same as w i t h i n the c l a s s o f functions
1.3
fi Boolean
.
SUBSTITUTION OECCMPOSITION FOR SET SYSTEMS
Set systems are systems T o f subsets o f a g i v e n base s e t A. p o s i t i o n , w.1.o.g.
With regard t o decan-
one can r e s t r i c t oneself t o s o - c a l l e d normal s e t systems, i . e .
systems T f u l f i l l i n g
u
TEJ
T = A.
Set systems belong t o t h e most i n t e r e s t i n g and
r i c h e s t classes o f s t r u c t u r e s i n d i s c r e t e mathematics, due p a r t i c u l a r l y t o the many i n t e r e s t i n g subclasses o f t h i s c l a s s .
(IS) [88],
Among them are independence systems
i.e. s e t systems T where 8 ' 5 B, B c T i m p l i e s B ' e T, dependence
systems ( D S ) , i. e . s e t systems T where
B
C_
B'
,B eT
implies B ' a T , regular
independence systems and the more s p e c i a l t h r e s h o l d independence systems [76] with B e T iff e.g.
r_ a d
w(a) c K ) .
u I01 and a t h r e s h o l d v a l u e
K E. W O f importance f o r c a n b i n a t o r i a l o p t i m i z a t i o n are
(meaning the existence o f a weighting w: A
-t
IN
the q u i t e s p e c i a l m a t r o i d s and, o f course, c l u t t e r s
[lo],
[47]
( i . e . systems
o f m u t u a l l y incanparable subsets o f a base s e t A ) , which i n i t i a t e d most research on decomposition o f s e t systems.
Often, problems concerning (in-)dependence
systems can be traced back t o c l u t t e r s by going over t o t h e (c-maxima1)g-minimal elements ( s o - c a l l e d bases) o f t h e (in-)dependence system. c l u t t e r s deserve special a t t e n t i o n , e.g. clutters w.r.t.
s e t i n c l u s i o n LBl],
C e r t a i n classes o f
the c-maximal and c-minimal (normal)
where t h e former class covers a l l k-out-of-n
s y s t m s i n r e l i a b i l i t y theory [73, w h i l e the l a t t e r covers p a r t i t i o n s . a t t e n t i o n w i l l then a l s o be devoted t o conformal c l u t t e r s [8],
Particular
d e f i n e d as the
systems o f c-maximal c l i q u e s o f u n d i r e c t e d graphs, c f . Example 1.1.1.
I n fact,
t h i s class o f c l u t t e r s may e s s e n t i a l l y be i d e n t i f i e d w i t h graphs and leads t o a b a s i c type o f i n t e r f a c e i n 1.5. For a r b i t r a r y s e t systems, t h e r e i s o n l y work a v a i l a b l e on t h e f i n i t e s p l i t decomposition [34], c l u t t e r s [16]. sets (e.g.
[38j and p a r t l y on the s u b s t i t u t i o n decomposition o f i n f i n i t e
Otherwise most research i s r e s t r i c t e d t o c l u t t e r s on f i n i t e base
theorems o f the Jordan-Holder type and unique f a c t o r i z a t i o n theorems
(P6)). One o f t h e aims o f t h i s paper i s t o i n c l u d e the i n f i n i t e case, and a r b i t r a r y s e t systems. While o n l y r a t h e r "weak" p r o p e r t i e s remain v a l i d , they are s u f f i c i e n t f o r a l l t h e e s s e n t i a l r e s u l t s and m o t i v a t e t h e general model o f Section
111.
Somehow s u r p r i s i n g l y i t turned o u t t h a t a r b i t r a r y s e t systems could be
handled more e l e g a n t l y than c l u t t e r s , and so we present ( P l )
-
(P5) f o r t h i s case,
and remark on s p e c i a l i z a t i o n f o r the sub-classes mentioned above.
A n a t u r a l m o t i v a t i o n f o r t h e d e f i n i t i o n o f s u b s t i t u t i o n ( P l ) , a r i s e s from a p p l i c a t i o n s o f f i n i t e c l u t t e r s , e.g. from simple n-person games i n GAME THEORY, where
Substitution decomposition for discrete structures t h e c l u t t e r o f $-minimal, [137]
,
[138]
213
w i n n i n g c o a l i t i o n s i s t o be d e s c r i b e d .
Here, Shapley
asked f o r a d i s j u n c t i v e d e s c r i p t i o n by means o f o t h e r s i m p l e games
w i t h s m a l l e r s e t s o f p l a y e r s , y i e l d i n g t h e subsequent n o t i o n o f s u b s t i t u t i o n f o r clutters.
The a s s o c i a t e d autonomous s e t s , h e r e s u g g e s t i v e l y termed committees,
may be i n t e r n a l l y c h a r a c t e r i z e d b y an exchange p r o p e r t y , g i v e n below.
The same
d e c o m p o s i t i o n p o s s i b i l i t i e s (termed modules) o c c u r f o r coherent systems i n RELIABILITY THEORY [7],
[14].
T h i s i s discussed h e r e as i n t e r f a c e i n 1.5.
Furthermore, i n t r y i n g t o d e s c r i b e a r i s i n g p o l y t o p e s ( c f . [6],
[36])
and t o
f a c t o r i z e o p t i m a l v a l u e f u n c t i o n s i n CQYBINATORIAL OPTIMIZATION o v e r ( i n - ) d e p e n dence systems and c l u t t e r s , t h e same decompositions a r e a g a i n f o u n d [81],
For f u r t h e r a p p l i c a t i o n s see [15],
c f . S e c t i o n 11.
[lo],
[ll],
The b a s i c
[19].
d e f i n i t i o n s are:
L e t T' be a ( n o r m a l ) s e t system on A ' and f o r each
Definition:
( n o r m a l ) s e t system on A
6'
where t h e s e t s A
B
E
A ' l e t TB be a
a r e non-empty and p a i r w i s e d i s j o i n t .
Then t h e ( n o r m a l ) s e t system T :=
{U
TB
BET '
I
T'
E
T', T
B
e T
B
i s c a l l e d t h e c o m p o s i t i o n o f T ' and TB,
f o r each B e T ' 1 on A := E
u
@eT' AB
A ' , and i s denoted by T = T ' [TB, B €A'].
T i s a l s o s a i d t o be o b t a i n e d by s u b s t i t u t i o n o f t h e elements 6 e A ' by t h e s e t
systems T g i n
T I .
The c a n p o s i t i o n i s proper i f I A ' 1 > 1 and ] A B ( > 1 f o r sane B
E
A'.
A s e t system
T i s s a i d t o be decomposable i f i t has a r e p r e s e n t a t i o n as a proper c a n p o s i t i o n .
Otherwise, i t i s s a i d t o be indecomposable o r p r i m e .
Definition:
L e t T be a s e t system on A.
a ) A p a r t i t i o n T = {Bi I i E I}o f A i s c a l l e d a congruence p a r t i t i o n o f T i f i h r e e x i s t s e t systems Ti, i E I, on Bi and a s e t system T ' on A/T such t h a t T = T ' [Ti,
Bi E A/T].
I n t h i s case, T ' i s c a l l e d t h e q u o t i e n t o f T modulo
II
and
i s denoted by T / T I . V ( T ) denotes t h e system o f congruence p a r t i t i o n s o f T.
b ) A subset B o f A i s c a l l e d a (T-)autonomous s e t i f t h e r e i s T E V ( T ) w i t h B E IT. I n t h i s case, t h e s e t system T l B : = { T n B I T 6 T, T n B # 01 i s c a l l e d t h e autonanous subsystem o f T induced b y B.
A(T) denotes t h e system o f a l l
T-au tonomou s s e t s . D i f f e r e n t from Boolean f u n c t i o n s , congruence p a r t i t i o n s and autonomous s e t s of s e t systems have n i c e i n t e r n a l c h a r a c t e r i z a t i o n s ,
which f o l l o w immediately fran
R.H. Mohring and EJ. Radermacher
274 the above d e f i n i t i o n s .
Lemna 1.3.1:
Let T be a s e t system on A .
a) A p a r t i t i o n ~i = {Bi T E T and f o r a l l (Ti)ieI
belongs t o T as w e l l .
I
E I } o f A i s a congruence p a r t i t i o n o f T i f f f o r each c_ T w i t h Ti Bi # fl t h e s e t
i
n
(T* i s obtained by exchanging each non-empty subset
Tn Bi
o f T f o r Ti (I Bi and i s denoted by T* = Ex(T,n, (Ti)ieI).)
n
B # fJ, b ) A subset B o f A i s T-autonomous i f f f o r a l l T1 ,T2 r T w i t h Ti i = 1,2, a l s o T* := (T1\B) U ( T 2 n 6 ) E T . ( T * i s obtained by exchanging
Tln
B 6 TI f o r
T2nB
and i s denoted by Ex(T1,B,T2).)
A f u r t h e r imnediate consequence o f these i n t e r n a l c h a r a c t e r i z a t i o n s are t h e prop e r t i e s (52) and (S3), where (S3) even holds i n the stronger, i n f i n i t e v e r s i o n (S3)*. With regard t o t h e s p e c i a l s e t systems considered above, we say t h a t a class S o f
s t r u c t u r e s i s closed w . r . t .
decomposition i f a l l q u o t i e n t s and autonomous sub-
s t r u c t u r e s o f s t r u c t u r e s o f S a l s o belong t o S, and t h a t i t i s closed w . r . t . canposition i f the composition o f s t r u c t u r e s o f S belongs a l s o t o S. I t i s e a s i l y v e r i f i e d t h a t normal s e t systems, independence systems, c l u t t e r s and
conformal c l u t t e r s a r e closed w . r . t .
b o t h decomposition and composition, whereas
r e g u l a r or t h r e s h o l d independence systens, matroids (viewed as independence systems), and minimal o r maximal c l u t t e r s are i n general o n l y closed w . r . t . positions.
decom-
This shows, furthermore, t h a t i n a l l these cases t h e decomposition
p o s s i b i l i t i e s w i t h i n t h e r e s t r i c t e d c l a s s remain the same as w i t h i n the class o f
a l l set
systems.
(There i s another, more i m p o r t a n t a p p l i c a t i o n o f the s u b s t i t u -
t i o n decanposition t o matroids, which i s based on i t s "path c l u t t e r " , c f . t h e end o f Section IV.2. I n t h i s i n t e r p r e t a t i o n , m a t r o i d s a r e closed w . r . t . s i t i o n and composition, c f . 0 7 1 , [38]).
decompo-
We now t u r n t o t h e v e r i f i c a t i o n o f the o t h e r c o n d i t i o n s mentioned i n ( P 2 ) and (P3). I n f a c t , t h e i r v e r i f i c a t i o n i s r a t h e r complicated, due p a r t l y o f course t o the c o n s i d e r a t i o n o f the i n f i n i t e case. On the o t h e r hand, we o b t a i n u s e f u l i n s i g h t s i n t o the s t r u c t u r e o f decomposable s e t systems.
275
Substitution decomposition for discrete structures Lemma 1.3.2
D2 : = B1
L e t B1,B2 € A ( T ) o v e r l a p and l e t D1 : = B1 \ B2,
(EXTENSION LEFMA):
n B2 and D3
:= B2
\
B1.
Assume f u r t h e r t h a t some To
E
T meets a t l e a s t
i = 1,2,3. Then each T E T which meets BIU B2 b u t n o t Di can be extended on Di t o T u ( T ' n Di) f o r each T ' T w i t h T ' n Di # 0, i ~ { 1 , 2 , 3 } .
two o f t h e Di,
Proof:
The p r o o f i s s t r a i g h t f o r w a r d b u t l e n g t h y , s i n c e many s i m i l a r cases have
t o be d i s t i n g u i s h e d .
We show two o f t h e s e cases, t o i n d i c a t e t h e p r o o f method.
T meets D1 and D3, b u t n o t D2.
Case 1 :
r T, i
Then Ti := Ex(T,Bi,T') Case 2:
= 1,2.
TU ( T ' n
Thus a l s o
T meets o n l y D1 and i s t o be extended on D2to T
We show f i r s t t h a t t h e r e i s T*
U
( T ' n D2).
By Then To can be extended t o To*,
( T h i s f o l l o w s from t h e cases i n which a g i v e n T meets
which meets a l l t h r e e Di.
Since T meets o n l y D1, T* := Ex(T,B2,T:)
c f . Case 1 ) .
E T.
T w i t h T c T * which meets a l l t h r e e Di.
E
assumption t h e r e i s T o e T which meets two Di. two Diy
D2) = Ex(T2,B2,T1)
meets a l l Oi and
c o n t a i n s T.
With t h i s p r e l i m i n a r y s t e p , we o b t a i n t h a t T1 := Ex(T*,B2,T') T
U ( T ' n D2)
= Ex(T,B1,T1)
E:
Di
.
Tn ( B I U
Lemma 1.3.3.
D2 := B1 T2
E
n B2
B2) #
T and t h u s a l s o
T. m
F o r c l u t t e r s , t h i s lemma reduces t o Lemma 3.2 i n [106], T E T with
E
which s t a t e s t h a t a l l
e i t h e r a l l meet e x a c t l y one Di o r a l l meet a l l t h r e e
(RESTRICTION LEMMA): and D3 := B>B1.
L e t B1,B2 E A ( T ) o v e r l a p and l e t D1 := B1\B2,
L e t T1 E T meet a t l e a s t two o f t h e Di and l e t
T meet some ( b u t n o t a l l ) o f t h e Di met by T1.
Then T1\
U
T2ADi =g
Di E. T .
P r o o f : As i n t h e p r o o f o f Lmma 1.3.2 we o n l y i n d i c a t e t h e p r o o f method f o r two cases. Case 1 :
Note t h a t we may assume t h a t T, meets a l l Di because o f Lemma 1.3.2. T2 meets D1 and D2.
Then U1 := Ex(T1 ,B1 ,T2) Case 2:
E
T , U2 := Ex(U1 ,B2,T2)
E
T and T b D 3
= Ex(u2~B1
El
T.
Tp meets o n l y D1.
Extend T2 on D2 t o T; T i * := Ex(U2,B1,T2)
and c o n s t r u c t U2 as i n Case 1 w i t h T;
B T and
Ti* n ( B 1 u B2)
= T2
n (BIU
i n s t e a d o f T2.
B2), T;*\(BIU
So we may assume i n t h e f o l l o w i n g t h a t T1\(B1UB2) T1\(B1UB2). are i d e n t i c a l .
Then
B2) =
and T2\(B1L)B2)
R.H Mohringand EJ. Radermacher
2 16
Then V1 : = Ex(T2,B1 ,T1) E T.
:= B2\B,.
a)
D2 E A ( T ) :
b)
D1eA(T):
T1.T2
-
A(T) f u l f i l l s ( A l )
This f o l l o w s from Ex(T, ,D2,T2)
€
T. rn
D2 := B 1 n B2 and
= Ex(T1 ,B1 ,Ex(T1 ,B2,T2)).
L e t T1 ,T2 E T meet D1 and consider T := Ex(T1 ,B1 ,T2)
do n o t meet D 2 , Ex(T1 ,D1 ,T2)
Then
(A4)
Let B1,B2 E A(T) o v e r l a p and l e t D1 := B1\B2,
Proof: D3
n D3).
a T and thus a l s o T 1 \ ( D g D3) = Ex(T2,B1.T1\D2)
T ~ \ D =~ Ex(V,,B2,V2)
Theorem 1.3.4:
Extend T2 on D3 t o V2 := T2 U (Tl
G
T.
If
I n a l l o t h e r cases, apply t h e
= T El T.
Extension Lemna and/or t h e R e s t r i c t i o n Lenma. L e t T1,T2E T c ) BIU B2 E A(T): Assume f i r s t t h a t a l l T e T meet a t most one Di. meet BIU B2. If they meet BIU B2 i n t h e same Biy Ex(T1,B1UB2,T2) = Ex(T1 ,Bi,T2) r 7 . So l e t w.1.o.g. T1 meet D1 and T2 meet D3. Then Ex(T1,B1UB2,T2) = EX(Ex(T1,B1,T),B2,T2)
E
7.
n
If some To meets more than one D i , then extend T1 on a l l Di w i t h T1 Di = P b u t 0 and make the exchange w i t h T 2 on these Di. Afterwards, r e s t r i c t t h e
T 2 Q Di #
obtained member o f T t o those Di with T1 d)
D1
u D2 a A(T):
0
Di #
B.
This i s shown s i m i l a r l y t o c)..
We now t u r n t o the p r o p e r t y ( S 5 ) , which f o r s e t systems does n o t f o l l o w from t h e p r o p e r t i e s of A(T) and (52) i n the i n f i n i t e case.
Theorem 1.3.5:
Let
be a congruence p a r t i t i o n o f T and B e A ( T ) .
TI
Then
E A ( T ) , i.e. T f u l f i l l s ( ~ 5 ) .
Proof:
L e t n := !Ci
and C : = [B]
TI
=
u
I Ci.
i €11, I1 := {i€ 1
I CinB# 01, I 2
:=
Ii E I , C +~ B#lal,
We d i s t i n g u i s h f o u r cases:
itIl Casel:
TflC=TnBforall
Then Ex(T1,C,T2)
= Ex(T1,B,T2)
TET
e T.
Tn (Ci\B)
f o r some i 6 12. L e t T1,T2 C j e A(T) and meet C, say T1 i n C i \ B and T2 i n CAB f o r i,j E 12. Then CiU J Ex(T1.C,T2) = Ex(T1,Ci Cj,T2) e: T. Case 2:
If T
Q
T meets C, then T n C =
u Bu
Bu
Substitution decomposition for discrete structures
2 77
i E 12, say C i \ B and Ci\B. Then 1 Lemma 1.3.2, a p p l i e d t o t h e overlapping T-autonomous s e t s Ci U 8 , C i B, y i e l d s 1 2 t h a t Case 3 i s contained i n Case 4. Case 3:
There i s Toe T which meets two Ci\B,
Case 4:
There i s To e T which meets B and C\B.
3
The proof o f t h i s n o n - t r i v i a l case i s based on two claims. For each i E 12, the assunptions o f the Extension Lemma are s a t i s f i e d
Claim 1:
and B.
f o r Ci
and l e t i E 12.
L e t To meet C\B i n C i
I f i # i,, B.
I f i = io, Claim 1 i s obvious.
0
U
i t f o l l o w s from t h e a p p l i c a t i o n o f t h e Extension Lemma t o To and Ci
0
C #
Given T1,T2 e T w i t h T i n
Claim 2: (i)
For a l l i
(ii) (iii)
For a l l i r 11,
~l
12,
Tln Ci T2nCi
0,
T1\C = T\C.
n Ci
T2
t h e r e e x i s t s T r T such t h a t
# j3 i f f Tn Ci # b . Then a l s o T n (Ci\B) # fl. # 0 i f f T f l Ci # 0. Then a l s o Tn (Gin B) # lo.
Cn i
For each i E. 11, choose Ui e T which meets
II
i = 1,2,
( i e 11) o f T2 by UiU Ci.
e V(T), T i
Replace each non-empty subset
L e t T 5 denote t h e r e s u l t i n g s e t .
T$ meets the same Ci
ET.
B.
E IT
as T2, b u t a l s o i n B.
TP E T be obtained from T1 by r e p l a c i n g a l l non-empty subsets Vi fl Ciy
where Vi
E
T meets CbB.
T1nCi
(i
12) by
I f necessary, extend TP on B
(which is possible because o f Claim l),and l e t T := Ex(TT*,B,T$).
t o Ti*
€
Then T? meets the same Ci as T1 does, b u t f o r
Furthermore, TP\C = T,\C.
a l l i E I2 a l s o i n Ci\B.
Since
Similarly, l e t
Then
T
f u l f i l l s Claim 2. We show now t h a t C
E
A(T).
the p r o p e r t i e s o f Claim 2. T2 T*
n Ci n Ci
To t h i s end, l e t T1,T2 E T meet C and l e t T E T have Let T* be obtained from T by r e p l a c i n g T n Ci by
f o r each i r I1 w i t h T2 (I Ci
n Ci
= T2
f o r a l l i e I, w i t h
# p.
Since
T2nCi
II
D,
#
c V ( T ) , T* E T.
Then
b u t T* may s t i l l meet other
i E I1.However, by c o n s t r u c t i o n o f T, these classes Ci a r e then a l s o We claim t h a t f o r these classes Ciy t h e r e i s Ui E T which meets Ci m e t on C$B. only i n Ci fl B ( i . e . , not i n Ci\B). To see t h i s , extend ( f o r f i x e d such Ci) T* classes Ci,
to
Tt
on B
n Ci
T2nC j
and ( f o r some C j w i t h
possible because o f Claim 1) and p u t Ui classes Ci,
r e p l a c e t h e subsets
T*n Ci
# p) T 2 t o T i on
by
Uin
Ci.
Cjn
B (which i s Then, f o r each o f these
:= Ex(T;,B,T!).
Since
~i
E V ( T ) , the r e s u l t i n g
s e t T** belongs t o T and meets a l l the classes Ci which are met by T* b u t n o t by
T2 o n l y i n B
n Ci.
Then Ex(T1 ,C,T2)
= Ex(T**,ByT2) E 7 .
I t remains t o show ( S l ) , (S4) and ( S 6 ) , ( 5 7 ) .
f o r f i n i t e c l u t t e r s , cf.
c106, Th. 3.31.
Now ( S l ) f o l l o w s i n t h e same way as
F i n a l l y , t a k i n g i n t o account the internal
R.H. Mohring and F.J. Radermacher
278
c h a r a c t e r i z a t i o n o f congruence p a r t i t i o n s i n Lemma 1.3.1
, (S4)
i s shown s t r a i g h t -
(S6) and ( S 7 ) a r e obvious.
forwardly.
There are t h r e e p r i n c i p l e s o f i n v a r i a n c e (P4); one on t h e connection between a s e t system and the c l u t t e r s o f i t s minimal o r maximal elements, the o t h e r two f o r blockers and a n t i b l o c k e r s o f s e t systems. T i s c a l l e d !onvex
L e t T be a s e t system on A.
aml T, 5 T c T2 imply t h a t T c T .
Let
?ln and
ma1 and r-maximal s e t s o f T, r e s p e c t i v e l y . T 4 T' i f each T
T cT.
E
i n c l u s i o n ) i f T1,T2 E T
(w.r.t.
Faxdenote
the systems o f c-mini-
Furthermore, w r i t e (as f o r p a r t i t i o n s )
T i s contained i n some T'
E
T ' and each T ' e T ' c o n t a i n s sane
We then e a s i l y o b t a i n t h e f o l l o w i n g i n v a r i a n c e p r i n c i p l e .
L e t T be a s e t system on A.
Theorem 1.3.6:
n v(Fax).
a)
V ( T ) 5 v(Fin)
b)
I f T i s convex and
c)
If T i s If T i s
Fax, then a) holds w i t h e q u a l i t y . an independence system w i t h T 6 Fax, then V ( T ) = V ( F a x ) . a dependence system w i t h Fins. T, then V ( T ) = V(Fin). pin6
T 6
I n p a r t i c u l a r , c l u t t e r s have t h e same decompositions as t h e i r associated dependence and independence systems. Let T be a s e t system on A.
Then b[T]
T, w h i l e a[T]
c a l l e d the b l o c k e r o f
c a l l e d the a n t i b l o c k e r o f T.
:= {U C A ( ( U
:= I U
€
A\
n TI
IU n TI
>C
1 for a l l T
.r< 1 f o r a l l T
TI is is
These d e f i n i t i o n s are s l i g h t l y d i f f e r e n t from the
usual ones f o r f i n i t e c l u t t e r s T, i n which one considers b[T]
:= b[TImin
and
t o be t h e b l o c k e r and a n t i b l o c k e r o f T, r e s p e c t i v e l y , c f . [47],
a[T] := a[T]""
[57].
[5q, [56],
E
E TI
We use t h i s sonewhat d i f f e r e n t n o t i o n here, as i t i s q u i t e
n a t u r a l i n t h e framework o f a r b i t r a r y s e t systems and a l s o avoids non-existence, which can occur f o r i.[T]
i n t h e i n f i n i t e case.
Indeed, f o r i n f i n i t e T , b[TImin
may n o t e x i s t , whereas a[TImaX always e x i s t s , since a[T]
i s the independence
system o f the c l i q u e s o f the canplementary graph G(T)' o f G(T), where G(T) has node s e t A and edges Obviously, b[T] = b[Fi3
if
dence system.
(a,B)
f o r a l l a , E~ A w i t h { a , B } c T f o r some T E T.
i s convex and normal, if T i s normal.
Fin6 T.
Also, T c b[b[T]],
I n particular,
Fin=
Furthermore, b[T]
=
where e q u a l i t y holds i f T i s a depen-
b[b[TImiTmin
f o r f i n i t e s e t systems, which
i s the well-known i d e n t i t y C = b [ b ( C ) ] f o r b l o c k e r s o f f i n i t e c l u t t e r s c f . [47], With regard t o the decomposition p o s s i b i l i t i e s o f b[T], we note: [El].
219
Substitution decomposition for discrete structures
L e t T b e a (normal) s e t system on A w i t h T = b[b[T]].
Theorem 1.3.7: V(b[T]),
i.e.
i n p a r t i c u l a r A(T) = A(b[T]).
t i o n r u l e s b[TlB]
= b[T]l
B for B
E
Furthermore, we have t h e t r a n s f o r i n a -
A(T) and ~ [ T / I T ] = b[T]/a
F o r f i n i t e c l u t t e r s T and b[TImin,
Proof:
Then V ( T ) =
Pl],
cf.
for
[Sl].
IT
e V(T).
The p r o o f methods can
be extended t o t h e case considered here, s i n c e t h e c r u c i a l p r o p e r t i e s t h a t t o each T e T ( U
E
b[T])
and a E T ( a E. U ) t h e r e e x i s t s U
= {a} remain v a l i d f o r normal s e t systems w i t h
5 = b[b[T]]
= a[PaXJ
i f T 6 Tmax.
Thus T = a[a[T]]
i f f T i s conformal, t o o .
Theorem 1.3.8:
L e t T be a s e t system on A.
C_
A(a[T]).
I
Even s t r o n g e r , a[T]
Then V ( T ) c V(a[T]),
E q u a l i t y h o l d s i f T i s conformal.
t h e t r a n s f o r m a t i o n r u l e s a[TIB] 77 E
*
Tn U
with
i s convex
is a
may be i n t e r p r e t e d as t h e system o f c l i q u e s o f a graph.
conformal s e t system, i.e.
u l a r A(T)
(T e T)
b[T]
t i s obvious t h a t a[T]
From t h e above remarks on t h e a n t i b l o c k e r a[T], and normal and t h a t a[T]
E
= a[T]
IB for B
E
i.e.
i n partic-
Furthermore, we have
A(T) and a[T/a]
= ~[T]/II
for
V(T), i f T i s conformal. F o r f i n i t e c l u t t e r s c f . [81].
Proof:
s i d e r e d here.
See a l s o t h e r e l a t i o n s h i p between T and G ( T ) i n 1 . 5 . a
Finally, w.r.t.
(P5), i t was shown t h a t " a l m o s t a l l " l a b e l e d
systems a r e p r i m e [99]. and c l u t t e r s .
The method c a r r i e s o v e r t o t h e case con-
independence
T h i s r e s u l t extends i m e d i a t e l y t o dependence systems
The p r o o f o f t h e theorem i s b y p u r e c o m b i n a t o r i a l arguments and
uses s t r o n g bounds on t h e number o f independence systems on an n-element base s e t [69], (i.e.
[85].
As t h e c l a s s o f independence systems t u r n s o u t t o be r i g i d r99]
almost a l l s t r u c t u r e s have a t r i v i a l automorphism g r o u p ) , t h i s a s y n p t o t i c
behaviour also c a r r i e s over t o t h e unlabeled structures are identified.
below) and m - c l u t t e r s ( I T 1 = m for a l l T cases a r e proved i n [99] t i o n s [Zl],
1.4
case, i . e .
also holds i f i s m o r p h i c
The same i s t r u e f o r conformal c l u t t e r s ( s e e graphs
via 0
-
E
T).
The r e s u l t s i n these s p e c i a l
1 laws induced b y f i r s t - o r d e r l o g i c c o n s i d e r a -
[51].
SUBSTITUTION DECOMPOSITION FOR RELATIONS
k F i n i t e and i n f i n i t e k - a r y r e l a t i o n s ( i . e . subsets o f A ) were t h e f i r s t s t r u c t u r e s f o r which t h e i n v e s t i g a t i o n o f t h e s u b s t i t u t i o n decomposition has l e d t o r e s u l t s o f t h e Jordan-Holder t y p e and t o t h e c h a r a c t e r i z a t i o n o f t h e a s s o c i a t e d congruence These i n c l u d e d t h e i n f i n i t e case, due t o t h e [122]. p a r t i t i o n l a t t i c e s [98],
280
R.H MGhring and EJ. Radermacher
f a c t that rather strong properties are v a l i d f o r relations. The discussion of aspects (Pl) - ( P 5 ) therefore i s much easier t h a n f o r s e t systems. The situation here i s rich f o r applications, since undirected (simple) graphs (which can be identified w i t h symnetric, irreflexive binary relations) and p a r t i a l orders ( i . e . reflexive, asymnetric and t r a n s i t i v e binary relations) are covered. The l a t t e r play, i n the f i n i t e case, a n important role in the description of the Below we technological structure underlying p r o j e c t networks [48], [77] , [80]. give a l i s t of structural aspects related t o the substitution decomposition, such as clique determination and perfectness of graphs and dimension or Moebius function computation in partial orders. Higher level problems o f that s o r t in COMBINATORIAL OPTIMIZATION over graphs and partial orders (such as detennination o f the shortest project duration i n networks), will be considered in Section 11. Another area of application involving k-ary instead of binary relations comes fran canputer science and concerns the decomposition of non-deterministic automata, cf. [149] , [150] , [151] . All these applications have - in a long historical developnent - led t o the same concept of substitution (X-join El321 or ordinal sum [74]) and t o the same autonomous s e t s (temed e.g. closed s e t s [58] , clumps [3] , [20] , externally related sets [ Z q , p a r t i t i v e s e t s [64] and stable s e t s [141]), g i v e n below. This concept o f s u b s t i t i t u t i o n r e s u l t s from replacing elements of a given relation by other relations, where elements from d i f f e r e n t relations are related t o each other i n the same way in which the replaced elements were. From the algebraic point of view, the resulting homanorphisms between k-ary relations R and R ' over base s e t s A and A ' are surjective homanorphisms h: A + A ' such t h a t , f o r a l l a l ,. .. ,ak E A with I { h ( a , ) , ...,h ( a k ) l l > 1 , ( a l ,...,ak) E R i f f ( h ( a , ) , ...,h ( a k ) ) e R' ( i . e . they are "almost" the strong relational homomorphisms, cf. [118], [llg]).
Definition: Let R' be a k-ary relation ( k a 2 ) on A' and l e t , f o r each B e A ' , R g be a k-ary relation on AB, where the s e t s A6 are non-empty and pairwise d i s j o i n t . Let A := U P u t A := I ( B , ...,6) E A t k B € A ' } and s e t
U
"
WA'
A!3'
I
A x...xA , Then R i s called the canposition (61, ..., B k ) RYL' B1 'k of R ' and the Re, 8 6 A ' , and i s denoted by R = R'[RB, B E A ' ] . R i s said t o be
R :=
BEA'
RB
obtained by substitution of the elements 5 e A ' by the relations R B in R ' . The canposition i s proper i f \ A ' \ > 1 and [ A B [ > 1 f o r some 6 t A'. A relation B i s said to be decomposable i f i t has a representation as a proper composition. Otherwise, i t i s said t o b e indecanposable or prime.
28 1
Substitution decomposition for discrete structures
a) A p a r t i t i o n 'TI = {Bi I i e I } o f A i s c a l l e d a congruence p a r t i t i o n o f R i f t h e r e e x i s t k-ary r e l a t i o n s Ri on Bi, Definition:
L e t R be a k-ary r e l a t i o n on A.
i E I , and a k - a r y r e l a t i o n R ' on A/.
such t h a t R = R ' [Ri ,Bi
R ' i s c a l l e d the q u o t i e n t o f R modulo
IT
E
A/T].
and i s denoted by R/'TI.
I n t h i s case,
V(R) denotes the
system o f congruence p a r t i t i o n s o f R. b ) A subset B o f A i s c a l l e d an R-autonanous s e t i f t h e r e i s 'TI t V(R) w i t h B E P I n t h i s case, t h e r e l a t i o n R I B := Rn B k i s c a l l e d the autonomous s u b - r e l a t i o n of R induced by B.
.
A(R) denotes the system- o f a l l R-autonanous sets.
As f o r s e t systems, congruence p a r t i t i o n s and autonanous sets o f r e l a t i o n s have nice internal characterizations.
Lemma 1.4.1: a)
L e t R be a r e l a t i o n on
A p a r t i t i o n IT = IBi 1 i Q 1) o f A i s a congruence p a r t i t i o n o f R i f f each ( a 1 ,.. ,ak) E R w i t h elements a . from a t l e a s t two d i f f e r e n t classes o f T
.
i m p l i e s t h a t (B1, b)
A.
...,B k )
J
R f o r a l l B~
E
A subset B o f A i s R-autonmous i f f
f o r scme i # j i m p l i e s
(al
(a1
=
,. .. , a k )
,...,aj-l,B,aj+l
i = 1, ...,k .
[ai]r, E
,...,ak)
R, E.
aj E
B, B e B and
ai E. ?SB
R.
S i m i l a r l y t o the case o f Boolean functions, the q u o t i e n t R/'TI i s n o t u n i q u e l y detk I n order t o o b t a i n
etmined b u t o n l y up t o r e f l e x i v e t u p l e s ( a , . . . ,a) e (A/'TI)
.
uniqueness, we w i l l t h e r e f o r e i d e n t i f y r e l a t i o n s which o n l y d i f f e r i n r e f l e x i v e t u p l e s (which e.g. f o r (undirected, simple) graphs and p a r t i a l orders i s no r e s triction at all). With regard t o t h e o t h e r c o n d i t i o n s i n (P2)
Theorem 1.4.2:
-
(P3), we o b t a i n :
Relations f u l f i l ( A l ) , (A2)* and (A3), ( S l ) , (S2)*,
( S 3 ) * , (S4)
-
(S7), ( b u t i n general n o t (A4)).
Proof
( A l ) i s obvious.
( c f . [98]):
autonanous sets w i t h
n
cI
n
Bi P 8.
icI
To show (A2)*,
L e t (al
,. .. ,ak) s
l e t (Bi)icI
be a f a m i l y o f R-
R and, w.1 .o.g.,
a e B := 1
Bi I a2 $ B, . In order t o show t h a t B 6 A(R), we must (because o f i€1 Lemna 1.4.1 j show t h a t (B,a2 ,... ,ak) e R f o r any B E B. Since a2 6 B, t h e r e e x i s t s io E I w i t h e 2 6 Bi But al, ~ E 5 B B . and B i E A(R). Hence (B,a23.*.,"k)
&
0
R.
To show t h a t C :=
u
1QI
.
'0
Bi E A ( R ) , assume again t h a t (a1
0
,...,a k ) E
R,
al E
Bi
1
,
282 a2
R.H. Mohringand EJ. Radermacher
# C , and fl
d
Bi
2
.
Let
obtain t h a t (Y,a2 , . . . , a k ) that
( 6 , a2 , . . . , a k ) E
Y E E
R.
n
i CI
Bi.
Since B i
Similarly, B i z €
1
6
A(R), a l , Y e B i
A(R), f l , y e B i
2'
1'
a2
a2 B B i
# Bi 2
1
we
yields
R.
(A3) i s shown s i m i l a r l y .
A(R). I f C .e A(R) and C c B, then obviously C E A(R1B). I n In the opposite d i r e c i t o n , l e t C e A ( R l B ) , ( a l ,..., a k ) 6 R , w.1.o.g. a1 E C, C , and 6 E C . If some a j + B ( j = 2 ,..., k ) , ( 6 , a 2 ,..-,a k ) e R because of a2 the R-autonomy of B. Otherwise, ( a l ,..., a k ) e R I B and we obtain ( 6 , a 2 , ...,a k ) e R f r m the RIB-autonomy of C. This shows t h a t C c A ( R ) . To show ( S l ) , l e t B
(S2)* follows immediately from the c h a r a c t e r i z a t i o n of congruence p a r t i t i o n s and autonomous sets in Lemma 1.4.1.
(S3)* follows from (S2)* and ( S l ) (54) i s e a s i l y v e r i f i e d with Lemma 1.4.1. F i n a l l y , (S5) follows immediately from (S2)* and (A2)*,
since [C].
=
U (BU C).w
Bell
W+b Going over t o (P4) there a r e two p r i n c i p l e s of invariance. For the complement R C := A k\R of a k-ary r e l a t i o n , we have A(R) = A(Rc) and V(R) = V(Rc) and the f o r any 6 € A(R), and RC/* = (R/T)' f o r any transformation r u l e s R C I B = IT Q
!AR).
The o t h e r p r i n c i p l e i s r e s t r i c t e d t o p a r t i a l orders and means t h e v a l i d i t y o f A ( R ~ ~= ) (A(R))sy, where RSy := RU R-l ( w i t h R-l : = {(y,x) I (x,y) 6 R } ) denotes t h e symnetric closure o f a r e l a t i o n (here: of a p a r t i a l order) and ASY the symmetr i c closure of the s e t system A(S) ( i n the sense of condition (A4)). The given i d e n t i t y , which c h a r a c t e r i z e s t r a n s i t i o n from p a r t i a l orders t o comparability graphs, i s not e a s i l y obtained and i s treated a s an i n t e r f a c e i n 1.5.
Finally, w.r.t. (P5), we mention r e s u l t s on t h e r e l a t i v e frequency of prime r e l a t i o n s . In [YY] i t i s shown t h a t i n each non-trivial c l a s s of parametric k-ary r e l a t i o n s [114], "almost a l l " members a r e prime. This includes as special cases t h a t "almost a l l " binary antisymnetric r e l a t i o n s , tournaments and p a r t i c u l a r l y , graphs, a r e prime. The r e s u l t follows form the f a c t t h a t i n these cases primeness i s a consequence of 0 - 1 laws f o r f i r s t order l o g i c p r o p e r t i e s f o r these s t r u c A l l these c l a s s e s a r e again rigid, so t h a t these t u r e s ; c f . [21], [51], [115]. r e s u l t s extend t o the unlabeled case, too. T h e s i t u a t i o n f o r t r a n s i t i v e relations,
283
Substitution decomposition for discrete structures
quasi-orderings and p a r t i a l orders (which are a l l n o t parametric) i s much harder t o deal w i t h .
Here r e s u l t s f o l l o w from pure combinatorial considerations [99],
using strong bounds on the number o f s t r u c t u r e s o f the r e s p e c t i v e types [49],
[86]. As i t i s n o t c l e a r whether these classes a r e r i g i d , the u n l a b e l e d case i s here n o t y e t s e t t l e d , b u t we conjecture t h a t here, too, almost a l l unlabeled s t r u c t u r e s are prime.
APPLICATIONS OF THE SUBSTITUTION DECOMPOSITION TO GRAPHS
-
Determination o f cliques, independent sets and o f the blocker [47]
o f these
sets, as w e l l as o f c l i q u e coverings and c o l o u r i n g s by means o f decomposition, c f . [28]
-
and Section 11.
Determination o f (maximal) matchings by means o f decomposition. Determination o f the automorphism group o f a graph v i a the associated automorphism groups f o r subgraphs and q u o t i e n t graph [72].
-
The classes o f e.g. p e r f e c t p8] graphs [64] Interval
w. r.t
-
,
[ZZ]
, superperfect,
t u r n o u t t o be closed w . r . t .
chordal and c o m p a r a b i l i t y
decomposition and composition.
graphs (as w e l l as proper i n t e r v a l graphs) [60]
are closed ( o n l y )
. decanposi t i o n .
For c o m p a r a b i l i t y graphs G(o) o f a p a r t i a l order o ( c f . Example 1.1.1 f o r t h e d e f i n i t i o n ) i t is known t h a t G ( o ) = G(0')[G(oi),
i
E
I],f o r B = o'[Oi,i
E
A'].
Furthermore, the uniquely p a r t i a l l y orderable graphs (UP0 gra hs [l],i.e.
rp } ) , are e s s e n t i a l l y
c a n p a r a b i l i t y graphs f o r which G(o*) = G ( o ) i f f o* E CO,O prime [141],
[158].
This has the i n t e r e s t i n g consequence t h a t "almost a l l "
Comparability graphs are prime
[loll.
A b a s i c observation i n t h i s c o n t e x t i s
t h a t i n v e r t i n g the o r i e n t a t i o n on some classes o f a congruence p a r t i t i o n o f G and/or on the associated q u o t i e n t leads again t o a ( a p a r t from t r i v i a l cases new) o r i e n t a t i o n o f G.
Another i n t e r e s t i n g consequence i n the case o f V ( G )
being f i n i t e i s t h a t G(o) = G(o') i m p l i e s t h a t o and O' have the same dimension [107],
[158],
a r e s u l t r e c e n t l y also shown i n t h e i n f i n i t e case
[Z], 0651.
APPLICATIONS OF THE SUBSTITLITION DECOMPOSITION TO PARTIAL ORDERS
-
Counting p a r t i a l orders, i t e r a t i v e l y
b u i l t up from c e r t a i n prime p a r t i a l orders
( i n p a r t i c u l a r s e r i e s - p a r a l l e l networks [129]),
-
c f . [lll].
For p r o j e c t networks, where the technological s t r u c t u r e i s i n t e r p r e t e d as a p a r t i a l order, i t i s known [77]
t h a t f o r the t o p o l o g i c a l s o r t i n g o f t h e a c t i v i -
t i e s , as w e l l as f o r the c o n s t r u c t i o n o f a c t i v i t y - o n - n o d e and a c t i v i t y - o n - a r c
R.H. Mohrmg and F.J. Radermacher
284
diagrams, the substitution decomposition may be used. Due t o the d i f f i c u l t i e s i n finding such diagrams and t o t h e i r frequent use in applications the l a s t case i s particularly interesting. The basis for the use of decomposition i s the easy identification of autonomous s e t s in such diagrams, a s was described i n connection with Example 1.1.1.
-
Concerning the so-called dimension (dim(oj) [46], [74] of partial orders, i t i s known that partial orders of dimension l e s s than some fixed cardinal number are closed w.r.t. to decomposition and composition. In particular, f o r partial orders with V ( G ) of f i n i t e length, dim(@) = maxIdim(o/n),dim(olLi), i = l , . . . , r J f o r any TI = { L 1 , ...,L r } E V ( O ) , i . e . dim(o) i s j u s t the maximum o f the dimension of a l l factors (canpare Section 111) of 8 [74], [107]. This implies t h a t (dim)-irreducible partial orders are prime. O f course, w i t h regard t o the reversibility of partial orders o 1461, which i s equivalent to dim(o) 6 2 [5], [46], t h i s implies that r e v e r s i b i l i t y i s also closed w.r.t. decanposition and composition. (1 i f a = B -c u(a,Y) if a
1
M@Y<$
lo
otherwise
[130] of a ( l o c a l l y ) f i n i t e partial order 8. Given a congruence partition { Bl , . . . , B r ] of 0 , i t can be shown t h a t i f we extend each O ( B i t o Ole; by introducing a l e a s t element a i and a greatest element b i , define f ( a ) := ,,o,Bf(a,bi) as well as g ( a ) := u O l B t ( a i , a ) f o r a l l a 6 B?, i = 1,...,rY 1
,=
1
11
together w i t h ~ ( 6 =~ - )[ ~ ~ , ~ ~ ( a+ ~ , bf o~r )fli
l o
Q
A/n
and p u t ( w i t h 8'
:= O / r )
otherwise,
we obtain that lJ0(a,6) = f ( a ) ' 'JOo
(Bi36j).g( 6)
holds for a l l a c B i , 6 Q BJ. , i # j , with a < 8 6 [126]. I n fact, this result extends to a wider class of elements of the so-called incidence algebra [13q over 0, thereby also integrating the above function V .
1.5
INTERFACES
In t h i s part we will deal w i t h three interfaces between the three types of discrete sturctures treated above, which will f a c i l i t a t e understanding o f the
Substitution decomposition for discrete structures
285
s u r p r i s i n g l y analogous decomposition b e h a v i o u r o f these c l a s s e s . As we w i l l see, t h e monotone Boolean f u n c t i o n s and c l u t t e r s can e s s e n t i a l l y be i d e n t i f i e d and l e a d t o e x a c t l y t h e same system V i n e i t h e r i n t e r p r e t a t i o n . t o be t r u e f o r conformal c l u t t e r s and graphs.
The same t u r n s o u t
We a l s o show t h e c l o s e c o n n e c t i o n
between p a r t i a l o r d e r s and c o m p a r a b i l i t y graphs. A l t o g e t h e r , i t t u r n s o u t t h a t t h e c l a s s o f ( c o m p a r a b i l i t y ) graphs c o n s t i t u t e s a common " k e r n e l " o f a l l s t r u c t u r e c l a s s e s considered, f o r which, i n a l l i n t e r p r e t a t i o n s , congruence p a r t i t i o n s and autonomous s e t s ( e s s e n t i a l l y ) c o i n c i d e .
COHERENT SYSTEMS: BOOLEAN FUNCTIONS VS. CLUTTERS
[TI,
I n RELIABILITY THEORY
m a j o r i n t e r e s t concerns t h e f u n c t i o n i n g o f complex
systems composed o f components, whose performance i s determined b y t h e f u n c t i o n i n g of these components. F: { O , l } n
Such systems can be r e p r e s e n t e d by Boolean f u n c t i o n s
which d e s c r i b e t h e o p e r a t i o n of t h e whole system as a f u n c t i o n
+ {O,l},
o f t h e o p e r a t i n g s t a t e s o f t h e components, i . e . of a s t a t e - v e c t o r x t { O , l l n . Here, i t i s reasonable t o r e s t r i c t o n e s e l f t o c o h e r e n t systems, i . e . w i t h o u t i n e s s e n t i a l components f u l f i l l i n g F ( 0 , F(l,.. . , l )
= 1.
...,0)
=
0, x
y-F(x)
4
systems 6 F(y),
The d i s j u n c t i v e r e p r e s e n t a t i o n o f c o h e r e n t systems b y s m a l l e r
ones ( h e r e c a l l e d modules [7]),
which i n f l u e n c e t h e b e h a v i o u r o f t h e whole system
o n l y as an e n t i t y , l e a d s e x a c t l y t o t h e d i s j u n c t i v e decomposition o f t h e associ a t e d Boolean f u n c t i o n s , as discussed i n 1.2. I n t h e i r fundamental work on t h i s s u b j e c t , Birnbaum and Esary
[la]
gave a second
i n t e r p r e t a t i o n wherein a c o h e r e n t system F i s e q u i v a l e n t l y d e s c r i b e d by t h e (normal) c l u t t e r CF o f a s s o c i a t e d p a t h s ( W = 1 b u t F ( l W , ,OA,w,)
=
sA
b e i n g termed a
path, if F(lw,OA,w)
0 f o r a l l W ' c S ) o r t h e system C; o f a s s o c i a t e d c u t s = 0 b u t F ( l T , ,OA,T, ) = 1 for all TI=
(T c A b e i n g termed a &, i f F(lT,OA,T)
T).
I n t h i s c o n t e x t : paths ( c u t s ) may be seen as p r i m e i m p l i c a n t s o f t h e a s s o c i a t e d Birnbaum and Esary
Boolean f u n c t i o n F ( a s s o c i a t e d d u a l Boolean f u n c t i o n F * ) .
t h e n showed t h a t A(F) = A(C,-) and V(F) = V(CF), c f . a l s o [g]
and Example 1.1.1.
Furthermore, CFln = ( C F ) / r f o r any 1~ e V(F) and (ClB), = CFIB f o r any BE. A(F) and so f o r t h . I n a d d i t i o n , we even have A(F*) (J)A(F) = A(CF) (2) - A(CF*), where ( 1 ) i s t h e i n v a r i a n c e w . r . t . d u a l i z a t i o n and ( 2 ) t h e i n v a r i a n c e w . r . t . b l o c k i n g . I n f a c t , f o r c o h e r e n t systems d u a l i z a t i o n and b l o c k i n g a r e i d e n t i c a l o p e r a t i o n s i n different interpretations.
CONFORMAL CLUTTERS VS. GRAPHS Given a graph G = (V,E),
l e t C(G) be t h e ( c o n f o r m a l ) c l u t t e r o f maximal c l i q u e s of
R.H. Mohring and F.J. Rademcher
286
G ( c f . Example 1 . 1 . 1 ) . Conversely, given a c l u t t e r C on V , l e t G ( C ) denote the graph G with vertex s e t V and edge s e t E = I(vl,v2) I Ivl,vzl c T f o r some T 6 C ) . Then G ( C ( G ) ) = G and C G C ( G ( C ) ) , where equality holds, i f f C is conformal [8]. I t i s easily seen t h a t A ( C ) 6 A ( G ( C ) ) , V ( C ) c ( G ( C ) ) , where in the conformal case!, again equality holds. Furthermore, C ( G l B ) = C ( G ) l B f o r a l l B E A ( G ) and C(G/n) = C ( G ) / n f o r a l l n t V(G). Of course, complementation of graphs, i . e . going over from G t o Gc, corresponds t o antiblocking of the (conformal) c l u t t e r C ( G ) . So again, going over t o Gc o r to a[C(G)] a r e identical operations in different interpretations.
Of course, the above notions can be extended t o a r b i t r a r y s e t systems J , where G(T) i s defined in the same way as for c l u t t e r s , and T(G) denotes the system of of G . Then T i s conformal ( i . e . the system of cliques of some graph) i f f T = T(G(T)) ( c f . also the remarks following Theorem 1.3.7). Because of the above and Theorem 1.3.6, we obtain that A ( T ) s A ( G ( J ) ) and V ( T ) c V(G(T)), where equality holds i f T i s conformal. Also the other remarks on c l u t t e r s made above carry over t o a r b i t r a r y s e t systems.
a l l cliques
PARTIAL ORDERS V S . CmPARABILITY GRAPHS The connection between a partial order o and the associated Comparability graph G ( o ) i s a rather close one. Its importance l i e s e.g. i n the f a c t t h a t i t allows insights into the degree of similarity between autonomous s e t s of different trans i t i v e orientations of the same comparability graph (where indeed different s e t s A and V may a r i s e ) . Another important aspect, particularly w i t h regard t o applications i n the theory of project networks, is the equality between the system C(o) ofs-maximal chains of o and C(G(e)). Of course, given the invariance results f o r blocking and antiblocking we have A(C(o)) = A(C(G(e))) = A ( C ( o ) ) = A(b[C(o)]) = A(a[C(o)J). Taking into account t h a t b[C(o)] and a[C(o)] denote the systems of separating sets (cuts) and independent s e t s of o ( o r of the graph G(o)), respectively, these i d e n t i t i e s are the very reason why many interesting problems in network theory ( c f . Section 11), s u c h as determining shortest or longest paths [77], chain coverings o r colourings, minimal or maximal flows [79] , bottleneck extrema and others Pl], @4], lead to the same decomposition p o s s i b i l i t i e s i n a l l cases, v i z . a l l n e V ( G ( o ) ) . The close connection between A(G) and A(G(o)) i s made precise i n the following theorem, cf. also Example 1.1.1:
Theorem 1.5.1, c f . [24], 1.
A(o) [cz,B],
n06],
= {B c A(G(o)) = {Y e A
I
I
poll:
B i s o-convex), where B i s @-convex, i f f B. &o fij 5 B f o r a l l a,E
a .s@Y
Substitution decomposition for discrete structures
2.
L e t G b e a c o m p a r a b i l i t y graph and C i: A(G).
287
Then t h e r e a r e 6, (Bi)irI
E. A(G)
such t h a t 1.
2.
3.
I}i s a p a r t i t i o n o f B such t h a t f o r any two Bi,Bj, i f j, each IBi I i a E. Bi i s a d j a c e n t t o each 6 E B j t h e r e i s J E I such t h a t C = B jeJ j B and a l l Bi, i E. I belong t o A(o) f o r each o such t h a t G = G ( o ) .
u
I n p a r t i c u l a r o i s prime, i f f G(o) i s prime. Note t h a t Theorem 1.5.1 i m p l i e s t h a t A(G(o)) = ( A ( O ) ) ' ~ i n t h e sense of (A4), i . e . A(G(o)) a r i s e s f r o m A(o) b y a d j o i n i n g a l l s y m n e t r i c d i f f e r e n c e s
B
A C with
Bn
B,C e A ( @ ) , B \ C # 0, C # p , C\B # b. (Remember t h a t w.r.t.(A4), graphs a r e symmetrically closed w h i l e p a r t i a l orders are not). This implies t h a t the f a c t o r s i n c a n p o s i t i o n s e r i e s as d e s c r i b e d i n S e c t i o n I11 a r e t h e same i n b o t h cases, showing t h a t t h e decomposition w i t h r e g a r d t o o and G(o) i s e s s e n t i a l l y t h e same. Note t h a t t h i s r e l a t i o n s h i p between p a r t i a l o r d e r s and c a n p a r a b i l i t y graphs w i l l become even more s t r i k i n g when seen i n c o n n e c t i o n w i t h t h e c a n p o s i t i o n t r e e i n Theorem 4.1.4.
I n f a c t t h i s i s t h e b a s i s f o r a simultaneous t r e a t m e n t o f algo-
r i t h m i c computation o f autonomous s e t s i n p a r t i a l o r d e r s and c o m p a r a b i l i t y graphs, p o s s i b l e i n O(n3) time; c f . S e c t i o n I V . As a l a s t consequence o f Theorem 1.5.1 we m e n t i o n t h e f a c t t h a t " a l m o s t a l l " C o m p a r a b i l i t y graphs a r e UP0 ( u n i q u e l y p a r t i a l l y o r d e r a b l e ) , c f .
[loll.
This
r e s u l t i s c l o s e l y r e l a t e d t o t h e above mentioned f a c t t h a t " a l m o s t a l l " p a r t i a l o r d e r s a r e p r i m e and, i n p a r t i c u l a r , i m p l i e s t h a t t h e number o f c o m p a r a b i l i t y graphs a s y m p t o t i c a l l y equals h a l f t h e number o f p a r t i a l o r d e r s .
1.6
CONNECTIONS WITH THE SPLIT DECCNPOSITION
The p r o p e r t i e s o f t h e s u b s t i t u t i o n decomposition d e r i v e d f o r t h e t h r e e c l a s s e s o f s t r u c t u r e s i n t h e p r e v i o u s s u b s e c t i o n s show t h a t t h e s u b s t i t u t i o n i n v o l v e s a n a t u r a l asymnetry which leads t o t h e two (asymmetric) n o t i o n s o f congruence p a r t i t i o n ( q u o t i e n t ) and autonomous s e t (autonomous s u b s t r u c t u r e ) .
I n f a c t , these two
n o t i o n s w i l l f o r m t h e b a s i s f o r t h e a l g e b r a i c d e c o m p o s i t i o n model i n S e c t i o n 111. Another approach, taken by Cunningham and Edmonds [34]
,
[38], c o n s i d e r s a symmet-
r i c e x t e n s i o n of t h e s u b s t i t u t i o n o p e r a t i o n , i n which a s u b s t r u c t u r e and i t s canplement (which e s s e n t i a l l y corresponds t o t h e q u o t i e n t ) occur s y m m e t r i c a l l y i n a
split, which i s t h e b a s i c n o t i o n o f t h i s approach.
For i n s t a n c e , f o r a graph G
on A, a s p l i t o f G i s a p a r t i t i o n IA1,A21 of A w i t h ( A 1 / 5 2 6 ( A 2 / such t h a t whenever a . E A1 i s a d j a c e n t t o Bi E A2 ( i = 1,2), t h e n al and B 2 and a2 and 61 1 are a l s o adjacent. This s p l i t decanposition includes the s u b s t i t u t i o n
R.H. Moliring and F.J. Rademzacher
288
d e c m p o s i t i o n ( i n the f i n i t e case), since each G-autonomous s e t B ( w i t h 161
2 4 \A\BI)
induces a s p l i t ( v i z . {B,A\BI)
b u t n o t vice-versa.
Similarly
t o t h e s u b s t i t u t i o n decomposition, t h e s p l i t decanposition may o f t e n be viewed as For graphs, t h i s composition i s as f o l l o w s
the inverse o f a unique canposition.
U
U
[ a ) , A2 { a } w i t h A, f) A2 = kl and edge sets 1351. L e t G1 ,G2 be graphs on A1 E1,E2. Then the c a n p o s i t i o n G = G1*G2 o f G1 and G2 i s the graph w i t h v e r t e x s e t
A 1 U A 2 and edge s e t ( ( E 1 l J (y,a) t
E21.
The node
a
I B ~ A , U A ~ II)( 6U, v ) I
E2)\{(6,a)
(6,a) Q El
each s p l i t tA1.A2} o f a graph G corresponds t o two graphs G1 on A1
and
O f course,
i s c a l l e d the marker o f t h i s composition.
u { a ) , G2
on
A2 U l a ) w i t h t h e c m o n marker a . I f A1 i s 6-autonanous, then G1 \ A 1 i s t h e o f G modulo the autonanous subgraph belonging t o A1, and G2 i s t h e q u o t i e n t G / I T A1 congruence p a r t i t i o n TI = {Al,ta) I a c A \ A 1 ? , The s u b s t i t u t i o n decanposition A1 can be expressed by t h i s composition, i f one o f t h e graphs i s assumed t o be p o i n t e d , i . e . o f the form vGi,
where v i s adjacent t o each node o f G i .
This con-
s t r u c t i o n means t h a t f o r every graph G, t h e r e i s a graph G ' such t h a t applying the s p l i t decanposition theory t o G' i s e q u i v a l e n t t o applying the s u b s t i t u t i o n decanposition theory t o G.
T h i s type o f t r a n s f o r m a t i o n i s general, b u t n o t
canonical, and n o t e q u a l l y successful i n a l l a p p l i c a t i o n s .
Sometimes i t produces
a n a t u r a l and s t r o n g g e n e r a l i z a t i o n (e.g. s t r o n g l y connected digraphs, s e t system) sanetimes i t produces sanething which i s n a t u r a l b u t n o t r e a l l y more general (e.g. m a t r o i d s ) , and sanetimes i t seems (as y e t ) t o produce n o t h i n g n a t u r a l o r u s e f u l (e.g.
k-ary r e l a t i o n s f o r k
2).
Yet t h e r e are a p p l i c a t i o n s o f the s p l i t
theory w i t h no apparent s u b s t i t u t i o n analog (e.g.
submodular f u n c t i o n s ) .
Based on t h i s n o t i o n o f s p l i t , Cunningham and Edmonds [38]
developed an a b s t r a c t
decanposi t i o n theory which shows t h a t under c e r t a i n axioms, each s t r u c t u r e has an u n r e f i n e d unique decomposition i n t o indecanposable ( w . r . t .
the s p l i t decanposi-
t i o n ) s t r u c t u r e s and two s o r t s o f h i g h l y decomposable s t r u c t u r e s ( s o - c a l l e d b r i t t l e and s e n i - b r i t t l e s t r u c t u r e s ) .
The s t r u c t u r e s t o which t h i s theory has i n
t h e meantime been a p p l i e d are ( s t r o n g l y connected) b i n a r y r e l a t i o n s [35], t i v e l a t t i c e s [54], s e t systems ( i n c l u d i n g matroids, c l u t t e r s , e t c . ) l i n e a r systems [39]
and submodular f u n c t i o n s [37] ,[53].
[34],
distribu[38],
I n a l l cases, the res-
p e c t i v e b r i t t l e and s e n i - b r i t t l e s t r u c t u r e s have been ccmpletely characterized. These unique decomposition r e s u l t s are c l o s e l y r e l a t e d t o t h e r e s u l t s on t h e canp o s i t i o n t r e e f o r the s u b s t i t u t i o n decanposition i n 111.4, and we s h a l l continue t h e i r d i s c u s s i o n i n 111.5. Canpared w i t h the s u b s t i t u t i o n decomposition, t h e g r e a t e r g e n e r a l i t y o f t h e s p l i t decanposition has a c e r t a i n p r i c e .
So f o r instance, t h e p r i n c i p l e s o f i n v a r i a n c e
(P4) and the i n t e r f a c e s f o r c l u t t e r s and graphs discussed i n 1.3-1.5 are l o s t , as i s t h e a l g e b r a i c i n t e r p r e t a t i o n o f t h e decanposition as developed i n S e c t i o n 111.
289
Substitution decomposition for discrete structures
Also, t h e a p p l i c a b i l i t y t o c o m b i n a t o r i a l o p t i m i z a t i o n problems seems p r e s e n t l y t o be more r e s t r i c t e d , compared w i t h t h e s u b s t i t u t i o n deccmposition ( c f . 1 . 4 and 1 1 ) . I n p a r t i c u l a r , p a r t i a l o r d e r s and p r o j e c t networks can o n l y be t r e a t e d by t h e s u b s t i t u t i o n d e c a n p o s i t i o n , s i n c e f o r r e l a t i o n s s t r o n g c o n n e c t i v i t y must be assumed i n o r d e r t o a p p l y t h e s p l i t decomposition t h e o r y . Also, i n s p i t e o f t h e g r e a t e r g e n e r a l i t y , i t t u r n s o u t t h a t f o r t h e s p l i t decomposition, t h e decanpoI n fact, also w.r.t.
s i t i o n p o s s i b i l i t i e s are q u i t e r e s t r i c t e d , too.
this
approach, " a l m o s t a l l " graphs, b i n a r y r e l a t i o n s , c l u t t e r s , and s e t systems a r e indecanposable.
T h i s f o l l o w s f r o m a s t r a i g h t f o r w a r d e x t e n s i o n of t h e techniques
used i n [99] f o r t h e s u b s t i t u t i o n decomposition. There i s some evidence t h a t o t h e r a p p l i c a t i o n s o f t h e s p l i t decomposition t o canb i n a t o r i a l o t p i m i z a t i o n may be developed, p o s s i b l y a l s o i n cases where no s u b s t i t u t i o n analogue i s known.
One such example i s t h e v e r y r e c e n t a p p l i c a t i o n t o t h e
d e c a n p o s i t i o n o f submodular f u n c t i o n s [37J,
[XJ, t h e consequences o f w h i c h a r e
n o t y e t f u l l y understood.
11.
UNIQUE CHARACTERIZATION OF THE SUBSTITUTION DECOMPOSITION FOR RELATIONS AND SET SYSTEMS FROM THE VIEW-POINT OF COMBINATORIAL OPTIMIZATION
I n t h i s S e c t i o n we demonstrate how a n a t u r a l approach t o t h e f a c t o r i z a t i o n o f o p t i m a l v a l u e f u n c t i o n s i n COMBINATORIAL OPTIMIZATION n e c e s s a r i l y l e a d s t o t h e s u b s t i t u t i o n d e c a n p o s i t i o n o f t h e u n d e r l y i n g s t r u c t u r e s , such as graphs, p a r t i a l orders,
(in-)dependence systems and c l u t t e r s .
Such r e s u l t s p r o v i d e a s t r o n g
m o t i v a t i o n f o r t h e s t u d y o f t h e s u b s t i t u t i o n decomposition and d i s t i n g u i s h e s i t u p t o now f r o m t h e o t h e r approaches l i k e t h e s p l i t d e c a n p o s i t i o n .
I n fact, i n
s e v e r a l cases these uniqueness r e s u l t s a l s o formed t h e r a i s o n d ' e t r e f o r t h e use o f s u b s t i t u t i o n d e c a n p o s i t i o n , p a r t i c u l a r l y i n t h e case of p r o j e c t networks, whose s t r u c t u r e n a t u r a l l y corresponds t o p a r t i a l o r d e r s . I n t h e f o l l o w i n g , we s t a r t i n 11.1 w i t h t h e d i s c u s s i o n o f graphs ( .e. symmetric, i r r e f l e x i v e r e l a t i o n s ) , t h e n c o n t i n u e i n 11.2 w i t h p a r t i a l o r d e r s p r o j e c t networks) and t r e a t c l u t t e r s ((in-)dependence systems) i n 11.3.
P s h o r t sumnary
w i t h a g e n e r a l d i s c u s s i o n o f t h e r e s u l t s , t o g e t h e r w i t h sane h i n t s on t h e s p l i t decomposition, w i l l c l o s e t h i s S e c t i o n i n 11.4.
11.1
CCMBINATORIAL OPTIMIZATION OVER GRAPHS
One o f t h e b a s i c f u n c t i o n s i n c o m b i n a t o r i a l o p t i m i z a t i o n over graphs G = (V,E) g i v e n by t h e s o - c a l l e d weighted c l i q u e number o G ( x ) : IR)
RL with
is
290
R.H. Mohring and F.J. Radermacher
mG(x) :=
1
max x ( v ) , where C(G) denotes t h e system of (s-maximal) c l i q u e s CfC(G) VEC
i n G and x = ( Xl,...,~n)
..., v &
V = {vl,
i s a weighting f u n c t i o n on V , i . e . xi
: = x(vi)
for
gives the (non-negative) weight o f vi.
Closely r e l a t e d i s the s o - c a l l e d weighted independence number aG(x): IRn
3
iR1 w i t h
> , 2 ,
aG(X) : =
max 1 x ( v ) , where I(G) denotes t h e system o f ( + m a x i m a l ) independent UeI(G) veU
sets i n G and x i s again a w e i g h t i n g f u n c t i o n . Other i n t e r e s t i n g f e a t u r e s are ( i n the i n t e g e r case) t h e c l i q u e covering number
G ( x ) ( i . e . the s m a l l e s t nunber o f c l i q u e s t h a t cover each v c V a t l e a s t x ( v ) times) and the c h r a n a t i c number x G ( x ) ( i . e . t h e s m a l l e s t number o f independent
p
s e t s covering each v
t
V a t l e a s t x ( v ) times).
Note t h a t the d e f i n i t i o n s o f wG and Further
aG
=
wGC
aG may
can r e s t r i c t ourselves w.1 .o.g.
t o the c o n s i d e r a t i o n of t h e weighted c l i q u e number.
There i s a s i m i l a r connection between
x.
be extended t o i n c l u d e n e g a t i v e weights
holds, where Gc denotes t h e complementary graph o f G, i . e . we p
and x; here we w i l l r e s t r i c t ourselves t o
Note t h a t the weighted case may be traced back t o t h e case x
each node v E V by x ( v ) i d e n t i c a l copies.
=
1 by r e p l a c i n g
O f course, we have aG(x)
e.g. perfectness would y i e l d e q u a l i t y [64].
6
xG(x), where
Even then, t h e covering problem i s o f
s p e c i a l i n t e r e s t i n the sense t h a t an optimal c l i q u e does s t i l l n o t s t r a i g h t forwardly y i e l d a b e s t covering. I n t h e f o l l o w i n g , we w i l l deal w i t h s o l v i n g the problem o f determining wG, which i s (even f o r x : 1) i n general an NP-canplete problem [59],
by deccmpos
Subsequently, sane h i n t s on t h e covering problem a r e added. F a c t o r i z a t i o n Problem: w i t h fi : R!LiI-lRk
A partition
71
= {Ll,
...¶Lml, a
and a normal c l u t t e r C ' over {B1,
function f = ( f l
...,@,1 c o n s t i t u t e
s o l u t i o n t o the f a c t o r i z a t i o n problem f o r wi; w i t h G = ( V , E ) w
( x ) = max
1
y(6)
iff
i = 1,
w i t h ~ ( 6 := ~ )f i ( x l L i ) ,
...
TeC' 6eT
holds. Note t h a t t h i s approach i s q u i t e n a t u r a l f o r a step-by-step computation o f graph parameters v i a decomposition. Gi
The idea i s t o p a r t i t i o n G i n t o d i s j o i n t subgraphs
:= G / L i and t o a l l o w an a r b i t r a r y computation on G(Li,
f u n c t i o n x / L i there.
using t h e weighting
The r e s u l t i s a r e a l number, v i z . f i ( x I L i ) ,
serves as the weight o f an a r t i f i c i a l l y introduced element gi G/Li.
Concerning fi,
which then
t h a t w i l l replace
n o t h i n g i s r e q u i r e d o t h e r than t h a t i t be independent fran
29 1
Substitution decomposition for discrete structures the weights o u t s i d e Li.
This r e f l e c t s a step-by-step computation w i t h o u t the use
o f (and need f o r ! ) e x t e r n a l i n f o r m a t i o n (LOCALITY o f INFORMATION TRANSFER).
i s then assumed t h a t on the s e t V / T = { B ~ ..., , 6l,
It
o f these a r t i f i c i a l l y i n t r o -
duced elements a n o t too complicated f u n c t i o n ( a l l o w i n g e.g. the weighted c l i q u e number w . r . t .
any graph on t h i s s e t ) determines the o b j e c t i v e wG(x). This i s n e.g. i n order t o cover t h e x E Rb,
r e q u i r e d t o h o l d n o t j u s t f o r one, b u t f o r s t o c h a s t i c case (canpare Theoren 2.2.6).
all
I n f a c t , a l l these c o n d i t i o n s mean a
r e p r e s e n t a t i o n o f wG i n a s o r t o f f a c t o r i z a t i o n , d e f i n e d on s m a l l e r domains, f o r one other.
substituting certain functions,
Now, as the f o l l o w i n g theorem shows,
the s o l u t i o n s t o t h i s problen are very s p e c i a l and c l o s e l y r e l a t e d t o the s u b s t i t u t i o n decunposi t i o n .
Theorem 2.1.1:
f = (f,,
71,
PROBLEM f o r 6, i f f
...,fm)and
C ' g i v e a solu-tion t o t h e FACTORIZATION
i s a congruence p a r t i t i o n o f G, fi = w
TI
C(G') w i t h G ' = G/T.
Proof :
1.
" <="
E
Let x
lRn be a r b i t r a r y and C
E
3
V ( G ) then i m p l i e s t h a t
x(v). v c C fl Li # $ as w e l l as C ' : = n,(C) T
2.
r
1
=
...,r;
1
I
1.
For x
1
1
c
s 0 i t i s uG(x0) = 0.
Obviously, f o r a normal c l u t t e r
# $1
r
such t h a t wGI ( y ) =
Consequently wGl(f(x))
1
1
=
1
BEC '
y ( ~ )
y ( 6 ) = 0 i m p l i e s y(B) = 0 f o r a l l
Using t h e l o c a l i t y assumption, we o b t a i n fi(xo]B)
i = l,.. .,m,
Consequently
E C(G/.n).
By assumption t h i s i m p l i e s w G , ( f ( x o ) ) = 0.
TeC' BET
...,.6,
=
i s a c l i q u e i n GILi i f
x(v) 6 wG(x), concluding t h i s p a r t
0
B1,
and C' =
C(G) be such t h a t w,(x)
,. .. , ~ ) eiC(G')
r C(G).
(x/Li)
j=l
W
1J
-
a l s o C := lJ Cij
x(v) =
j=l v4..
E
=: Ci
= I BC ~ f l Li
L e t y E lRr be a r b i t r a r y and C ' = {bi
f o r j = 1,
'1711
Cn Li
GI L i
i . e . zero weights f o r a l l elements o f sane s e t Li
= 0 for a l l 71
imply a zero
weight f o r Bi.
2.
L e t x be a r b i t r a r y .
assume x
E
0 on V\Li.
have wG(x) =
o
~
0
We want t o study f i By l., f i ( x )
0
f o r sane f i x e d io.W.1.o.g.
= 0 f o r a l l i # io( * ) .
we
Consequently, we
( x ( L, i ) ~which, ~ by assumption, equals w G , ( f ( x ) ) , which, due t o 0
292
R.H. Mohring and FY. Radermacher Thus we have proved t h e f i r s t p a r t o f t h e statement,
(*), i s j u s t fi ( x ( L i ) . 0
v i z fi(xILi) 3.
0
= wGILi
Assume now t h a t
autonunous.
(xlLi)
TI
f o r a l l i = 1,
...,m.
i s n o t a congruence p a r t i t i o n , i . e . t h a t sane Li i s n o t G-
Then t h e r e are vl,v2
E Li
and some v
( i . e . are contained i n some c l i q u e ) b u t (v,,v)
L . # Li such t h a t (vl,v) E E J E ( i . e . a r e n o t contained i n a 6
Depending on I B . ,c_~ T f.o} r sane T r C ' (Case 1) o r I B ~6,T ~ f o r. ) 1 J J 1 i n Case 1 ( l e a d i n g w i t h 1. and 2. t o IV2.,V 1 w ( x ) = 1 # 2 = max 1 y ( 6 ) ) and x = 1I V l ,V) i n Case 2 ( l e a d i n g t o wG(x) = 2 # 1 G T e ' 6eT clique).
all T
E
C ' (Case 2), choose x =
1
= max y ( ~ ) ) ,b o t h a c o n t r a d i c t i o n . TCC' g r T
F i n a l l y assume C' # C(G/v).
4.
with T'
G
(Here l B ( a ) :=
We show f o r any T '
E
A{
Thus
C ' t h a t t h e r e i s T E C(G/TI)
T and vice-versa, which, b y t h e i n c o m p a r a b i l i t y o f c l u t t e r members,
y i e l d s e q u a l i t y o f b o t h systems. f i r s t case.
By symmetry we may r e s t r i c t ourselves t o the
Now p u t t i n g f o r T ' = I B ~ ~1 ,x .= .1. , B ~ r
, it
a r b i t r a r i l y chosen o u t of Li
IVil
i s (by 1,Z) y ( 6 ) =
j quently wG(f(x)) < I T ' I =
1
I
6eT
9 . .
with v i
.a V i r
j
1I 6 i ,..., B i I 1 r
and conse-
y ( 6 ) = max 1 y ( ~ ) a, c o n t r a d i c t i o n . m TtC ' 6rT
Note t h a t t h i s r e s u l t leads n o t o n l y t o the n o t i o n o f congruence p a r t i t i o n s , b u t a l s o t o q u o t i e n t s ( v i a C(G/n)) and autonunous s u b s t r u c t u r e s ( v i a w
). Also GI L i note t h a t i t extends t o d e t e r m i n a t i o n o f a l l (x-)maximal weighted c l i q u e s . I n
f a c t these are e x a c t l y obtained by s u b s t i t u t i n g elements 6 i
from ( f ( x ) - ) maximal j weighted c l i q u e s i n G/TI by a r b i t r a r y ( X ILi -)maximal weighted c l i q u e s . j Now w h i l e i n Theorem 2.1.1 the t r a n s f o r m a t i o n r u l e s f = ( f l,...,fm) were allowed t o be a r b i t r a r y f u n c t i o n s , we asymmetrically asked f o r a r a t h e r s p e c i a l f u n c t i o n on V j n .
O f course, a r e l a x a t i o n t o
oG(x) = H ( f ( x ) )
would be even more s a t i s f y i n g .
w i t h H:
5 +R:
arbitrary
Note, however, t h a t no reasonable r e s u l t s a r e t o
be expected i n such a general f o r m u l a t i o n , due t o t h e f a c t t h a t t h e r e e x i s t b i j e c t i o n s fi: t R i L i l + R 1 f o r any l L i [ . T
= lL1
fil(y,,,))
,...,I,,!
Therefore, t a k i n g any p a r t i t i o n and associated b i j e c t i o n s fi, H(yl ym) := w ( f -1(yl), G 1 >,
,...,
y i e l d s a t r i v i a l s o l u t i o n t o t h e f a c t o r i z a t i o n problem.
...,
These a r e o f
course s o l u t i o n s w i t h o u t any p r a c t i c a l o r c m p u t a t i o n a l i n t e r e s t , as they j u s t
293
Substitution decomposition for discrete structures mean t r a n s i t i o n t o a d i f f e r e n t (and i n f a c t even more d i f f i c u l t ) encoding o f
w >,I L i I . As we w i l l see, i t i s already s u f f i c i e n t t o exclude such a behaviour f o r the fi i n order t o o b t a i n a c h a r a c t e r i z a t i o n analogous t o t h a t o f Theorem 2.1.1.
I n fact,
we w i l l o n l y use t he weak assumption t h a t no fi transforms a s e t homeomorphic t o Cer t ainly, such a condition means no actual r e s t r i c t i o n f o r a p r a c t i c a l decomposition procedure, since the excluded functions
IR2 b i j e c t i v e l y 6nto a subset of R1. are t o o complicated w . r . t .
computational o r s t o c h a s t i c aspects.
, but
f u n c t i o n s may s t i l l be Bore1 measurable [117] nor order-preserving)
Theorem 2.1.2:
'TI,
are, e.g.,
( I n general, such
n e i t h e r continuous
.
f = ( fl,...,fm)
( w i t h no fi transforming a s e t homeomorphic t o
R* b i j e c t i v e l y onto a subset o f R1) and H g i v e a s o l u t i o n t o the (modified) FACTORIZATION PROBLEM f o r G , i f f fi( zIL i )
implies that
(xlLi) GlLi wG(x*) w i t h x*(v) := xi(v) f o r v
Proof:
"<="
i s a congruence p a r t i t i o n f o r G, fi(xILi)
TI
= uGlL.(ZILi)
w
=
and H(fl(X,ILl)y...,fm(XmILm))
=
1
6
i = 1 ,...,m.
Li,
i s clear , as H i s def ined independently o f representatives, due t o
the a d d i t i o n a l assumption concerning the fi and the f a c t t h a t
TI
i s a congruence
p a r t i t i o n (together w i t h Theorem 2.1.1). 'I-"
1.
Note t h a t once t he fi are given, H must necessarily be defined i n the
way described, because o f the l o c a l i t y condit ion. 2.
Now concerning the a d d i t i o n a l assumption on the fi, we may assume w.1.o.g. implying f . ( x l L . ) = f . ( z l L . ) f o r a l l j # i. J J J J = fi(zILi) im plies f ( x ) = f ( z ) , i.e. wG\Li(xILi) = wG(x) =
both x : 0 and z z 0 on A\Li, Consequently, fi (x( Li)
H ( f ( x ) ) = H ( f ( z ) ) = %( z) = u,-.,,.(zIL~),
i.e.
the above statement.
1
3.
F i n a l l y assume
=
TI
{L1,
...,$,}
Then
n o t t o be a congruence p a r t i t i o n o f 6.
some Li i s n o t G-autonomous, i . e . t her e are v l , v 2 ~ Li and sane v* E L . # Li such J E. We then consider the set X o f weights on Li t h a t (v,,v*) c E b u t (v,,v*)& ILi I d e fi ned by X: = { x E IJl
I
x( v) = 0 f o r v # v1,v2;
Obviously, X i s homeomorphic t o R2.
0
6
1 x(vl)
<
x ( v 2 ) 4 101.
We w i l l show t h a t fi must map X b i j e c t i v e l y
onto a subset of R1, thereby obt aining a c o n t r a d i c t i o n . To t h i s end, assume f i ( x ( L i )
x ( v2 ) =
wGIL
(xlLi) i
=
= fi(zILi)
wGILi(z(Li)
f o r sane x,z e X.
= z(v2).
By Z . , we obtain
So we o n l y have t o show x(vl)
=
z(v,).
R. H. Mohring and F.J. Radennacher
294
To t h i s end, consider the extension o f xlLi x(v) = z(v) = 0 f o r a l l v obviously, fi(xILi)
E
V\(LiU
= fi(zIL.)
!v*})
t o weights on V by p u t t i n g
and zlLi
and x(v*) = z(v*) = 10.
Then,
i m p l i e s f ( x ) = f ( z ) and consequently 10 t x(vl)
1
uG(x) = H ( f ( x ) ) = H ( f ( z ) ) = wG(z) = 10 + z(vl),
i m p l y i n g x(vl)
= z(vl),
=
i . e . the
claimed b i j e c t i v i t y . m Due t o the strong c h a r a c t e r i z a t i o n s (Theorems 2.1.1
and 2.1.2)
the n o t i o n o f con-
gruence p a r t i t i o n e n t e r s almost i n e v i t a b l y and n a t u r a l l y i n c a n b i n a t o r i a l o p t i m i z a t i o n over graphs.
This extends t o many o t h e r questions i n v o l v i n g c-maximal
c l i q u e s o f minimal weight, b o t t l e n e c k extrema, problems w i t h l e x i c o g r a p h i c o b j e c t i v e s , extrema i n v o l v i n g o r d i n a r y m u l t i p l i c a t i o n and, f i n a l l y , problens over d i f f e r e n t s e t systems having the same congruence p a r t i t i o n s , i . e . such as t h e b l o c k e r o r a n t i b l o c k e r o f C(G) (which correspond t o t h e systems o f maximal independent sets o r minimal c l i q u e separating sets, c f . the i n v a r i a n c e r e s u l t s i n 1.4). Such r e s u l t s , though i n a somehow weaker v e r s i o n , w i l l f o l l o w i n t h e general approach d e a l t w i t h i n 11.3.
We w i l l c l o s e t h i s s e c t i o n w i t h a h i n t on COLOURING
PROBLEMS (and thus a l s o c l i q u e covering problems, which can be t r e a t e d c a n p l e t e l y analogously).
There again i t t u r n s o u t (though i n a somewhat weaker f o r m u l a t i o n
s i m i l a r t o those i n 11.3) t h a t congruence p a r t i t i o n s y i e l d the o n l y decanpositions. I n f a c t , the proof (which i s a n i t t e d ) a l s o g i v e s a c o n s t r u c t i v e method f o r f i n d i n g minimal (weighted) c o l o u r i n g s o f a graph v i a s u b s t i t u t i o n d e c a p o s i t i o n , u s i n g minimal (weighted) c o l o u r i n g s o f autonanous subgraphs and q u o t i e n t s and necess i t a t i n g vertex mu1ti p l i c a t i o n .
Theorem 2.1.3:
Let
G
=
(V,E)
l e t G ' be a graph on V ' = V/I
iff
11.2
T
be a graph,
...,6,).
= { B ~ ,
II
= {L1,...,Lr)
be a p a r t i t i o n o f V and
Then
i s a congruence p a r t i t i o n o f G and G' = G/II.
COMBINATORIAL OPTIMIZATION OVER PARTIAL ORDERS
The d i s c u s s i o n o f p a r t i a l orders i s i n t e r e s t i n g f r a n t h e view-point o f operations research and i t s a p p l i c a t i o n s , because p a r t i a l orders n a t u r a l l y correspond t o proj e c t networks ( p a r t i c u l a r l y the CPM-case), which are b a s i c f o r d e a l i n g w i th schedu l i n g theory and i t s p r a c t i c a l a p p l i c a t i o n s , e.g. i n b u i l d i n g and c o n s t r u c t i o n i n d u s t r y and i n p r o d u c t i o n planning and machine scheduling. i s due t o the f a c t t h a t a (CF'M-) order B = (A,O),
This correspondence
p r o j e c t network may be i n t e r p r e t e d as a p a r t i a l
where A i s t h e s e t af a c t i v i t i e s and (a,B) E S O
295
Substitution decomposition for discrete structures
I
(:= O\{(a,a)
a E A } ) means t h a t a must b e completed b e f o r e 5 can be s t a r t e d .
DETERMINISTIC PROJECT NETWORKS When d e a l i n g w i t h d e t e r m i n i s t i c p r o j e c t networks, one o f t h e i n t e r e s t i n g param e t e r s i s t h e s h o r t e s t p r o j e c t d u r a t i o n AO(x), depending on a c t i v i t y d u r a t i o n s n It i s known t h a t A o ( x ) = max [ESo[x](a) + x ( a ) ] , where ES, i s t h e x e 07,. a d
e a r l i e s t s t a r t schedule d e f i n e d i t e r a t i v e l y as
T h i s r e p r e s e n t a t i o n l e a d s t o an O(n2) procedure t o canpute A,(x)
f o r any x c€R>”.
I n f a c t , i t extends t o t h e more g e n e r a l s i t u a t i o n o f m i n i m i z i n g any ( w e l l behaved) r e g u l a r c o s t f u n c t i o n ( r e g u l a r measure o f performance [93] ) K : R: which i s a m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n o f t h e c a n p l e t i o n t i m e s tl the a c t i v i t i e s a,
,. . . ,a,
and g i v e s t h e performance c o s t
w i t h t h e s e c a n p l e t i o n times. K(@;x) := ~(Es,[x](a~) scheduling
o
+ x(al)
K(
tl ,.
. . ,tn)
+
W,,1
,. .. ,tn o f
associated
It i s e a s i l y obtained t h a t
,...,ESo[x](an)
f o r a c t i v i t y durations x
E
+ x ( a n ) ) gives the lowest c o s t f o r
Wr. Note t h a t p r o j e c t d u r a t i o n i s
i n c l u d e d h e r e as a s p e c i a l case b y p u t t i n g
= max.
K
However, f o r t h i s s p e c i a l
o b j e c t i v e , a f u r t h e r , d u a l r e p r e s e n t a t i o n can be g i v e n , v i z . A0(x) :=
max KcC( 0)
c x ( a ) , where C(O) denotes t h e system ofG-maximal c h a i n s i n o ( r e g a r d e d as s e t s a€ K
r a t h e r than l i n e a r orders). Thus t h e s h o r t e s t p r o j e c t d u r a t i o n equals t h e l e n g t h o f a l o n g e s t c h a i n , t h e soc a l l e d c r i t i c a l path length.
G i v e n t h e f a c t t h a t t h e s m a x i r n a l c h a i n s i n EI a r e
e x a c t l y t h e S-maximal c l i q u e s i n t h e a s s o c i a t e d c a n p a r a b i l i t y g r a p h G(O), i . e . C ( o ) = C(G(o)), we may e q u a l l y w e l l i n t e r p r e t A,(x)
t o t h e s i t u a t i o n i n 11.1.
Theorems 2.1.1 and 2.1.2
as w
( x ) , i . e . we a r e back G(0) t h u s extend t o t h e problem o f
f a c t o r i z a t i o n o f t h e s h o r t e s t p r o j e c t d u r a t i o n f u n c t i o n o f a network and t e l l us t h a t , g i v e n t h e l o c a l i t y c o n d i t i o n as d e s c r i b e d , t h e congruence p a r t i t i o n s o f t h e a s s o c i a t e d c o m p a r a b i l i t y graph d e s c r i b e
fl e x i s t i n g
i n t e r f a c e r e s u l t s i n 1.5 (Theorem 1.5.1),
decanpositions.
Given the
these a r e e s s e n t i a l l y (up t o c e r t a i n
symmetric d i f f e r e n c e s ) t h e congruence p a r t i t i o n s o f t h e u n d e r l y i n g p a r t i a l o r d e r . I t i s , however, p o s s i b l e t o g i v e a s t r o n g e r f o r m u l a t i o n , which l e a d s e x a c t l y t o
t h e congruence p a r t i t i o n s o f t h e p a r t i a l o r d e r O; compare [77],
0211.
This i s i n
f a c t an immediate consequence o f Theorem 2.1.1.
C o r o l l a r y 2.2.1:
L e t 0 = ( A , O ) be a poset,
IT
= IL1,
.... Lr)
be a p a r t i t i o n o f A
296
R. H. Mohring and F. J. Radermacher
and A ' := A/T := { ~ ~ , . . . , 6 ~ ] . Then: 1 ' 1,
f = (fl
,...,f m )
and 0' = ( A ' , O ' )
I 0'1,
(cx&) e 0 i m p l i e s ( B . , B . ) J
h g ( x ) = n,,(y)
iff
TI
1
[where o ' i s such t h a t a e Li,
i = 1 ,..., r, f o r a l l x ER'
w i t h ~ ( € 3 : ~= )f i ( x I L i ) ,
i s a congruence p a r t i t i o n f o r 0, fi =
i # j,
a ' e L., 3
y i e l d s the e q u a l i t y
>/
(xlLi). i
i = 1,
...,r
and o ' = o / n .
Corol1ar.y 2.2.1 s t i m u l a t e d study o f the s u b s t i t u t i o n decomposition f o r p a r t i a l orders ( p r o j e c t networks), c f . a l s o [42], e.g.
[145],
a l l t h e more so as i t extends t o
s h o r t e s t path canputation, d e t e r m i n a t i o n o f t h e weighted independence number
(which d e f i n e s f e a s i b i l i t y i n scheduling problems p r o j e c t networks and many other s u b j e c t s [793, i n 11.3.
[loo],
[127];
[125]),
f l o w problems i n
see a l s o the g e n e r a l i z a t i o n
For p r o j e c t networks C o r o l l a r y 2.2.1 means t h a t a t l e a s t one b a s i c net-
work parameter, v i z . s h o r t e s t p r o j e c t d u r a t i o n , can be handled v i a s u b s t i t u t i o n decanposition.
Fortunately,
i t turned o u t t h a t t h i s i s e q u a l l y t r u e f o r most
other aspects o f p r o j e c t networks ( w i t h t h e exception o f many problens i n v o l v i n g resource c o n s t r a i n t s ) .
For instance, we have already p o i n t e d o u t t h a t a c t i v i t y -
on-node and a c t i v i t y - o n - a r c diagrams as w e l l as t h e d e t e r m i n a t i o n o f a t o p o l o g i c a l sorting, can be obtained t h i s way.
The same i s t r u e f o r t h e c h a r a c t e r i s t i c a c t i v i t y
times (such as e a r l i e s t s t a r t and l a t e s t f i n i s h ) and the r e s u l t i n g f l o a t s ( o r s l a c k s ) (such as t o t a l f l o a t (TF), f r e e f l o a t (FF), backward and independent f l o a t and path f l o a t ) , canpare 0 2 4 1 . congruence p a r t i t i o n n = {L, ESoILi[x/Li](a)
f o r a e Li,
For exanple, i t t u r n s o u t t h a t , given a
,..., L r } o f o i = 1, ...,r .
and y = f ( x ) , ESo[x](a)
= ESolGY](~i)
O f course, t h i s y i e l d s the p o s s i b i l i t y
o f determining l e a s t p r o j e c t costs K ( O ; X ) v i a s u b s t i t u t i o n decomposition. very analogous behaviour, v i z . TFo[x] ( a ) = TF,CS;] total float.
t
( Bi) t TF,l
Li
[XI
We see
Li] (a), f o r t h e
This has the consequence t h a t t h e c l u t t e r o f c r i t i c a l paths over
the c r i t i c a l elements i s j u s t the s u b s t i t u t i o n o f t h e corresponding c l u t t e r s belonging t o the r e s t r i c t i o n o f n t o t h e s e t o f c r i t i c a l elements. f o r the o t h e r f l o a t s i s more canplicated. FF&]
The s i t u a t i o n
For instance,
FFo,[y](~i)+FFolLi[x(Li](a)
a maximal i n olLi
FFolLi C x I L i l ( a )
otherwise.
(a) =
This l a s t r e s u l t shows t h a t the f r e e f l o a t o f an a c t i v i t y a e A i s already d e t e r mined by any (s-minimal) autonanous s e t , c o n t a i n i n g a as a non-maximal element. A l t o g e t h e r , these observations show t h a t the c l a s s i c a l time a n a l y s i s o f p r o j e c t networks as a whole allows t h e employment o f the s u b s t i t u t i o n decanposition. f a c t , t h i s extends ( c f . [62])
even t o t h e VPM case [131]
In
and i t s g e n e r a l i z a t i o n s
291
Substitution decomposition far discrete structures
where a r b i t r a r y time c o n s t r a i n t s between s t a r t i n g time and c a n p l e t i o n t i m e o f any two a c t i v i t i e s a r e allowed.
These g e n e r a l i z a t i o n s imply a s t r a i g h t f o r w a r d t r a n s -
i t i o n t o r e l a t i o n a l systems, each r e l a t i o n d e s c r i b i n g another type o f time cons t r a i n t s i n the MPM-network.
TIME-COST TRADE OFF I N PROJECT NETWORKS The s i t u a t i o n considered here i s s t i l l d e t e r m i n i s t i c , i.e. a c t i v i t y d u r a t i o n s are n o t random.
However, durations may be v a r i e d t o some e x t e n t by f i n a n c i a l i n p u t s ,
where s h o r t e r d u r a t i o n s r e q u i r e a higher i n p u t .
A general model f o r such a s i t u -
a t i o n i s g i v e n by p r o j e c t networks w i t h costs systems (o,K) where
:= (A.O,(ka)aeA),
ka, f o r any a E A , i s a R ' , where IaC Wl g i v e s t h e p o s s i b l e
= (A,O) describes t h e p r o j e c t s t r u c t u r e , w h i l e
EI
m o n o t o n i c a l l y decreasing f u n c t i o n k a :
Ia
-*
d u r a t i o n s f o r a , a n d k , ( x ( a ) ) f o r x ( a ) E I(a) denotes the associated c o s t ( r e q u i r e d financial input).
Given (A,O,(ka)aeA),
the main i n t e r e s t concerns the f o l l o w i n g
two problems: Determination o f t h e minimal c o s t f u n c t i o n H ( t ) : Given a f i x e d time l i m i t t
1.
f o r t h e p r o j e c t d u r a t i o n , what i s the l e a s t c o s t H ( t ) f o r achieving t h i s task? Determination o f the minimal time f u n c t i o n H*(k): Given a f i x e d budget k,
2.
what i s the s h o r t e s t p r o j e c t d u r a t i o n H*(k) obtainable w i t h t h i s budget? From now on we w i l l concentrate on H, although H* behaves s i m i l a r l y .
under s u f f i c i e n t l y strong assumptions (e.g.
a l l ka s t r i c t l y monotonically decreas-
ing and convex on closed i n t e r v a l s Ia, canpare [12]), of H.
X Ia and n,(x)
6
H* i s the i n v e r s e f u n c t i o n
More d e t a i l e d , H i s d e f i n e d on t h e s e t o f possible s h o r t e s t p r o j e c t
d u r a t i o n s J := { A ~ ( X )
x
I n fact,
6
x E X I a l and i s g i v e n by H ( o , K ) ( t ) := i n f I c k a ( x a ) 1 @A CLeA t), t Q J. Note t h a t the infimum may n o t be a t t a i n e d and
I
aeA
t h a t canputation o f H ( t ) may be a d i f f i c u l t problem.
I n f a c t , f o r piecewise
a mixed l i n e a r o p t i m i z a t i o n l i n e a r cost f u n c t i o n s ka on closed i n t e r v a l s Ia, problem w i t h b i n a r y i n t e g e r c o n s t r a i n t s has already t o be solved.
However, i f t h e
k a are a l s o convex, a l i n e a r d e s c r i p t i o n i s p o s s i b l e t h a t does n o t r e q u i r e i n t e g e r
variables, f a c i l i t a t i n g e f f i c i e n t canputation.
As, furthermore, the l a r g e class
of a l l convex c o s t f u n c t i o n s can be (smoothly) approximated t h i s way, t h e r e are good reasons f o r concentrating on t h i s class o f piecewise l i n e a r , convex cost functions.
An a d d i t i o n a l argument i n t h i s d i r e c t i o n f o l l o w s below (Theorem 2.2.3).
But f i r s t , we w i l l t r e a t the decanposition problem f o r t h i s case.
I n f a c t , one
obtains the f o l l o w i n g theorem as a g e n e r a l i z a t i o n o f Theorem 2.1.1
and C o r o l l a r y
2.2.1;
cf.
[lz]:
R.H Mohringand F.J. Radermacher
298 Theorem 2 . 2 . 2 :
Given ( o , K ) = (A,O,(ka)acA), a p a r t i t i o n
TI
p a r t i a l order 0' on A ' = C13~,...,6~1, and c o s t f u n c t i o n s k have H ( 0 , K ) = H(Oi , K ' ) i f f
TI
o f A, a
= { L l,...,Lr} = fi((ka)aeLi)
1 3 i
we
i s a congruence p a r t i t i o n o f G(o) ( o r even o f EI i n
the stronger v e r s i o n ) , G ( o ' ) = G ( o ) / i r ( o r even 8 ' = @/IT)and ksi = H(o,K),Li. So again, o n l y s u b s t i t u t i o n decanposition a l l o w s i t e r a t e d d e t e r m i n a t i o n o f the
minimal c o s t f u n c t i o n under t h e above assumptions, and t h i s i s done by canputing f i r s t the minimal c o s t f u n c t i o n s o f t h e associated autonanous suborders and then u s i w these f i i n c t i o n s as c o s t f u n c t i o n s f o r t h e image elements.
Following t h i s
approach, i t i s o f course i m p o r t a n t f o r p r a c t i c a l a p p l i c a t i o n s t h a t t h e considered class o f cost f u n c t i o n s i s closed w . r . t .
time-cost t r a d e o f f , i.e.
f o r each n e t -
work w i t h a c t i v i t y cost f u n c t i o n s f r a n t h e g i v e n class, t h e minimal c o s t f u n c t i o n shculd a l s o belong t o t h e c l a s s .
For such classes o f c o s t f u n c t i o n s , decanpo-
s i t i o n w i l l then n o t l e a d t o more " c a n p l i c a t e d " c o s t f u n c t i o n s .
Fortunately,
the piecewise l i n e a r , convex c o s t f u n c t i o n s behave t h i s way; i n f a c t , an even stronger i n s i g h t i s p o s s i b l e [lZ]:
Theorem 2.2.3:
The c l a s s o f piecewise l i n e a r and convex c o s t f u n c t i o n s on
closed i n t e r v a l s i s t h e
least closed
c l a s s o f c o s t f u n c t ons c o n t a i n i n g the l i n e a r
c o s t f u n c t i o n s on closed i n t e r v a l s . Theorem 2.2.3
t e l l s us t h a t l i n e a r c o s t f u n c t i o n s do n o t form a closed class.
However, i f we r e s t r i c t ourselves t o
[.,-[- l i n e a r
k a ( x ( a ) ) := -a.x(a) + b, a,b I 0,
= [c,-[,
Ia
c o s t functions,
i.e. functions
c % 0, then a t l e a s t w i t h regard
t o the two two-element prime p a r t i a l orders, closedness i s obtained. t h i s then extends t o a l l s e r i e s - p a r a l l e l networks; i n f a c t we have
Theorem 2.2.4: systems K o f
Given a f i n i t e poset
[. ,-[- l i n e a r
or
H
(0.K)
cost f u n c t i o n s over
O f course,
PZ]:
[.,-[- l i n e a r f o r a l l c o s t o iff o i s series-parallel. is
The p r o o f o f t h i s theoren i s e s s e n t i a l l y based on t h e r e s u l t t h a t a p a r t i a l order i s s e r i e s - p a r a l l e l i f f i t does n o t c o n t a i n a suborder i s a n o r p h i c t o (A,O) w i t h A = (1,2,3,419 S o = { ( 1 9 3 ) 3 ( 1 , 4 1 3 ( z 9 4 ) } ; cf. a l s o [12], [80], D60]. Theorem2.2.4 r e f l e c t s s p e c i a l f e a t u r e s o f s e r i e s - p a r a l l e l networks, due t o t h e simple n a t u r e
o f t h e prime s t r u c t u r e s frcm which they are i t e r a t i v e l y b u i l t up.
This special
nature a l s o plays a r o l e i n another c o n t e x t which d e a l s w i t h one o f t h e few known a p p l i c a t i o n s o f s u b s t i t u t i o n decanposi t i o n t o scheduling theory.
Substitution decomposition for discrete structures
299
ONE MACHINE SCHEDULING Consider t h e NP-complete problem o f s c h e d u l i n g n j o b s w i t h a r b i t r a r y p r o c e s s i n g t i m e s on one machine s u b j e c t t o a r b i t r a r y precedence c o n s t r a i n t s among t h e j o b s such as t o MINIMIZE TOTAL WEIGHTED COMPLETION TIME [93].
I n [92],
p44]
i t was
shown t h a t t h i s problem can be t r e a t e d v i a s u b s t i t u t i o n d e c m p o s i t i o n , d u e t o t h e f a c t t h a t any o p t i m a l sequence f o r t h e j o b s o f an autonomous s e t can be extended t o an o p t i m a l sequence f o r a l l j o b s .
Theorem 2.2.5:
T h i s has t h e f o l l o w i n g consequence:
m2
There i s an O(n ) a l g o r i t h m f o r t h e m i n i m i z a t i o n o f t h e t o t a l
weighted c o m p l e t i o n t i m e i n a one machine s c h e d u l i n g problem w i t h a r b i t r a r y p r o c e s s i n g t i m e s and precedence c o n s t r a i n t s , p r o v i d e d t h a t t h e precedence r e l a t i o n i s i t e r a t i v e l y o b t a i n e d v i a s u b s t i t u t i o n d e c a n p o s i t i o n from prime p a r t i a l o r d e r s
EN
w i t h a t m o s t m elements, m
fixed.
The a l g o r i t h m f i r s t decides whether t h e precedence r e l a t i o n i s o f t h e d e s c r i b e d t y p e . T h i s can be done i n O(n3) t i m e w i t h t h e methods presented i n S e c t i o n I V . Proceeding by i n d u c t i o n , we may t h e n assume t h a t t h e precedence r e l a t i o n has a congruence p a r t i t i o n w i t h a t most m b l o c k s , and t h a t each b l o c k i s a l r e a d y o p t i m a l l y o r d e r e d i n a l i n e a r o r d e r . We t h e n c o n s i d e r a l l subproblems induced by f i x e d p o s i t i o n s f o r t h e l a s t j o b i n each b l o c k and f i x e d numbers o f j o b s f r o m each b l o c k t o precede these l a s t j o b s .
F o r each such subproblem, t h e o p t i m a l sequence Since
can be determined by methods f o r p a r a l l e l c h a i n s i n O(n l o g n ) t i m e [92].
The d e t a i l s
t h e r e a r e nm(m-1)-2 such subproblems, we o b t a i n t h e above c o m p l e x i t y . o f t h i s a l g o r i t h m w i l l be p u b l i s h e d s e p a r a t e l y .
STOCHASTIC PROJECT NETWORKS There a r e good reasons f o r t r e a t i n g t h e s t o c h a s t i c v e r s i o n s o f t h e u s u a l q u e s t i o n s c o n c e r n i n g p r o j e c t networks.
We w i l l r e s t r i c t o u r s e l v e s h e r e t o t h e case o f
f i x e d p r o j e c t s t r u c t u r e s and random a c t i v i t y d u r a t i o n s . more g e n e r a l GERT-Networks [113], p o s i t i o n [113]
[120]
,
[164],
We h i n t however a t t h e
f o r which s e r i e s - p a r a l l e l decom-
as w e l l as g e n e r a l s u b s t i t u t i o n decomposition [96]
A s t o c h a s t i c p r o j e c t network i s g i v e n by (A,O,P),
where
i s possible.
P i s the j o i n t d i s t r i b u t i m
o f a c t i v i t y d u r a t i o n s , i . e . a p r o b a b i l i t y measure on ( R , ! A l , B F l ) ; where B,!Al I * \ . L e t t h e r e a l random v a r i a b l e X . denotes t h e system o f Bore1 s e t s o f R>, J
describe the duration o f
C X . and
J
a . e A , e x i s t . For any B 6- A, 3 w i t h t h e X j . a . r B. We w i l l J
assume t h a t t h e expected d u r a t i o n s E(X . ) , J
l e t PB denote t h e m a r g i n a l d i s t r i b u t i o n a s s o c i a t e d
-
except f o r sane h i n t s on bounds
g i v e n below -
R. H. Mohrimg and EJ. Radermacher
300
concentrate on t h e independence case, i . e . assume t h a t P
x Pa.
=
A main o b j e c t
a A
o f i n t e r e s t is then the d i s t r i b u t i o n o f t h e s h o r t e s t p r o j e c t d u r a t i o n P more generally, f o r measurable and l i n e a r l y bounded o f the l e a s t p r o j e c t cost). distribution function F
hO
P K(O,.)
o r s e c u r i t y q u d n t i l e s ) the needed i n f o r m a t i o n concerning This i s even more t r u e as, i n general,
,...,
E(Xn)) < E [{J,i . e . d e t e r m i n i s t i c p l a n n i n g techiiiques - as does P - s y s t e m a t i c a l l y underestimate t h e expected p r o j e c t d u r a t i o n Q7], [97], cl2].
no(E(X1)
-
the d i 3 t r i b u t i o n
(or,
These d i s t r i b u t i o n s g i v e (e.g. v i a the associated
planning, scheduling and d e c i s i o n making. PERT
K,
AO
Unfortunately, t h e treatment o f t h e s t o c h a s t i c case t u r n s o u t t o be d i f f i c u l t because o f inissiny data and r a t h e r i n v o l v e d computational requirements [75], [135].
So, a p a r t from q u i t e s p e c i a l cases [77]
,
[146]
and from bounds f o r t h e
d i s t r i b u t i o n f u n c t i o n s such as t h e ones discussed below, m a i n l y s i m u l a t i o n cdn p r a c t i c a l l y be used t o c a l c u l a t e F
A0
, i n general.
O f course, i n view of these
d i f f i c u l t i e s , t h e q u e s t i o n o f decomposition becomes even more urgent.
Again we
have the f o l l o w i n g theorem c h a r a c t e r i z i n g s u b s t i t u t i o n decomposition. Given a network 8 = ( A , O ) ,
Theorem 2.2.6: ~1
=
t L ,,..., L r i o f A, a poset
0'
0 Pa, a p a r t i t i o n Ci€A and d i s t r i b u t i o n s
a distribution P =
on A ' = tB1
,...,By),
), we have P = P'!,@,' i f f TI i s a congruence p a r t i t i o n o f G(0) ( o r Li even o f c i n the s t r o n g e r v e r s i o n ) , G(0') = G ( a ) / n ( o r even 0' = O / T ) and r P ' = .@ P w i t h P; . = (PILi), . I = I 'i 1 01 Li Pai
= fi(P
L,
Theorem 2 . 2 . 6 shows t h a t again e x a c t l y the s u b s t i t u t i o n decomposition a l l o w s determination o f the s h o r t e s t p r o j e c t d u r a t i o n as r e q u i r e d , by f i r s t doing t h i s f o r a1 1 associated suborders and subsequently for the q u o t i e n t , where marginal d i s t r i b u t i o n s are those obtained p r e v i o u s l y on the r e s p e c t i v e classes. there are more general versions [123]
Note t h a t
n o t r e q u i r i n g s t o c h a s t i c a l independence.
However, these do n o t h e l p much i n p r a c t i c a l a p p l i c a t i o n s .
Theorem 2.2.6 may
h e l p considerably i n t h e exact computation of l a r g e r examples (e.g. t e s t examples
[78],
[l46]),
and can a l s o be used when s i m u l a t i o n i s employed.
This i s s i m i l a r l y
t r u e i f bounds f o r t h e d i s t r i b u t i o n f u n c t i o n FA o f t h e p r o j e c t d u r a t i o n ( o r O
analogously o f c e r t a i n p r o j e c t costs) are t o be determined. bounds a r i s e s
-
besides from computational aspects
frcm p o s s i b l e s t o c h a s t i c dependences.
-
I n t e r e s t i n such
from m i s s i n g d a t a as w e l l a s
For b o t h cases, s u i t a b l e bounds are a v a i l -
able, which, furthermore, t u r n out t o be compatible w i t h s u b s t i t u t i o n .
We w i l l
g i v e h i n t s t o b o t h approaches and s t a r t w i t h t h e case t h a t a c t i v i t y d u r a t i o n s are independent and a l l variances V(Xa),
a E
A, e x i s t .
The aim then i s , among other
t h i n g s , t o g e t around the d a t a problem, which i s o f t e n c r u c i a l i n a p p l i c a t i o n s , by p r o v i d i n g bounds depending on the p a i r s (E(Xa),U(Xa))aeA o n l y .
To t h i s end
Substitution decomposition for discrete structures
t h e f a c t i s used t h a t t h e random v a r i a b l e s YK :=
X(a),
30 I
g i v i n g t h e random
a K
l e n g t h o f t h e maximal chains K
IZ C ( O ) , a r e a s s o c i a t e d random v a r i a b l e s [7] ,[SO]. For l a r g e p r o j e c t networks, any such v a r i a b l e Y K may ( a p p r o x i m a t e l y ) be assumed t o
t o be n o r m a l l y d i s t r i b u t e d w i t h meanE(YK) =
1
E ( X a ) and v a r i a n c e V(YK)
=
aeK
1 V(Xa) - i n d e p e n d e n t l y o f t h e r e s p e c t i v e a c t i v i t y d u r a t i o n d i s t r i b u t i o n s a€ K [ 1 4 g on a s s o c i a because of t h e C e n t r a l L i m i t Theorem. Now a b a s i c r e s u l t [50], r~
t e d random v a r i a b l e s g i v e s FA ( t ) 5
F K ( t ) , FK b e i n g t h e d i s t r i b u t i o n f u n c -
KEC(0)
0
t i o n o f YK.
I t i s worth noting t h a t i n a l l a v a i l a b l e p r a c t i c a l applications t h i s stochastic upper bound (which can t h e o r e t i c a l l y be a r b i t r a r i l y bad) proved t o be q u i t e near t o t h e d i s t r i b u t i o n f u n c t i o n FA , c f . [7:78), [146]. I t i s t h e r e f o r e , when combined 0 w i t h a s t r a i g h t f o r w a r d l y o b t a i n e d s t o c h a s t i c lower bound, a u s e f u l i n s t r u m e n t f o r h a n d l i n g a p p l i c a t i o n s i n v o l v i n g s t o c h a s t i c p r o j e c t networks. These bounds can be f u r t h e r improved, i f s u b s t i t u t i o n decomposition can be used
[i'],
i . e . i f t h e bounds o b t a i n e d f o r t h e autonomous suborders induced b y t h e
c l a s s e s o f a congruence p a r t i t i o n
V ( o ) a r e used as a c t i v i t y d u r a t i o n d i s t r i b -
II E
u t i o n s f o r t h e elements o f t h e q u o t i e n t
O/T.
F i n a l l y , we d e a l w i t h t h e q u i t e n a s t y case o f s t o c h a s t i c dependences between a c t i v i t y d u r a t i o n s . To t h i s end, l e t m a r g i n a l d i s t r i b u t i o n s Pa., a c A be g i v e n J J and l e t Q denote t h e s e t o f a l l p r o b a b i l i t y measures Q over (RR,Bn) w i t h m a r g i n a l d i s t r i b u t i o n s Q,.
= Paj,
aj E A = 11,. ..,n}.
For any t clR1 and x = (xl
J EIRn
,... ,xn)
put n
$ ( t ) :=
i n 1 {(A,(X) X€R
I t can be shown [95]
-
t)+
+
f (€(Xi) i=l
-
xi)+},
where a+ := max {O,ai.
E(Z - t ) + Z i s a convex t o a l l p r o j e c t d u r a t i o n d i s t r i b u t i o n s PA ,
t h a t t h e r e i s a r e a l random v a r i a b l e
Z
such t h a t
=
+ ( t ) f o r a l l t eIR1 and, f u r t h e r m o r e , t h a t t h e d i s t r i b u t i o n o f upper bound ( i n t h e sense o f [148])
P e p.
T h i s bound i s even t i g h t i n t h e sense t h a t t o any t
such t h a t Ept(AO
-
t)+ =
E(Z - t)'.
R1 t h e r e i s P t
0
E
2
For s e r i e s - p a r a l l e l networks (but p o s s i b l y n o t
f o r a r b i t r a r y a) t h e r e i s even sane P* j u s t P;
E
E
Q such t h a t t h e d i s t r i b u t i o n o f Z i s
. 0
Using t h i s approach, i t i s f o r example p o s s i b l e t o g i v e an upper bound f o r t h e expected p r o j e c t d u r a t i o n i n t h e s i t u a t i o n d e s c r i b e d , r e g a r d l e s s o f t h e p o s s i b l e s t o c h a s t i c dependences [84],
[162].
A l s o t h e d e t e n n i n a t i o n o f t h e f u n c t i o n ji
seems t o be e f f i c i e n t l y p o s s i b l e and shows a c l o s e s i m i l a r i t y t o t h e t r e a t m e n t of
R. H. Mohriilg arid KJ. Radermachcr
302
time-cost trade o f f i n the case o f piecewise l i n e a r convex c o s t f u n c t i o n s [log].
So a l t o g e t h e r , t h e use o f t h i s method f o r p r a c t i c a l a p p l i c a t i o n s looks q u i t e promising.
A f u r t h e r n i c e f e a t u r e i s g i v e n by t h e o b s e r v a t i o n [162]
that the
employment o f the s u b s t i t u t i o n decomposition f o r a g i v e n congruence p a r t i t i o n ~f
E V ( a ) leads t o an i t e r a t i v e method t o compute $ by f i r s t l y determining the
convex upper bounds
aB
elB, and afterwards
f o r the associated autonomous suborders
using t h e d i s t r i b u t i o n s of t h e associated random v a r i a b l e s ZB as marginal d i s t r i b u t i o n s on the q u o t i e n t e/n.
Note t h a t t h i s i s s u r p r i s i n g as t h e i n v o l v e d d i s t r i b -
u t i o n s o f the randan v a r i a b l e s
ZB may never occur as a p r o j e c t d u r a t i o n
distribution.
11.3
COMBINATORIAL O P T I M I Z A T I O N OVER (IN-)DEPENDENCE SYSTEMS AND CLUTTERS
A c m o n type o f o p t i m i z a t i o n problem over independence systems I S i s g i v e n by
ma% 1 x ( a ) , x being a weighting f u n c t i o n on the base s e t A. UaIS a d systems DS i t i s g i v e n by
1
min x(a). UEDS aeU
However, i n b o t t l e n e c k problems,
replaced by e i t h e r min o r max, w h i l e i n r e l i a b i l i t y theory, f o r "+" plays a r o l e .
Over dependence
'I.
"
"+"
is
as replacement
Obviously, e.g. f o r non-negative weights, a l l these prob-
lems may e q u i v a l e n t l y be considered on the c l u t t e r o f e i t h e r c_-maximal independent o r s - m i n i m a l dependent sets, where w.1.o.g.
n o r m a l i t y may a l s o be assumed ( i . e .
any a c A can be assumed t o belong t o sane s e t i n t h e c l u t t e r ) .
So by v a r y i n g
the e x t e r n a l composition max/min, t h e i n t e r n a l canposi t i o n +/max/min/- or the c l u t t e r (e.g. maximal c l i q u e s o r independent sets o f ( c o m p a r a b i l i t y ) graphs, c-minimal cuts i n networks, and o t h e r s ) , o p t i m i z a t i o n problems as discussed i n 11.1 and 11.2, as w e l l as s h o r t e s t p a t h problems [go],
minimal o r maximal f l o w
canputation ( v i a t h e theorem o f Ford and Fulkerson [52],
[go]),
and problems i n
re1 iabi 1it y t h e o r y are i n c l u d e d . These STANDARD CASES, as w e l l as o t h e r s [25],
[166]
are covered by considering
a r b i t r a r y normal c l u t t e r s C on s e t s A and p a i r s (m, @ ) o f e x t e r n a l (m) and i n t e r n a l ( @ ) canpositions on ( s u i t a b l e ) sets o f r e a l numbers, where m = max o r min, and
0
i s assumed t o be a s s o c i a t i v e , comnutative and m o n o t o n i c a l l y
increasing w . r . t .
t o both canponents ( i . e . x1 ,x2, y1,y2 E
i m p l i e s x1 @ x 2 6 y1
r C,m, @I ( x )
:=
m
0 y2).
W 1 , x1
6 yl,
x2
B
y2
O f i n t e r e s t i s then the o p t i m a l value f u n c t i o n
@ x ( a ) , where x belongs t o a s e t o f "admissible" r e a l
TEC ~ E T
weighting f u n c t i o n s over A.
The r e s u l t i n g problem o f decomposition discussed i s :
303
Substitution decomposition for discrete structures
A partition
F a c t o r i z a t i o n Problem:
71
= { L ly...,Lm},
c l u t t e r s Ci o v e r Li and a
normal c l u t t e r C ' o v e r A/n a r e c a l l e d a s o l u t i o n t o t h e f a c t o r i i a t i o n problem f o r
r
C,myQ
iff
rc,m,O(x)
where Y ( B ~ =)
= rcl,m,a(~),
r Ci,m,O(~ILi),
i = 1,
...,m .
Note t h a t , due t o t h e d i f f e r e n t cases covered, t h i s f o r m u l a t i o n i s somewhat more r e s t r i c t i v e t h a n those i n 11.1 and 11.2. [127]
, generalizations
Though f o r a t l e a s t some o b j e c t i v e s
pa,
o f t h e s t r o n g e r v e r s i o n s t o t h e p r e s e n t case a r e p o s s i b l e ,
t h e r e i s l i t t l e hope f o r a s i m i l a r r e s u l t i n such a g e n e r a l s e t t i n g t h a t covers a l l functions
r
C,m,O
Note a l s o t h a t , w h i l e a l a r g e s e t o f a d m i s s i b l e
as described.
w e i g h t i n g f u n c t i o n s i s wanted f o r a r e s u l t on f a c t o r i z a t i o n , i t i s t h e o t h e r way round f o r a uniqueness r e s u l t . Concerning t h e f a c t o r i z a t i o n problem mentioned, B i l l e r a and B i x b y Birnbaum and Esary [14]
showed f o r t h e s t a n d a r d cases i n network and r e l i a b i l i t y
t h e o r y t h a t t a k i n g a congruence p a r t i t i o n
71
o f A, and p u t t i n g Ci
r
v e r s i o n o f these r e s u l t s , c o n c e r n i n g a l l f u n c t i o n s
as described, was sub-
C,m,@ I n t h a t paper a uniqueness r e s u l t was a l s o f o r m u l a t e d ,
s e q u e n t l y g i v e n i n [81]. 71 E
:= CILi,
A more general
C ' := C/n, a s o l u t i o n t o t h e f a c t o r i z a t i o n theorem i s g i v e n .
which s t a t e s t h a t
El] , and
V(C), Ci
f a c t o r i z a t i o n problem i f (m,@)
= CILi and C ' = C/T g i v e t h e
only s o l u t i o n s r
t o the Cymy@ Note t h a t a l l
i s r e g u l a r ; r e g u l a r i t y meaning t h a t
f o r a l l admissible weighting functions implies C = C ' . 'cI,m,@
s t a n d a r d cases (max,t),
(max,min),
(max;),
(min,+),
(min,max),
(min;)
(even w i t h r e s t r i c t i o n t o v e r y s p e c i a l w e i g h t i n g f u n c t i o n s [Sl]), and (min,min)
are not regular.
I n fact,
rCYmaxyGx) = max x ( a )
=
ae A
f o r any normal c l u t t e r C ' o v e r A ( t h e same i s t r u e f o r (min,min)), f a c t o r i z a t i o n problem i s s o l v e d b y
each p a r t i t i o n
71
o f A.
are regular
w h i l e (max,max)
rc
1
,max
,&A
and t h u s t h e
So (max,max) and (min,
m i n ) f o r m two q u i t e i r r e g u l a r (and u n i n t e r e s t i n g ) e x c e p t i o n s ( i n f a c t , e s s e n t i a l l y t h e o n l y such i r r e g u l a r i t i e s p r o v i d e d t h a t @ induces a p o s i t i v e l y o r d e r e d semigroup p66] on t h e a d m i s s i b l e w e i g h t s ) , w h i l e i n o t h e r cases, decompositions a r e - g i v e n C - t h e same f o r glJ cases, and i n p a r t i c u l a r independent o f t h e T h i s adds t o t h e understanding o f t h e occurrence o f t h e s u b s t i t u t i o n p a i r (m,@). decomposition i n a v a r i e t y o f q u e s t i o n s i n g r a p h t h e o r y , networks, f l o w networks, r e l i a b i l i t y t h e o r y and o t h e r areas, a l l t h e more so when combined w i t h t h e i n t e r face r e s u l t s discussed i n 1.5. I n t h e f o l l o w i n g , we g e n e r a l i z e a s t e p f u r t h e r , t h e (max,+)-algebra i n [33],
M o t i v a t e d by t h e d i s c u s s i o n o f
i t t u r n e d o u t t o be u s e f u l t o view ( m , @ )
m u t a t i v e s e m i - r i n g on (a subset o f ) r e a l numbers.
as a
com-
I n fact, d i s t r i b u t i v i t y f o r
m = max (min) f o l l o w s f r o m t h e o b s e r v a t i o n t h a t , g i v e n t h e monotony o f
0,
R. H. Mohring and F.J. Radermacher
304
Q
x @ max (y,z) = max ( x (min) (min)
I f we extend t h e s e t o f admissible weight-
y, x @ z ) .
i n g f u n c t i o n s accordingly, we even g e t a semi-ring w i t h zero O* and one 1* i n a l l standard cases, e.g. by proceeding according t o Table 2.1.
Here, from an appli;
c a t i o n a l p o i n t o f view, the assumption o f a r t i f i c i a l elements
"-"
and
"-ml'
can be
replaced by l i m i t arguments o r by r e s t r i c t i o n t o "extreme" upper and lower bounds, c f . C81-J. admissible weights
zero
(max,+) (min,+)
one 0 0
( max ,mi n )
+-
(min,max)
- m
(niax, .)
1
(min, .)
1
TABLE 2.1 Note t h a t types (max,max) and (min,min)
a l s o l e a d t o semi-rings, b u t these semi-
r i n g s cannot c o n t a i n a O* and a l*simultaneously, because t h e r e i s then no way t o d i s t i n g u i s h between these two elements. a r t i f i c i a l elements +m, +m* w i t h
+-
(Note t h a t even t h e i n t r o d u c t i o n o f two
< t-*
would n o t h e l p ! )
A l t o g e t h e r the above remarks m o t i v a t e t h e c o n s i d e r a t i o n o f a r b i t r a r y f u n c t i o n s
r c,
ffJ ,@ with
cases
c,+,
$36
CY
( x ) :=
@
6
x ( a ) , thereby i n t e g r a t i n g a l l standard
k C acT
mentioned above, and a l s o t h e n a t u r a l composition *
(x
:=
1 .
( + , a ) ,
i.e. the function
x ( a ) , which i s e.g. o f i n t e r e s t i n r e l i a b i l i t y t h e o r y .
Now
TEC aeT
adapting the f a c t o r i z a t i o n problem t o t h i s general case, we have t h e f o l l o w i n g theorem:
Theorem 2.3.1:
1.
If
(8.6) forms
f a c t o r i z a t i o n problem i s given i f n
2.
If
(@,a)forms
a c o m u t a t i v e semi-ring, a s o l u t i o n t o t h e
E.
V ( C ) , Ci
= CILi
and C ' = C/n.
a c o m u t a t i v e semi-ring w i t h O* and 1*, t h e o n l y s o l u t i o n s t o
the f a c t o r i z a t i o n problem a r e those given i n l.,even i f weights a r e r e s t r i c t e d t o the set {0*,1*}. Proof: - 1. L e t 71 We w i l l show t h a t
E
r
V ( C ) , Ci = CILi and C ' = C/a, i . e . C = C '
c,o,a(x)
= rcl,O,O(~),
where ~ ( f 3 i = )
[ti, i
= 1
,..., 4 .
(xlLi)
r ci ,@ ,6
and
305
Substitution decomposition for discrete structures
x i s any a d m i s s i b l e w e i g h t i n g f u n c t i o n . d i s t r i b u t i v i t y (see s t e p
2a.
*
T h i s i s e s s e n t i a l l y a consequence o f
i n the proof).
We have:
We show t h a t ( @ , @ )
L e t t h e r e be a O* and 1* f o r ( @ , @ ) .
pair, i.e.
that %,@&l
= rC * , @ , @
w e i g h t s O* and 1* a l o n e ) .
C = C* ( t h i s i s a l r e a d y t r u e w . r . t .
Assume
rC , O , @
=
rc*,@,@
w i t h C # C*.
W.1.o.g. 1* a d o
T* $ C* f o r a l l T* s To.
P u t x o ( a ) := O*
1*, w h i l e 2b.
r
Ci,@,@
Finally l e t
IT,
t h e r e i s some T o € C such t h a t
.
Then o b v i o u s l y
"$To
C'
3
8
,@
(y). (y).
. . ,r
(xo) =
0
C i , i = 1,
...,r,
and C ' be any s o l u t i o n t o t h e f a c t o r i z a t i o n
P u t C* = C ' cCi , i = 1,. Consequently, we have r
regularity, implies
rC,$,@
( x ) = 0*, a c o n t r a d i c t i o n .
problem f o r some r e g u l a r p a i r ( Q , @ ) . Then by assumption,
r C',O,QI r
i s then a r e g u l a r
f o r c l u t t e r s C,C* on t h e base s e t A i m p l i e s
..,r] .
C,@,@
c
= C*,
i.e.
c
= C'
[Ci,
rC,Q,@(')
=
Then, by 1 ., r C*,Q,@(') = which, because o f C*,@,@' i = 1 ,r], i . e . C . = cIL., i =
= r
,...
1
1
I,..
and C' = C / IT, c o n c l u d i n g t h e p r o o f . I
Note t h a t though t h e f o r m u l a t i o n o f t h i s r e s u l t i s more general t h a n f o r t h e o r i g i n a l s t a n d a r d cases, t h e p r o o f t u r n e d o u t t o be s i m p l e r , i . e . t h e a l g e b r a i s a t i o n and t h e s e m i - r i n g i n t e r p r e t a t i o n o f (m,@)
seem t o be w o r t h w h i l e .
R.H. Mohring and F.J. Radennacher
306
We would l i k e t o add t h a t Theorem 2.3.1 extends ( f o r measurable@,@) t o t h e stochastic case completely analogously t o Theorem 2.2.6. Furthermore, again r e s t r i c t i n g ourselves t o the o p t i m i z a t i o n case (my 8 ), t h e bounding approach f o r the d i s t r i b u t i o n f u n c t i o n of
D471,
rC
,m, @
i n the independence case a l s o c a r r i e s over
due t o the f a c t t h a t because of
random weights o f elements T
6
@ being monotonically increasing, the
C are again associated random variables.
questly, l e t t i n g FT denote t h e d i s t r i b u t i o n f u n c t i o n o f the weight t h e d i s t r i b u t i o n f u n c t i o n o f t h e optimal value, we have: F, 6
n
FT
i f m = max
Conse-
o f T and F,
and
T6.C
The use o f t h i s approach has, a p a r t from stochastic p r o j e c t networks, been part i c u l a r l y f r u i t f u l i n RELIABILITY THEORY, where most o f the concepts i n v o l v e d have o r i g i n a l l y been introduced and where f u r t h e r improvements have been obtained, c f . e.g.
[llO].
Also i n t h i s more general s e t t i n g , the use o f t h e s u b s t i t u t i o n decom-
p o s i t i o n improves the bounds i n a way analogously t o 11.2, c f [7]. Me would f i n a l l y l i k e t o mention t h a t t h e r e have been obtained [84],
[162]
also
extensions o f the bounds f o r the case o f s t o c h a s t i c dependences, mentioned i n 11.2.
I n f a c t , f o r t h e case (min,+),
concave lower bound i n t h e sense o f
t h e r e i s completely analogously obtained a
p4g, w h i l e f o r
t h e cases (max,min) and (min,
max), there are s t o c h a s t i c upperjlower bounds, r e s p e c t i v e l y .
I n a l l these cases,
the bounds can be computed v i a t h e s u b s t i t u t i o n decomposition, c f . p62].
11.4 SUMMARY AND HINTS ON SOME OTHER APPROACHES TO DECOMPOSITION OF CERTAIN COMBINATORIAL OPTIMIZATION PROBLEMS W e would l i k e t o sumnarize t h a t the r e s u l t s i n t h i s p a r t i n d i c a t e t h a t using a
n a t u r a l but q u i t e strong approach t o f a c t o r i z a t i o n o f optimal value f u n c t i o n s i n combinatorial optimization, one i n e v i t a b l y a r r i v e s a t t h e s u b s t i t u t i o n decomposition.
The reason f o r t h i s l i e s i n asking f o r a simultaneous s o l u t i o n f o r a l l the
weighting functions considered, together w i t h t h e l o c a l i t y c o n d i t i o n and an i n f o r mation t r a n s f e r f o r each class c o n s i s t i n g i n j u s t one r e a l number. Given t h e indecomposability o f "almost a l l " s t r u c t u r e s (P5), such comfortable decomposition p o s s i b i l i t i e s are a q u i t e r a r e s i t u a t i o n .
However, as mentioned before, t h e s i t u a t i o n i n p r a c t i c a l a p p l i c a t i o n s o f t e n seems t o be the o t h e r way round: because o f h i e r a r c h i c a l planning, coarse models are subsequently r e f i n e d from one l e v e l t o
another, thereby n a t u r a l l y generating autonomous sets.
Furthermore, one obtains
307
Substitution decomposition for discrete structures many a d d i t i o n a l p r o p e r t i e s o r i g i n a l l y n o t r e q u i r e d , e.g.
the possibility o f hier-
a r c h i c a l computation o f many parameters o f i n t e r e s t , as shown above.
Also i t i s
p o s s i b l e t o e f f i c i e n t l y compute and r e p r e s e n t a l l autonomous s e t s i n t h e most wanted cases o f r e l a t i o n s and bounded c l u t t e r s . s i t i o n t r e e d e a l t w i t h i n S e c t i o n s I11 and I V .
T h i s i s done by u s i n g t h e compoFurthermore, t h e Jordan-Holder
theorem deduced i n S e c t i o n I 1 1 shows t h e independence o f m u l t i - s t e p decomposition f r o m t h e o r d e r and s t a r t i n g p o i n t o f t h e s t e p w i s e minimal decompositions, t h u s l e a d i n g t o a r a t h e r s i m p l e concept f o r p r a c t i c a l a p p l i c a t i o n s . N a t u r a l l y , g i v e n t h e l i m i t a t i o n s e x i s t i n g f o r t h e s u b s t i t u t i o n decompositions, o t h e r concepts have been t r i e d . and r e l i a b i l i t y t h e o r y [48],
[97],
T h i s i s p a r t i c u l a r l y t r u e f o r p r o j e c t networks p16],
@43],
F5q,
c5q].
The m a j o r i t y
o f t h e s e methods e s s e n t i a l l y aim a t r e p l a c i n g an a r b i t r a r y connected subnetwork by i t s e n t r a n c e and e x i t nodes, where an e n t r a n c e and an e x i t node a r e j o i n e d by an 2dge i f f t h e y a r e connected by a d i r e c t e d p a t h i n t h e o r i g i n a l network.
Then, f o r
i n s t a n c e , t h e d u r a t i o n o f such an a r t i f i c i a l edge w i l l be t h e l o n g e s t p a t h l e n g t h between t h e corresponding o r i g i n a l nodes.
I n t h i s way, t h e s h o r t e s t p r o j e c t
d u r a t i o n and some o t h e r parameters can be computed. depend on t h e w e i g h t s x valued functions. one.
E
However, these computations
R,” i n a c o m p l i c a t e d way which cannot be expressed by real-
I n f a c t , t h e “reduced“ network may even be l a r g e r than the o r i g i n a l
Therefore, h i g h e r - l e v e l aspects, e.g.
time-cost trade o f f s o r stochastic
g e n e r a l i z a t i o n s can s c a r c e l y be handled t h i s way.
I n t h e s t o c h a s t i c case, f o r
i n s t a n c e , s t o c h a s t i c dependences between t h e r e s u l t i n g d i s t r i b u t i o n s o f t h e new a c t i v i t y d u r a t i o n s would have t o be t a k e n i n t o account, a l s o i n t h e case o f s t o c h a s t i c a l l y independent a c t i v i t y d u r a t i o n s . We c l o s e t h i s s e c t i o n w i t h some h i n t s on t h e s p l i t decomposition, which may i n t h e l o n g r u n prove, t o some e x t e n t , t o be a f u l l c o u n t e r p a r t o f t h e s u b s t i t u t i o n decomposition i n t h e f i e l d o f c o m b i n a t o r i a l o p t i m i z a t i o n .
There a r e by now e.g.
a p p l i c a t i o n s t o t h e d e t e r m i n a t i o n o f t h e weighted independence number a G ( x ) o f
[3g.
graphs
I n f o r m a t i o n t r a n s f e r h e r e i s more i n v o l v e d , and n e c e s s i t a t e s t h e
comparison o f a t l e a s t two d i f f e r e n t a l t e r n a t i v e s , depending on t h e p o s i t i o n o f t h e marker.
I n p a r t i c u l a r , no such i n s t r u m e n t as a simultaneous t r e a t m e n t o f a con-
gruence p a r t i t i o n i s a v a i l a b l e .
S t i l l , given a proper organization, e f f i c i e n t
s o l u t i o n s may be p o s s i b l e and may perhaps extend t o some o f t h e o t h e r cases handled above f o r t h e s u b s t i t u t i o n decomposition.
111.
AN ALGEBRAIC MODEL OF DECOMPOSITION
I n t h i s s e c t i o n , we p r e s e n t a general a l g e b r a i c decomposition t h e o r y which
generalizes and u n i f i e s t h e basic p r o p e r t i e s o f t h e s u b s t i t u t i o n decomposition f o r r e l a t i o n s , s e t systems and Boolean f u n c t i o n s . Subsection 111.1 i n t r o d u c e s t h e assumptions o f t h e general model and shows how t o embed the s u b s t i t u t i o n decomposition i n t o t h i s framework.
This w i l l be done by
i n t e r p r e t i n g the n a t u r a l l y a r i s i n g n o t i o n s o f q u o t i e n t s and autonomous subs t r u c t u r e s i n an " a l g e b r a i c " way, i . e . as a l g e b r a i c q u o t i e n t s o r substructures f o r s u i t a b l y d e f i n e d homomorphisms.
The p r o p e r t i e s t h a t c h a r a c t e r i z e t h e s u b s t i -
t u t i o n decomposition then correspond ( a p a r t from the usual a l g e b r a i c p r o p e r t i e s (MI)
-
(M6)) t o a s p e c i a l s o r t o f i n t e r p l a y between these two n o t i o n s which have
no counterpart i n a l g e b r a i c t h e o r i e s ( c f . ( M 7 ) , (M8)). I n 111.2, we i n v e s t i g a t e the system V ( S ) o f congruence p a r t i t i o n s o f a s t r u c t u r e
S, when considered as a suborder o f the p a r t i t i o n l a t t i c e Z ( A ) o f t h e base s e t A
o f S.
As the main p r e p a r a t o r y step f o r t h e i n v e s t i g a t i o n o f composition s e r i e s
i n 111.3 we o b t a i n t h a t i f V ( S ) i s o f f i n i t e length, then i t i s already an upper semimodular s u b l a t t i c e o f Z ( A ) .
Furthermore, o t h e r l a t t i c e - t h e o r e t i c a l p r o p e r t i e s
o f V ( S ) such as being modular, d i s t r i b u t i v e o r complemented can a l s o be characteri z e d i n terms o f p r o p e r t i e s o f t h e given s t r u c t u r e
s.
Subsection 111.3 deals w i t h t h e f a c t o r i z a t i o n o f a s t r u c t u r e by means o f composit i o n s e r i e s , which correspond t o c h i e f s e r i e s i n u n i v e r s a l algebra.
Besides
c r i t e r i a f o r t h e i r existence, we o b t a i n a " c l a s s i c a l " Jordan-Holder theorem which s t a t e s t h a t any two composition s e r i e s o f a s t r u c t u r e have t h e same l e n g t h and, up t o isomorphism and rearrangement, a l s o t h e same f a c t o r s .
This c e n t r a l r e s u l t
on decomposition i s then r e l a t e d t o the Jordan-HUlder theorems i n u n i v e r s a l algebra. Under a d d i t i o n a l assumptions (which a r e f u l f i l l e d i n t h e a p p l i c a t i o n s considered i n 1.2
-
1 . 4 ) , i t i s p o s s i b l e t o represent t h e decompositions o f a s t r u c t u r e S i n
a t r e e , the composition t r e e
B(S) o f S.
This i s discussed i n 111.4.
The basic
methods f o r the t r e e c o n s t r u c t i o n a r e two m u t u a l l y e x c l u s i v e decomposition p r i n c i p l e s , which g e n e r a l i z e the methods observed f o r t h e s p e c i a l cases o f c l u t t e r s , Boolean f u n c t i o n s and c e r t a i n r e l a t i o n s ( c f . Section I V ) : E i t h e r t h e r e e x i s t s a maximal d i s j o i n t decomposition i n t o autonomous s e t s ( i n which case t h e q u o t i e n t s t r u c t u r e i s indecomposable), o r t h e r e e x i s t s a f i n e s t decomposition such t h a t the q u o t i e n t s t r u c t u r e belongs t o c e r t a i n , w e l l - c h a r a c t e r i z e d classes. Furthermore, t h i s t r e e represents a l l " e s s e n t i a l " decompositions of a s t r u c t u r e i n a polynomial ( l i n e a r ) number o f nodes, a f a c t which makes i t a s u i t a b l e data s t r u c t u r e f o r algorithms concerned w i t h decomposition.
Suhstitutioir decomposition fiir discrctc stnictiirc's
309
I n 111.5, we d i s c u s s t h e c o n n e c t i o n o f t h e s e r e s u l t s w i t h t h e s p l i t decomposition F i n a l l y , i n 111.6, we g i v e some h i n t s on t h e a l g o r i t h m i c d e t e r m i n a t i o n o f t h e c o m p o s i t i o n t r e e B(S) o f a s t r u c t u r e S.
It can be shown t h a t t h i s t a s k i s p o l y -
n o m i a l l y e q u i v a l e n t t o two a p p a r a n t l y weaker t a s k s , v i z . d e t e r m i n i n g t h e autonomous c l o s u r e o f a g i v e n s e t , o r (under a d d i t i o n a l assumptions) d e c i d i n g whether a g i v e n s t r u c t u r e i s decomposable o r n o t , and p r o d u c i n g a n o n - t r i v i a l autonomous set i f i t i s .
These polynomial r e d u c t i o n s f o r m t h e s t a r t i n g p o i n t f o r t h e
r e s u l t s on t h e c o m p u t a t i o n a l c o m p l e x i t y o f decomposing r e l a t i o n s and c l u t t e r s i n Section I V . A p a r t f r o m 111.2,
the presentation o f t h i s section follows that i n
[log,
so
t h a t we can r e s t r i c t o u r b i b l i o g r a p h i c a l notes and r e f e r t o D O 2 3 f o r f u r t h e r d e t a i l s on r e s u l t s n o t proved here.
111.1
THE ALGEBRAIC MODEL
I n t h e g e n e r a l decomposition model we c o n s i d e r ( c f . a l s o t h e h i n t s i n S e c t i o n I ) a " c o n c r e t e " c a t e g o r y K, whose o b j e c t s a r e c a l l e d s t r u c t u r e s and a r e denoted by S,T e t c .
"Concrete" means ( c f .
[73])
t h a t each s t r u c t u r e i s d e f i n e d on an under-
base s a t o f S), and t h a t each homomorphism ( o r morphism i n l y i n g s e t A = AS ( t h e ___c a t e g o r i c a l t e r m i n o l o g y ) f r o m S t o T i s a mapping f r o m As i n t o AT.
A s p e c i a l r o l e w i l l be p l a y e d by t h e s u r j e c t i v e and i n j e c t i v e homomorphisms which ( i n accordance w i t h t h e usual a l g e b r a i c t e r m i n o l o g y [30],
b u t d i f f e r e n t from
c a t e g o r i c a l t e r m i n o l o g y [73] ) w i 11 be r e f e r r e d t o as epimorphisms and monomorphisms, r e s p e c t i v e l y .
F o r s t r u c t u r e s S,T f r o m K, l e t Hom(S,T),
Epi(S,T),
and
Mono(S,T) denote t h e s e t s o f homomorphisms, epimorphisms, and monomorphisms f r o m S t o T, r e s p e c t i v e l y .
L e t S denote t h e c l a s s o f s t r u c t u r e s o f K.
Two s t r u c t u r e s
S and T a r e isomorphic i f t h e r e e x i s t s a b i j e c t i v e mapping f such t h a t f
E
Hom
(S,T) and f - l E Hom(T,S). I n a d d i t i o n t o t h e usual c a t e g o r i c a l p r o p e r t i e s (which e s s e n t i a l l y mean t h a t home morphisms a r e c l o s e d under composition, i . e . f, c Hom(S1,S2), f 2 E Hom(S2,S3) yields
f20
fl
E
Hom(S1,S3),
and t h a t , f o r each s t r u c t u r e S on A, t h e i d e n t i c a l
A f u l f i l l s i d A Hom(S,S) and f o i d A = di ,?, o f = f f o r each S T we impose two groups o f c o n d i t i o n s , ( M l ) - (M5) and (M6) - (M8). The f i r s t group p r o v i d e s us w i t h elementary a l g e b r a i c p r o p e r t i e s needed t o d e f i n e
mapping i d A : A f
E
+
Hom(S,T)),
q u o t i e n t s and s u b s t r u c t u r e s , w h i l e t h e second group d e a l s w i t h t h e r e l a t i o n s h i p between these n o t i o n s .
Note t h a t (Ml)
-
(M6) h o l d i n the f a m i l i a r algebraic
t h e o r i e s (e.g. t h e t h e o r y o f groups, r i n g s , e t c . ) and t h a t i t i s , i n f a c t ,
R.H. Mohring and F.J. Radermacher
310
o n l y c o n d i t i o n s (M7) and (M8) which a r e "non-algebraic".
These two c o n d i t i o n s
may thus be viewed as r e p r e s e n t i n g t h e s p e c i a l c h a r a c t e r o f t h e s u b s t i t u t i o n decomposi ti on. Each f
E
g e Epi(S,U),
h
E
-t
and f-'
there e x i s t U
S,
g i v e n a s t r u c t u r e S on A and a b i j e c t i o n
B, t h e r e e x i s t s a unique s t r u c t u r e T on B such t h a t f E
E
Mono(U,T) such t h a t f = hog
S t r u c t u r e i s a b s t r a c t , i.e., f: A
i.e.,
Hom(S,T) has an epi-mono-factorization,
E
Hom(S,T)
Hom(T,S).
Given a s t r u c t u r e S on A and a s u r j e c t i o n f from A onto a s i n g l e t o n t h e r e e x i s t s a s t r u c t u r e So on A,
such t h a t f
E
A,,
Epi(S,So).
I f h E E p i ( S , T l ) n Epi(S,T2) then T1 = T2. If g
Q
Mono(S1,T)
n Mono(S2,T)
then S1 =
s2.
D e f i n i t i o n a) A s t r u c t u r e T i s c a l l e d a q u o t i e n t s t r u c t u r e o r q u o t i e n t o f a s t r u c t u r e S on A i f t h e r e e x i s t s a p a r t i t i o n TI o f A such t h a t nTI E Epi(S,T), where n,, denotes the n a t u r a l mapping associated w i t h a l l a € A , where [a].
i s the class o f
CY
w.r.t.
TI
(i.e.,
= [a]* f o r
qTI(ct)
I n t h i s case,
n).
i s called a
IT
congruence p a r t i t i o n o f S, and t h e u n i q u e l y determined (because o f (M4)) q u o t i e n t T i s denoted by S / n .
A
V ( S ) denotes t h e system o f congruence p a r t i t i o n s o f S.
s t r u c t u r e S on A i s c a l l e d prime o r indecomposable i f V ( S ) c o n t a i n s no proper congruence p a r t i t i o n , i . e . i f C c
TI
c
V(S) implies C
=
A o r I C [ = 1.
b ) A s t r u c t u r e T i s c a l l e d a s u b s t r u c t u r e o f a s t r u c t u r e S on A, i f t h e r e e x i s t s a subset B o f A such t h a t i n c AB L Mono(T,S), where i n c AB denotes the i n c l u s i o n mapping associated w i t h B and A ( i . e . i n c i ( a ) =
CY
f o r a l l a e B).
I n t h i s case,
B i s c a l l e d an S-autonomous set, and t h e uniquely determined (because o f (M5)) substructure T i s denoted by S I B .
A ( S ) denotes the system o f S-autonomous s e t s .
Condition (142) i m p l i e s t h e ( i n a l g e b r a i c t h e o r i e s ) usual f a c t o r i z a t i o n o f epimorphisms and monomorphisms:
a) Each epimorphism h 0 Epi(S,S') has a f a c t o r i z a t i o n h = where o7 E Hom(S,T) i s the n a t u r a l mapping associated w i t h the p a r t i t i o n Lemna 3.1.1:
induced by h ( i . e . anB i f f h ( a ) = h ( B ) ) , f i s the q u o t i e n t S/TI.
E
foqT, T
o f AS
Hom(T,S) i s an isomorphism, and
T
311
Substitution decomposition for discrete structures b ) Each monomorphism h tsMono(S',S) has a f a c t o r i z a t i o n h = incg'o f, where f E Hom(S',T) i s an isomorphism, T = S I B i s t h e s u b s t r u c t u r e SIB induced by
B
= h ( A S , ) , and i n c ?
Proof:
E
Hom(T,S) i s t h e i n c l u s i o n o f T = SIB i n S.
Apply (M2) t o t h e b i j e c t i o n s gl:AS,
and g2:AS, +
B, d e f i n e d by g 2 ( a ' )
homomorphisms, nn = gloh
E
= h(a'),
-+
AS/r,
d e f i n e d by g,(h(a))
:= [a]~,
r e s p e c t i v e l y . Then, by c o m p o s i t i o n o f -1 = hog2 e Mono(T,S), which a l s o
Epi(S,T) and i n c ?
gives t h e claimed f a c t o r i z a t i o n . s So epimorphisms and monomorphisms correspond e s s e n t i a l l y t o q u o t i e n t s t r u c t u r e s and s u b s t r u c t u r e s , which i n t u r n can f o r a g i v e n s t r u c t u r e S be " i n t e r n a l l y " d e s c r i b e d by i t s system o f congruence p a r t i t i o n s V ( S ) and i t s system o f S-autonomous s e t s A ( S ) . The q u e s t i o n t h e n i s how t o embed t h e s u b s t i t u t i o n decomposition i n t o t h i s framework.
T h i s i s n o t q u i t e obvious, s i n c e , i n general, t h e r e i s no " n a t u r a l " n o t i o n
o f homomorphism.
There are, however, n a t u r a l n o t i o n s o f " q u o t i e n t " and "sub-
s t r u c t u r e " which may be used t o d e f i n e homomorphisms a p p r o p r i a t e l y .
To t h i s end,
l e t S be a s t r u c t u r e ( i . e . a r e l a t i o n , s e t system o r Boolean f u n c t i o n ) o b t a i n e d by s u b s t i t u t i o n , i . e . each B
A', S
B
S = SICS,,
B
E
A'],
where S ' i s a s t r u c t u r e on A ' and, f o r
i s a s t r u c t u r e on AB.
Then t h e r e l a t i o n s h i p between S and t h e " q u o t i e n t " S ' may be e q u i v a l e n t l y des+ A ' with h ( a ) = B i f a e A 8 induces a congruence p a r t i t i o n ( i n t h e sense o f t h e s u b s t i t u t i o n o p e r a t i o n , c f .
c r i b e d by t h e f a c t t h a t t h e s u r j e c t i v e mapping h:A
1.2
-
1.4).
S i m i l a r l y , t h e embedding o f each " s u b s t r u c t u r e " Sg i n t o S may be e q u i v a l e n t l y d e s c r i b e d by t h e p r o p e r t y t h a t t h e i n c l u s i o n i n c A s i s an isomorphism f r o m S o n t o B A6 S I A B ( i n t e r p r e t e d as t h e induced s u b s t r u c t u r e i n t h e g i v e n c l a s s , i . e . t h e r e s t r i c t i o n t o A ) and t h a t A 6
B
i s autonomous ( w . r . t .
the substitution
decomposition). These s u r j e c t i v e mappings and i n c l u s i o n s a r e t h e n t a k e n as " s p e c i a l " homomorphisms i n t h e general model and a r b i t r a r y homomorphisms a r e d e f i n e d as t h e c o m p o s i t i o n o f f i n i t e l y many o f t h e s e " s p e c i a l " homomorphisms. Based on t h e p r o p e r t i e s (Sl) - ( S 7 ) o f t h e s u b s t i t u t i o n o p e r a t i o n given a t t h e b e g i n n i n g o f S e c t i o n I , one o b t a i n s t h e f o l l o w i n g c h a r a c t e r i z a t i o n o f t h e homomorphisms d e f i n e d above.
R.H. Mohring and F.J. Radermacher
312 L e m a 3.1.2:
Each homomorphism h = hno ...Ohl has a f a c t o r i z a t i o n h = gof, where
f and g a r e a s u r j e c t i v e mapping and an i n c l u s i o n o f t h e s p e c i a l type described above , r e s p e c t i v e l y .
Proof:
I t f o l l o w s from ( S 4 ) ( i i ) t h a t t h e composition o f two s u r j e c t i v e mappings
of t h e special type i s again o f t h e special type.
Similarly, (Sl) implies t h a t
the composition o f two i n c l u s i o n s o f t h e special type i s again an i n c l u s i o n o f t h e special type. where hl e Hom(S1,S2) i s an
Thus i t remains t o be shown t h a t given f = h20hl, i n c l u s i o n and h2
Hom(S2,S3) i s s u r j e c t i v e o f t h e s p e c i a l type, t h e r e e x i s t an
i n c l u s i o n f2 and a s u r j e c t i o n fl o f t h e s p e c i a l type such t h a t f = L e t Ai be the base s e t o f Si, of S2 associated w i t h h2.
i = 1,2,3,
and l e t
be t h e congruence p a r t i t i o n
i s S2-autonomous Because o f ( S 4 ) ' ( i ) , B := h2( [A1]~)
Since hl i s an i n c l u s i o n , A1 = hl(A1)
and t h e r e f o r e , because of ( S 5 ) , a l s o [A1]v. i s S3-autonomous.
T
On t h e o t h e r hand, i t f o l l o w s from ( S 6 ) t h a t v l A 1 i s a congru-
ence p a r t i t i o n o f S21A1 = S1 which corresponds t o t h e r e s t r i c t i o n Since (57) i m p l i e s t h a t Sl/(nlA1) fl :=
h2
f20fl.
and ( S 2 1 [A1]T)/(Tl
[A,]n)
b2
o f hp t o A1.
= S31B a r e isomorphic,
and f 2 := i n c i 3 c o n s t i t u t e t h e claimed f a c t o r i z a t i o n o f f . m
Thus the n o t i o n o f homomorphism induced by t h e s u b s t i t u t i o n o p e r a t i o n f u l f i l s the f a c t o r i z a t i o n condition (Ml).
Furthermore, t h e epimorphisms and monomorphisms
are (up t o b i j e c t i o n s ) e x a c t l y t h e s p e c i a l s u r j e c t i v e mappings and i n c l u s i o n s associated w i t h t h e s u b s t i t u t i o n operation.
Hence, a l s o (M4) and (M5) a r e
s a t i s f i e d (up t o the i d e n t i f i c a t i o n o f r e l a t i o n s w i t h d i f f e r e n t r e f l e x i v e t u p l e s and Boolean f u n c t i o n s w i t h complemented v a r i a b l e s o r f u n c t i o n a l value i n t h e sense o f Section I.2,1.4).
O f course, S-autonomous s e t s and congruence p a r t i t i o n s
i n t h e a l g e b r a i c model correspond e x a c t l y t o t h e i r counterparts f o r t h e s u b s t i t u t i o n operation.
Since (M2) and (M3) a r e f u l f i l l e d t r i v i a l l y , t h e embedding o f
t h e s u b s t i t u t i o n decomposition i n t o t h e general model i s completed. This approach may seem r a t h e r l a b o r i o u s , i n p a r t i c u l a r w . r . t .
the v e r i f i c a t i o n o f
(Ml). However t h e obtained i n t e r p r e t a t i o n o f t h e s u b s t i t u t i o n decomposition i n terms o f an a l g e b r a i c homomorphism theory permits t o t h e a p p l i c a t i o n o f methods and concepts from u n i v e r s a l algebra.
From t h i s p o i n t o f view, (Ml) t u r n s o u t t o
be a powerful c o n d i t i o n and e s s e n t i a l l y e q u i v a l e n t t o c o n d i t i o n s ( S 5 ) , (S6) and
( S 7 ) introduced a t t h e beginning o f Section I ( c f . Theorem 3.1.3).
Furthermore,
there are examples o f the general model which cannot be obtained from a s u b s t i t u t i o n operation, i . e . exanples i n which i t i s n o t p o s s i b l e t o "uniquely r e c o n s t r u c t " a s t r u c t u r e S from a q u o t i e n t S / n and the substructures S I B , B E T ( c f . [IOZJ).
Substitution decomposition for discrete structures
313
The second group o f c o n d i t i o n s o f t h e a l g e b r a i c model r e f l e c t s p o s s i b i l i t i e s f o r c o n s t r u c t i n g new congruence p a r t i t i o n s f r o m a l r e a d y known ones. (M6)
'TI,U
(M7)
(i)
E:
V ( S ) and 'TI
V(S)
e
'TI
# 4.
4 u => Epi(S/*,S/u)
=>
B EA(S)
for all B
E
'TI.
n
If
(ii)
= I L 1,...,Ln,Iu31a
'TI
Li E A ( S ) , i = 1 (M8)
'TI
= EBi
=>
I
,..., n,
i E I}E V ( S ) ,
u = IBi,C.
J
I ic
E
A\U
t h e n 'TIBV(S).
[ j
= ICj
T
i s a p a r t i t i o n o f AS w i t h
Li}
i=l
E
J}
E
V(S
I Bi
) f o r some Bi 0
E
'TI
0
I\{i0}, j e J} C V ( S ) .
I n t h e s u b s t i t u t i o n decomposition, t h e s e c o n d i t i o n s correspond t o ( S 4 ) ( i ) , ( S 2 ) and (S3), c f . S e c t i o n I. (M6) i s known as t h e "Theorem o f Induced Homomorphism" i n u n i v e r s a l a l g e b r a . I t i m p l i e s by s t a n d a r d arguments f r o m u n i v e r s a l a l g e b r a ( c f . [66,
p.611) t o g e t h e r w i t h Lemma 3.1.1
t h a t V ( S / ' T I ) i s c o m p l e t e l y determined
by V ( S ) ( c f . a l s o Example 3.2.9):
L e t S be a s t r u c t u r e on A and
Theorem 3.1.3: {U E
V(S)
I
'TI
[a]lT(u/lr)[B]'TI
: <+
t. V ( S / T ) ,
ao'B : <=>
[a]'TIu"B]*).
U/T
(u c
and i t s i n v e r s e u'
+
u'
(u'
-
Contrary t o (Ml)
V(S). Then
~,*'],
+
(S/a)/(u/a)
E
V ( S / n ) and
['TI,T~]
Isomorphisms a r e g i v e n by t h e mapping
4 u} a r e order-isomorphic.
u
I n particular,
'TI
auB)
and S/u a r e isomorphic (Second Isomorphism Theorem).
(M6), (M7) and (M8) have no c o u n t e r p a r t i n common a l g e b r a i c
So i t i s t h e s e two c o n d i t i o n s t h a t r e f l e c t t h e s p e c i a l p r o p e r t i e s o f
theories.
t h e s u b s t i t u t i o n decomposition. We s h a l l now g i v e some immediate consequences o f t h e c o n d i t i o n s ( M l )
-
(M8).
They a r e weaker than t h e p r o p e r t i e s o b t a i n e d i n t h e s p e c i a l cases i n 1.2 b u t a l r e a d y c o n t a i n most o f them, v i z . ( A l )
Theorem 3.1.4: 1.
-
(A3) and ( S l )
L e t S and S ' be s t r u c t u r e s on A and
-
-
1.4,
(57).
A', r e s p e c t i v e l y .
A ( S ) has t h e f o l l o w i n g p r o p e r t i e s :
a)
A
E
A(S), tn) e A(S) for a l l a
E
A
I f n e V ( S ) and B € A ( S ) , then [BIT E A ( S ) . I f B,C
E
A ( S ) such t h a t B n C f 4 , t h e n B n C
L
A ( S ) and.BU C E A ( S ) .
=
R.H. Mohring and F.J. Radermacher
314
t h e n A ( S 1 B ) = tC
If B f A(S),
b)
E
I
A(S)
C -c 61 =: A ( S ) l B .
I n particular,
(SlB)(C = SIC f o r a l l C e A(SlB).
I f h e Epi(S,S'),
c)
for all C'
2.
t h e n h(C) E A ( S ' )
nlB
E
c)
Proof:
I
B
Ln 8 = $1 IJ {[B]TI)
E V(S)
( S I B ) / ( T / B ) and ( S l [ B ] n ) / ( n I
[BIT)
2a):
rl,o
Because o f ( M l ) , f :=
induced by g one o b t a i n 6
T =
e Hom(SIB,S/a)
has a f a c t o r i z a t i o n T
e V(S\B)
A
f := h o i n c C has a f a c t o r i z a t i o n f =
The second s t a t e -
One t h e n o b t a i n s t h a t h(C) = C ' .
A(S').
E
inc:
nlB.
Because o f ( M l ) and Lemma 3.1.1,
A' i n c C 4o g, where C '
a r e isomorphic.
F o r t h e congruence p a r t i t i o n
f = hog, where g i s an epimorphism.
lc):
Then:
V(S1B)
:= I L c
b)
EA(S)
and h - ' ( C I )
L e t n c V ( S ) and B E A ( S ) .
V ( S ) has t h e f o l l o w i n g p r o p e r t i e s . a)
for a l l C =A(S),
A(S').
E
ment f o l l o w s f r o m (M7) and Theorem 3.1.3. I b ) f o l l o w s from (M7), 2a) and (M5). la):
The f i r s t s t a t e m e n t f o l l o w s f r o m (M2), (M3) and (M7), Let
because o f [BIB = Q;~(~,(B)). which i m p l i e s B
nC E A(S)
TI^
:= {C,{a}la
EA\C}.
because o f (M7), 2a) and 2b).
t h e second f r o m l c ) Then B fl C
E
nC[B,
F i n a l l y , B UC = [B]nce
A ( S ) because o f (M7) and t h e above. a p p l i e d t o u' = {nn(B),{L}
2b) f o l l o w s from Theorem 3.1.3,
I
L
n , L n B = $1
which i s i n V ( S / r i ) because o f (M7) and 1.c. 2c) f o l l o w s s i m i l a r l y t o t h e above arguments.
I
T h i s concludes t h e p r e s e n t a t i o n of t h e g e n e r a l model and i t s b a s i c p r o p e r t i e s . They a r e a l r e a d y s u f f i c i e n t t o p r o v e t h e Jordan-Holder theorem f o r c o m p o s i t i o n
I n t h i s r e s p e c t , i t i s i n t e r e s t i n g t o know whether o r n o t t h e
s e r i e s i n 111.3.
s t r o n g e r s t r u c t u r a l p r o p e r t i e s o f V ( S ) and A ( S ) observed f o r t h e s u b s t i t u t i o n decomposition (e.g.
f o r r e l a t i o n s ) a r e i m p l i c i t l y c o n t a i n e d i n t h e g e n e r a l model.
These p r o p e r t i e s a r e : 1.
I n f i n i t e c o n s t r u c t i o n s i n v ( S ) ( i n f i n i t e v e r s i o n s o f (M7) and (M8)
(M7)*
71
eV(S) i f f L e A ( S ) f o r a l l L e
(M8)* I f
?I
=
{Li
1
i
E
1)
E
TI
V ( S ) and, f o r each i e I ,
T~
E
V(SILi
,
then
315
Substitution decomposition for discrete structures
2.
3.
Additional properties o f A ( S ) A ( S ) overlap, then C1\C2
(119)
If C1,C2
(MIO)
If ( c ~ ) ~ ,E. ~A ( S ) and
E
n
c A ( S ) and C2\C1
n ci
ci # +, then i d i €1 ( i n f i n i t e version of Theorem 3.4, l a ) ) .
E
A(S)
E A ( S ) and
i EI
ci
E
A(S)
Unique reconstruction property
(M11)
If S and T a r e s t r u c t u r e s on A such t h a t , f o r T/IT and SIL = T I L f o r a l l L e IT, then S = T.
Because of Theorem 3.1.4, (M7)* implies (M8)*.
IT E
V(S)
n v(T),
Sin
Relations f u l f i l (M7)*, (M9) (Mll), whereas c l u t t e r s only f u l f i l (M8)*, (M9) and (Mll), c f . 1.4 and 1.3.
=
-
Counterexamples showing t h a t the other conditions may not be f u l f i l l e d in the general model can be constructed by considering t h e category KO o f a l l s e t system S, whose members a r e the S-autonomous s e t s ( i . e . S = A ( S ) ) and in which homomorphisms a r e defined via the composition of surjective and i n j e c t i v e mappings which f u l f i l property 1 . c ) i n Theorem 3.1.4. This category f u l f i l s (M7)*, b u t not (Mll), This shows t h a t t h e r e a r e examples for the general framework which cannot be obtained from a s u b s t i t u t i o n operation. By taking special subcategories of KO, one can a l s o construct examples which f u l f i l only (Ml) - (M8), and examples which show t h a t (M9) and (M10) a r e independent of (M7)* and (M8)* and can be taken i n t o the model independent of each other.
rII.2
THE SYSTEMOF CONGRUENCE PARTITIONS
We s h a l l now investigate the system V ( S ) of congruence p a r t i t i o n s of a s t r u c t u r e S, viewed a s a suborder of t h e p a r t i t i o n l a t t i c e Z(A) of the base set A of S, where the ordering r e l a t i o n i s t h e refinement r e l a t i o n c f o r p a r t i t i o n s . In the following, A and v denote the l a t t i c e operations "meet" and "join" in Z ( A ) . Of 0 course, V ( S ) always contains IT = { I a } [ a e A ) and = {A} because of (M2) and (M3). I n general, V ( S ) need not be a l a t t i c e , c f . Example 4.2.2. following embedding i n t o Z ( A ) ( c f . Also Example 3.2.9).
We have, however, the
Lemna 3.2.1: Let I T , U E Y(S), where IT has only f i n i t e l y many non-singleton classes. Then li and u have a 1.u.b. and g.1.b. i n U(S), and l . u . b . ( ~ , o ) = ITVO, g . l . b . ( T , ~ ) =
TAU.
R. H. Mohring und F.J. Ruderniacher
316
Proof:
Since V ( S ) i s a suborder o f Z ( A ) ,
are congruence p a r t i t i o n s o f S. n54]),
L of
i t follows, t h a t
[154])
S i m i l a r l y , the r e p r e s e n t a t i o n o f
y i e l d s t h a t , since
-I
L
ITAU
ITVO
and
ITAO
i n Z(A) (p3],
i s obtained from r e f i n i n g the non-singleton classes
TAU
according t o OIL.
T
class
i t suffices t o show t h a t
From t h e r e p r e s e n t a t i o n o f
ITVO
i n Z ( A ) (Q3],
has o n l y f i n i t e l y many non-singleton classes, each
of iivc i s of the form L =
Pjn
PI U q1 U P2 u 42 U
. ..U
Pk-,
qk-,
u Pk w i t h
P j E T, Qj E o and Qj # f Q.fl P. Each such L i s 5-autonomous because J J+1' of Theorem 3.1.4, and t h e r e are o n l y f i n i t e l y many such classes L E (IIVU)\O.
Thus
TAU E
V ( S ) and
TVO
$J
E
V ( S ) because o f ( M 8 ) and Theorem 3.1.4,
2a) and Z.b).m
The p r o o f a l s o shows which a d d i t i o n a l c o n d i t i o n s should h o l d i n order t o make
v ( S ) a l a t t i c e ( c f . a l s o [98] f o r r e l a t i o n s ) .
Theorem 3.2.2: b)
a ) I f S f u l f i l l s (M8)*,
then V ( S ) i s a A-semi-sublattice o f Z ( A ) .
I f S f u l f i l l s ( M 7 ) * and ( M l O ) , then V ( S ) i s a complete s u b l a t t i c e o f Z ( A ) .
As a consequence o f Lemna 3.2.1, many non-singletons.
V ( S ) i s a l a t t i c e i f A ( S ) contains o n l y f i n i t e l y
The n e x t theorem shows t h a t t h i s i s a l r e a d y e q u i v a l e n t t o
the existence o f a f i n i t e maximal chain i n V ( S ) .
For a s t r u c t u r e 5, the f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t .
Theorem 3.2.3: 1.
A ( S ) contains o n l y f i n i t e l y many non-singletons.
2.
V(S) i s finite.
3.
V ( S ) i s o f f i n i t e length.
4.
V ( S ) contains a maximal c h a i n o f f i n i t e l e n g t h .
For the p r o o f o f t h i s theorem, we need t h e f o l l o w i n g c h a r a c t e r i z a t i o n o f t h e atoms i n v ( S ) , i . e . o f those
Lemna 3.2.4:
si
L U(S)
B E
V ( S ) which cover TO.
i s an atom i n V(S) i f f
B
=
I B , I a ) J a e A\BI, where \ B ) > 1
and S I 6 i s prime. Proof:
(M7),
( M 8 ) and Theorem 3.1.4,
Proof o f Theorem 3.2.3:
We only show"4=>2".
by induction on the l e n a t h V(S) i s finite. :i
# " 2 cover
iil
2a)
Gf
The i e s t f o l l o w s t r i v i a l l y .
a f i n i t e maximal chain t h a t each i n t e r v a l
This i s obvious i f
T~
covers
i n a maximal, f i n i t e chain i n
nl. [T,,IT,].
We show [T.,T,]
I n t h e i n d u c t i v e step, l e t By the i n d u c t i v e
of
Substitution decomposition for discrete stnictures
assumption,
317
i s f i n i t e and, because o f Theorem 3.1.3 isomorphic t o
[IT.IT,]
we may assume t h a t n1 = IT'. Lemma 3.2.4 t h e n y i e l d s [ n ' , ~ ~ ~ / n ~ ] .So w.l.o.g., t h a t 71 has only one n o n - s i n g l e t o n c l a s s , say 6. So, because o f Lemma 3.2.1 ITAU E V ( S ) and ITVO E V ( S ) f o r a l l u E V ( S ) . L e t K be a maximal c h a i n i n [r1,n2].
If there exists u
E
K such t h a t avn < n2,
then we can conclude f r o m t h e i n d u c t i v e h y p o t h e s i s a p p l i e d t o [n1,uv~] and
K i s finite.
that
[uvn,n2]
-
the i d e n t i t i e s
= r2,
ITVU
T h i s i m p l i e s t h a t [r1,n2] as
[~T",IT,]
[To,Tz],
=
and
I f t h e r e i s no such u , one o b t a i n s [ K I \< 3 because of
ITAU
=
= nl f o r a l l a E K \ { ~ T ~ , I T ~ I .
IT^,^^]
i s o f f i n i t e l e n g t h and may t h u s be w r t t e n
{ITOIUlJ[ui,r2], where ui, i E I, a r e t h e I i s f i niQ i i e because of Lemma 3.2.7 and 3.2.4.
p r o o f , s i n c e each
atoms o f V ( S ) i n T h i s concludes t h e
i s f i n i t e by t h e i n d u c t i v e h y p o t h e s i s .
[ u ~ , I T ~
I
and v.
L i s called
upper semimodular, i f avb covers a and b whenever a and b cover anb.
Upper semi-
L e t L be a l a t t i c e o f f i n i t e l e n g t h w i t h l a t t i c e o p e r a t i o n s
4
modular l a t t i c e s have t h e i m p o r t a n t p r o p e r t y t h a t t h e y f u l f i l t h e Jordan-Dedekind c h a i n c o n d i t i o n [13],
which s t a t e s t h a t any two maximal c h a i n s between any two
elements of L ( t h u s i n p a r t i c u l a r a l l maximal c h a i n s i n L ) have t h e same l e n g t h . For V ( S ) , we o b t a i n :
Theorem 3.2.5:
I f S f u l f i l s one o f t h e c o n d i t i o n s o f Theorem 3.2.3,
then V ( S ) i s
an upper semimodular s u b l a t t i c e o f Z(A).
Proof
(cf.
[98]
and Lemna 3.2.1, must show t h a t that
~
~
f o r r e l a t i o n s and PO61 f o r c l u t t e r s ) : V ( S ) i s a s u b l a t t i c e o f Z(A). n1
v nz covers r1 and
L e t n1,n2 e V ( S ) cover u .
1
B 2 ) > i f B 1 n B2 = $ and n 1 v n2 =
=
We
IT^. Because o f Theorem 3.1.3 we may assume
a r e, atoms n i~n V ( S ) , i . e . have t h e form n . = {Bi,{a}la
i s p r i m e because o f Lemma 3.2.4.(i
1,Z).
IBIU
Then
e A\Bi}
v n 2 = IBl,B2,{a}lcr
where SIBi
A\(BIU 1 A\(BIU B z ) > i f B 1 n B2 # $. 2a) and (M7) t h a t n1 v IT^ covers IT ?I
E
B2,{alIcre
I n b o t h cases, i t f o l l o w s f r o m Theorem 3.1.4, and n
Because o f Theorem 3.2.3
1
2'
Some f u r t h e r l a t t i c e t h e o r e t i c a l p r o p e r t i e s o f V ( S ) w i l l be g i v e n i n t h e s t r o n g e r case t h a t (M7)*,
(M9) and (M10) h o l d , which i s t h e case f o r r e l a t i o n s and c e r t a i n
c l u t t e r s , c f . Section I V .
We o m i t t h e p r o o f s and i n s t e a d r e f e r t o @05].
p a r t i a l o r d e r s , a r b i t r a r y r e l a t i o n s , and c l u t t e r s , see a l s o [122], [106],
respectively.
For
1981, D51],
R. Ii. Mfikring and F.J. Radermacher
318
We f i r s t c h a r a c t e r i z e complemented congruence p a r t i t i o n l a t t i c e s .
t h a t o n l y s t r u c t u r e s S f o r w h i c h A ( S ) i s degenerate ( i . e .
It turns o u t
A ( S ) = P(A)\{$I,
where
P(A) denotes t h e power s e t o f A ) or l i n e a r ( i . e . t h e r e i s a l i n e a r o r d e r i n g 6 on
A such t h a t A ( S ) = A ( $ ) ) , and which a r e o f importance i n c o n n e c t i o n w i t h (P6), can have complemented congruence p a r t i t i o n l a t t i c e s .
Under t h e assumptions (M7)*,
Theorem 3.2.6:
(M9) and (MlO), V ( S ) i s complemented
i f f one o f t h e f o l l o w i n g c o n d i t i o n s h o l d s : Then V ( S ) i s a 2-element Boolean a l g e b r a .
1.
S i s prime.
2.
S i s l i n e a r and t h e a s s o c i a t e d l i n e a r o r d e r 6 i s l o c a l l y f i n i t e ( i . e .
3.
S i s degenerate.
interval o f 4 is finite).
each
Then V ( S ) i s a Boolean a l g e b r a .
Then U ( S ) i s t h e p a r t i t i o n l a t t i c e Z ( A ) .
Under t h e above assumptions t h e f o l l o w i n g statements a r e e q u i v -
C o r o l l a r y 3.2.7: ilent
1. 2.
V ( S ) i s r e l a t i v e l y complemented
3.
V ( S ) i s a p a r t i t i o n l a t t i c e o r a Boolean a l g e b r a .
V ( S ) i s complemented
Theorem 3 . 2 . 6 , 2 i s a s p e c i a l case o f a theorem o f Hashimoto [70,
Th. 8-41 s t a t i n g
t h a t t h e congruence p a r t i t i o n l a t t i c e o f a d i s t r i b u t i v e l a t t i c e ( c o n s i d e r e d as an t h e j o i n and meet o p e r a t i o n s ) i s a Boolean a l g e b r a i f f t h e l a t t i c e
algebra w . r . t .
i s locally finite.
This f o l l o w s from t h e f a c t t h a t l i n e a r orders a r e l a t t i c e s ,
and t h a t t h e homomorphisms d e f i n e d v i a t h e s u b s t i t u t i o n decomposition c o i n c i d e f o r t h i s Lase w i t h t h e l a t t i c e homomorphisms.
T h i s means t h a t , f o r l i n e a r o r d e r s ,
l a t t i c e congruence p a r t i t i o n s a r e t h e same as t h o s e f o r t h e s u b s t i t l i t i o n decomposition. F o r t h e c h a r a c t e r i z a t i o n of modular and d i s t r i b u t i v e congruence p a r t i t i o n l a t t i c s we need t o s p e c i f y t h e degree t o which a s t r u c t u r e i s degenerate.
Definition:
L e t S be a s t r u c t u r e on A and l e t Deg(S) be t h e s e t o f a l l degeneraie
s t r u c t u r e s which a r e a s u b s t r u c t u r e o f S o r o f some homomorphic image o f S . t h e maximum c a r d i n a l i t y o f a base s e t o f t h e s t r u c t u r e s o f Deg(S) ( o r
-,
Then
i f the
maximum does n o t e x i s t ) i s c a l l e d t h e d e g e n e r a t i o n degree o f S and i s denoted by deg(S).
319
Substitution decomposition for discrete structures
Theorem 3 . 2 . 8 : deg(S) 6 3 (6
Under t h e above assumptions, V ( S ) i s modular ( d i s t r i b u t i v e ) i f f
2).
The p r o o f shows t h a t deg(S) = n i m p l i e s t h a t V ( S ) c o n t a i n s an i n t e r v a l i s o m o r p h i c t o t h e congruence p a r t i t i o n l a t t i c e o f a degenerate s t r u c t u r e on n elements, i . e . T h i s g i v e s t h e easy d i r -
isomorphic t o t h e p a r t i t i o n l a t t i c e Z(n) o f {l,...,n}. ection o f the proof.
I n t h e o t h e r d i r e c t i o n one shows t h a t deg(S) Q 3 ( 2 ) Because o f t h e repre-
i m p l i e s t h a t complements i n V ( S ) a r e incomparable ( u n i q u e ) .
s e n t a t i o n o f any i n t e r v a l [T,u] c V ( S ) as t h e d i r e c t p r o d u c t [n,.]
=
Y
V((S/n)IB),
Bw/n
a l s o r e l a t i v e complements i n V ( S ) a r e t h e n incomparable ( u n i q u e ) .
T h i s proves t h e
theorem by s t a n d a r d c h a r a c t e r i z a t i o n s o f m o d u l a r i t y ( d i s t r i b u t i v i t y ) , c f . p3],
n54-j. For f u r t h e r r e s u l t s on t h e r e p r e s e n t a t i o n o f congruence p a r t i t i o n l a t t i c e s as a s u b d i r e c t e d p r o d u c t o f s p e c i a l l a t t i c e s c f . [151].
1x1
n,&;;G)
0
1 . Besides the t r i v i a l congruence p a r t i t i o n s no and n l , G has t h e following congruence p a r t i t i o n s : n1 = {{l,Zi,{31,{41,t5l), n2 = {{11,{21,{3},{4,511, n 3 = {{1,21,{3},t4,5}}, n 4 = {{1,4,5l,{21,{3lI,
x5 = {{11,{3l,{2,4,51}, {t1,2,4,51,{3}}. V ( G ) i s a l s o given i n Figure 3.1. Note t h a t t h e meet and j o i n i n V ( S ) correspond t o the l a t t i c e operations i n t h e p a r t i t i o n l a t t i c e n6 =
Z({l,.. .,53).
2 . The quotient graphs G/.rri, i = 1,2,3, a r e given i n Figure 3.2. The associated congruence p a r t i t i o n l a t t i c e s V(G/ni) are given a s the dual principal ideal Gi,n’] of V ( G ) according t o Theorem 3.1.3 ( c f . t h e bold nodes and l i n e s in Figure 3.2).
R.H. Mohring and F.J. Radermacher
3 20
4Y75
11
1 4 3
&
Figure 3.2:
Q u o t i e n t graphs and associated dual p r i n c i p a l i d e a l s of V ( G )
None o f t h e l a t t i c e s W(G) and W(G/ni)
3.
follows alsc, from Theorem 3.2.6, o r degenerate.
i = 1,2,3,
s i n c e none
G f
i s complemented.
t h e graphs G,G1,G2,G3
This i s linear
Note f u r t h e r t h a t a l l these l a t t i c e s are modular, and t h a t V(G/nl)
and V(G/n3) are even d i s t r i b u t i v e .
This f o l l o w s a l s o from Theorem 3.2.8,
since
= deg(G/n3) = 2. For i n s t a n c e f o r G (and G/n2) t h i s f o l l o w s from t h e f a c t t h a t t h e 3-node subgraph G/n2)(11,2,451 of deg(G) = deg(G/np) = 3 and deg(G/nl) i s degenerate.
G/n2
111.3
THE JORDAN-HOLDER
THEOREM FOR COMPOSITION
SERIES
An important instrument o f an a l g e b r a i c theory i s t h e f a c t o r i z a t i o n o f a s t r u c t u r e by means o f homomorphisms, where s p e c i a l i n t e r e s t i s p a i d t o t h e successive f a c t o r i z a t i o n i n steps which cannot be r e f i n e d any f u r t h e r ( c h i e f s e r i e s of an algebra i n the sense o f [30]). Definition:
Here, we s h a l l i n t r o d u c e t h e same n o t i o n .
L e t S be a s t r u c t u r e on A.
f i n i t e sequence S = So,Sl,,..,Sn
A composition s e r i e s o f S i s a maximal o f p a i r w i s e non-isomorphic s t r u c t u r e s Si on Ai
w i t h ( A n [ = 1 such t h a t t h e r e e x i s t s an epimorphism hi i = 1, ..., n.
E
Epi(Si-l,Si)
f o r each
Since the sequence i s supposed t o be of maximal length, the congruence p a r t i t i o n V(Si-l)
ri
induced by hi i s an atom i n V(Si-l).
Thus, because o f Lemma 3.2.4,
hi maps e x a c t l y one n o n - t r i v i a l , prime s u b s t r u c t u r e Si-ll element o f Si,
and maps t h e elements n o t i n Bi b i j e c t i v e l y .
SoIB1, S1 JB2, ..., Sn-,IBn
s
=
Bi o f Si-l
so,sl ,.. . ,sn.
=
onto one
The prime s t r u c t u r e s
Sn-l are c a l l e d t h e f a c t o r s o f t h e composition s e r i e s
321
Substitution decomposition for discrete structures
(M6) and Theorem 3.1.3 i m p l y e a s i l y t h a t , g i v e n a c o m p o s i t i o n s e r i e s S = So,S1,
...,Sn,
t h e r e i s a maximal c h a i n no 4
rl 6
... 6
Si V ( S ) induces t h e c o m p o s i t i o n s e r i e s S,S/rl,.
vn =
TI^ i n V ( S ) w i t h
... .s
= TI in n (where, o f course, d i f f e r e n t
S / T ~ and t h a t , conversely, each maximal c h a i n T O 6
. .,S/TI,,
nl 6
maximal c h a i n s may induce isomorphic c o m p o s i t i o n s e r i e s ) .
So c o m p o s i t i o n s e r i e s correspond e s s e n t i a l l y t o t h e maximal c h a i n s i n V ( S ) .
This
g i v e s t h e f o l l o w i n g r e s u l t s , t h e second o f which i s a " c l a s s i c a l " Jordan-
-
H o l d e r - t y p e theorem f o r c o m p o s i t i o n s e r i e s o f s t r u c t u r e s f u l f i l l i n g ( M l )
Theorem 3.3.1:
(EXISTENCE CRITERION):
one o f t h e statements o f Theorem 3.2.3
Theorem 3.3.2:
(M8).
A s t r u c t u r e S has a c o m p o s i t i o n s e r i e s i f f holds.
(JORDAN-HOLDER THEOREM):
Any two c o m p o s i t i o n s e r i e s o f S have t h e
same l e n g t h and t h e same f a c t o r s up t o isomorphism and rearrangement. Proof:
.3.3.1
and t h e i n v a r i a n c e o f t h e l e n g t h i n 3.3.2 f o l l o w f r o m t h e i n t e r p r e -
t a t i o n o f c o m p o s i t i o n s e r i e s as maximal c h a i n s i n V ( S ) and t h e r e s u l t s f r o m 111.2. The i n v a r i a n c e o f t h e f a c t o r s i s t h e n shown by i n d u c t i o n on t h e l e n g t h o f a comp o s i t i o n s e r i e s , which because o f t h e upper s e m i m o d u l a r i t y o f V ( S ) (and t h e Given two atoms
symmetry of t h e s i t u a t i o n ) reduces t o t h e f o l l o w i n g argument: 0
EA\B}
= {B,{a)la
and
T
= { C , { a j l ~ r E A\C> o f V ( S ) , show t h a t S I B i s isomorphic
t o ( S / T ) I C ' , where C ' i s t h e o n l y n o n - t r i v i a l c l a s s i n
UVT/T
E
This i s
V(S/T).
done as f o l l o w s : Since B E A ( S ) , f := nToinc;
Because o f ( M I ) , and Lemma 3.1.1,
E HOm(S[B,S/r).
f has a f a c t o r i z a t i o n f = i n c i l o h , where h r Epi(SIB,S1), and S1 := S/T has t h e base s e t A1.
inc!lcMono(SIID,S1)
One e a s i l y v e r i f i e s t h a t D = h(B) =
rlT(B).
If
Bn C
:
If
Bn C
# 4 , t h e n I B f ) C I = 1 ( o t h e r w i s e S I C would be decomposable because o f
4 , t h e n n T maps a l l elements o f
B bijectively.
( M 7 ) and Theorem 3.1.4) and one a g a i n o b t a i n s t h a t n T maps a l l elements o f B bijectively. Thus h i s t h e r e s t r i c t i o n o f n T t o B and b i j e c t i v e . a unique s t r u c t u r e T w i t h h (M5) t h a t T
=
S1 I h ( B ) .
E
Hom(SJB,T) and h - l
Hence S I B and SIID
E
Because o f (M2) t h e r e e x i s t s Hom(T,SIB).
= S,[h(B)
I t remains t o be shown t h a t S1 [ h ( B ) = S1 I C ' .
I t f o l l o w s from
a r e isomorphic.
L e t g E Epi(S1,S/cm).
Then
R. II: Miihring and F.J. Radennacker
322
gonT = nuvi and t h u s 1gonT(6)l = 1. S i n c e [ r l T ( B ) I > 1, g maps rl,(B) o n t o one element. T h e r e f o r e t h e congruence p a r t i t i o n U V T / T induced b y g has t h e f o r m
{nT(E),{~llG
E
Al\ni(B)l
which p r o v e s n T ( B ) = C ' .
m
Theorems o f t h e Jordan-Holder t y p e a r e by no means s e l f - e v i d e n t ,
f o r i n s t a n c e , l a t t i c e s fi54] o r
hold i n well-structured algebraic theories, cf., semi-automata [ 6 l l .
and do even n o t
So t h e r e have been e x t e n s i v e i n v e s t i g a t i o n s w . r . t .
b r a i c p r o p e r t i e s under which such a theorem holds, c f . f o r example
t h e alge-
"1,
[45],
U s u a l l y t h e " c o m m u t a t i v i t y " o f t h e congruence r e 1 a t i o n s o f t h e [66], [128]. a l g e b r a o r c e r t a i n g e n e r a l i z a t i o n s t h e r e o f [63], [128] a r e b a s i c t o such a theorem. Such a p r o p e r t y does n o t h o l d f o r t h e s t r u c t u r e s c o n s i d e r e d h e r e .
T h i s becomes,
f o r i n s t a n c e , apparent b y t h e f a c t t h a t t h e congruence p a r t i t i o n l a t t i c e s a r e ( i n t h e f i n i t e case) o n l y upper semimodular, whereas f o r a l g e b r a s w i t h commuting congruence r e l a t i o n 5 t h e y a r e modular [13].
I n t h i s r e s p e c t , t h e s t r u c t u r e s con-
s i d e r e d h e r e c o n s t i t u t e a s e p a r a t e case, i n which a Jordan-Holder Theorem h o l d s under c o n d i t i o n s d i f f e r e n t f r o m t h e u s u a l ones.
A f u r t h e r i n d i c a t i o n as t o t h e d i f f e r e n t n a t u r e of t h i s r e s u l t i s an a d d i t i o n a l i n v a r i a n c e ( a Church-Rosser p r o p e r t y ) f o r c o m p o s i t i o n s e r i e s which i s known f o r r e l a t i o n s and c l u t t e r s , and which does n o t h o l d f o r a l g e b r a s , i n g e n e r a l : Any two c o m p o s i t i o n s e r i e s of S have t h e same l a s t f a c t o r up t o isomorphisni.
I n t h e general model, t h i s i n v a r i a n c e does n o t f o l l o w f r o m (Ml)
-
(M8), b u t
r e q u i r e s an a d d i t i o n a l assumption, w h i c h h o l d s i n t h e s p e c i a l cases o f 1.2 - 1.4. (M12)
L e t SIB and S I C be p r i m e s u b s t r u c t u r e s o f S such t h a t B fl C # $. Then S I B and S I C a r e i s o m o r p h i c .
I f t h e c a t e g o r y K f u l f i l l s ( M l ) - (M8) and (M12), t h e n any two c o m p o s i t i o n s e r i e s o f a s t r u c t u r e S have t h e same l a s t f a c t o r up t o isomorphism.
Theorem 3.3.3:
Proof:
I f , i n t h e p r o o f o f Theorem 3.3.2,
UVT
#
nl,
t h e n t h e l a s t f a c t o r i n any
c o m p o s i t i o n s e r i e s o f S i s i s o m o r p h i c t o t h e l a s t f a c t o r o f S / O V T by t h e i n d u c t i v e a s s u m t i on. If
OVT
= nl,
then
Bn C
#
$
and S I B = S I C because o f (M12).
Since
two c o m p o s i t i o n s e r i e s c o n s i d e r e d i n t h e p r o o f o f Theorem 3.3.2 and S, S / T , S/svr,
i.e.,
S / u and
S/T
are the l a s t factors.
t h a t SIB and S/T as w e l l as S I C and S/u a r e i s o m o r p h i c . a l l f o u r f a c t o r s S I B , SIC, S/o,
S/T are isomorphic. m
UVT
= nl, t h e
a r e S, S / u , S/UVT,
Theorem 3.3.2
shows
So, because o f (M12),
323
Substitution decomposition for discrete structures
S i m i l a r t o t h e examples i n 111.1, t h e r e e x i s t subcategories o f KO i n which (1:12) holds b u t n o t
Example 3.3.4:
(M11), and v i ce-versa.
L e t 0 and G be t h e p a r t i a l o r d e r and i t s a s s o c i a t e d c o m p a r a b i l i t y
graph o f F i g u r e 3.3.
A composition series o f
0 ( i n activity-on-arc
representa-
t i o n ) and a corresponding one f o r G a r e g i v e n i n F i g u r e 3.3 t o g e t h e r w i t h t h e composition series
composition s e r i e s
factors
factor.
0
*
I
0
in
c .
F i g . 3.3 Composition
W
Series for a P a r t i a l Order and f o r i t s Comparabi 1it y
0
0
Q
9-
m
.
.
Graph.
R.H. Mdhring and F.J. Radermacher
324
Note t h a t each composition s e r i e s f o r 0 induces one f o r G
associated f a c t o r s .
(because of Theorem 1.5.1),
b u t n o t vice-versa.
S i m i l a r l y , t h e g i v e n composition
s e r i e s of 0 a l s o induces composition s e r i e s f o r the c l u t t e r s C := C(G) o f c - m a x i ma1 cliques, arc] o f c-maximal independent sets, and b[C]
o f 5-minimal c l i q u e
separating sets, and the associated monotonic Boolean f u n c t i o n s ( c f . Example
1 . 1 . 1 ) v i a the connections given i n 1.5.
This a l s o holds f o r t h e r e s p e c t i v e
factors.
111.4
THE COMPOSITION TREE
We s h a l l show t h a t the r e p r e s e n t a t i o n o f t h e decomposition p o s s i b i l i t i e s i n a tree, which i s known f o r c l u t t e r s n38],
[2q,
p81 and
graphs [32],
p a r t i a l orders
i s a l s o v a l i d i n t h e general theory, i f , f o r A ( S ) , t h e a d d i t i o n a l c o n d i t i o n
(M9) i s imposed, which s t a t e s t h a t i f C1,C2
E
A ( S ) overlap, then C1\C2
E
A(S).
This c o n d i t i o n w i l l be assumed throughout 111.4 and 111.5 and i s v a l i d for t h e s t r u c t u r e s considered i n Section I .
Related r e s u l t s f o r t h e f i n i t e case based
o n l y on the p r o p e r t i e s o f A ( S ) a r e a l s o contained i n c49]. The c o n s t r u c t i o n o f the t r e e i s based on two decomposition p r i n c i p l e s , the f i r s t of wh'ich i s the "maximal d i s j o i n t decomposition".
Definition:
Let S be a s t r u c t u r e on A .
a partition
a* o f
A maximal d i s j o i n t decomposition o f S i s A i n t o c-maximal S-autonomous s e t s B # A.
The f o l l o w i n g decomposition p r i n c i p l e i s then obvious.
Theorem 3.4.1: I f S admits a maximal d i s j o i n t decomposition a*, then each Sautonomous s e t o t h e r than A i s SIB-autonomous f o r some B E u*. I f , furthermore, a* V(S)\{a?J,
E
V ( S ) , then u* i s the coarsest congruence p a r t i t i o n i n
and S/O* i s prime.
The existence o f a maximal d i s j o i n t decomposition o f S depends t o some degree on the non-existence o f special q u o t i e n t s o f S.
Definition:
A s t r u c t u r e S on A i s c a l l e d s e m i - l i n e a r i f t h e r e i s a l i n e a r order-
i n g 6 on A such t h a t A ( + ) c A ( S ) , i . e . i f A ( S ) contains a l l <-convex sets ( c f . a l s o t h e d e f i n i t i o n o f l i n e a r s t r u c t u r e s i n 1.1).
Recall t h a t B
C
P. i s ,<-convex,
Substitution decomposition for discrete structures i f a,B
~l
B and a 6 y
\c
B implies t h a t y
By L ( S ) we denote t h e system o f a l l n
E
E
325
B.
V ( S ) such t h a t S/a i s s e m i - l i n e a r .
A s t r u c t u r e S on A admits a maximal d i s j o i n t decomposition u* i f f
Theorem 3.4.2:
the f o l l o w i n g conditions are f u l f i l l e d 1.
Each a E A i s c o n t a i n e d i n an G-maximal S-autonomous s e t C(a) # A.
2.
In[
Proof:
Q
2 for all n eL(S).
Assume t h a t u* e x i s t s .
Then 1. i s obvious.
Let
Then n 6 u* and t h e r e e x i s t s a n o n - t r i v i a l S/+-autonomous c l a s s C ' of u*/a as a p r o p e r subset.
Then q i l ( B ' )
3
e L(S) w i t h
3 3.
set B' containing a
nil(C')
E u*,
a contradic-
t i o n t o t h e m a x i m a l i t y o f t h e c l a s s e s o f a*. I n t h e converse d i r e c t i o n , i t s u f f i c e s t o show t h a t t i t i o n o f A.
U*
:= { C ( a ) l a E A } i s a p a r -
I f n o t , t h e r e e x i s t C(a) and C(B) which o v e r l a p .
t h e m a x i m a l i t y o f C(a) and C(B), Theorem 3.1.4,la) { C ( a ) \ C ( ~ ) , C ( a ) f l C(B),C(B)\C(a)l
E
Then, because o f
and (M9) we o b t a i n t h a t n : =
L(S), a c o n t r a d i c t i o n t o 2.
T h i s means t h a t i f 1 . h o l d s and S admits no maximal d i s j o i n t decomposition, then S has n o n - t r i v i a l s e m i - l i n e a r q u o t i e n t s .
I n order t o i n v e s t i g a t e t h i s s i t u a t i o n
f u r t h e r , we w i l l exclude c e r t a i n " b i z a r r e " cases by t h e f o l l o w i n g assumptions which encompass a l l i n t e r e s t i n g cases and which guarantee a c e r t a i n f i n i t e n e s s o r supply us w i t h s t r o n r c o n s t r u c t i o n p r i n c i p l e s .
(Fl):
L(S) i s o f f i n i t e length ( i n V ( S ) ) .
(F2):
S f u l f i l l s (147)" and (M10).
Theorem 3.4.3:
Under ( F l ) o r (FZ), each s e m i - l i n e a r s t r u c t u r e S on A w i t h \ A / 3 3
i s e i t h e r l i n e a r o r degenerate.
PrQof: L e t 6 be a l i n e a r o r d e r on A such t h a t A ( c ) E A ( S ) . We s h a l l show t h a t A ( S ) i s degenerate i f A ( & ) # A ( S ) . So l e t A ( & ) # A ( S ) . Then t h e r e e x i s t s Co E A ( S ) , a1,a2 E Co and y {al , a 2 } E
A(S).
L e t C1 :=
Ll,y] n Co,
4 Co such t h a t
C2 =
h,a2] n Co,
o r d e r 6 . I t f o l l o w s t h a t C,,C2 ( c , U c,)\]a,,y]
=
al
<
Y < a2.
We s h a l l f i r s t show t h a t
where t h e i n t e r v a l s r e f e r t o t h e l i n e a r
and C1 U C2 a r e S-autonomous.
I a l l U C2 e A ( S ) .
I f C, # { a l l then
S i m i l a r l y , if C2 ic f a 2 } , t h e n
R.IL Mohring and F.J. Rudermacher
326
The next step i s t o show t h a t { a , a1. ) then B1 := ial,a21
instance, a >, ct2, also
= B1\B2
Ia,nl}
E. A ( S ) .
Then
(Ia,cxliu
6
Ia,all
If, f o r
A ( S ) ( i = 1,2) f o r a l l h E A.
U]y,a]
The cases
Now l e t a,B E A\Ial,x21. {B,al))\tul,a2~
E
A ( S ) and B2 := [y,a[
E
a, 6 a 4
E
A ( S ) . Hence
a2 and a Q al f o l l o w s i m i l a r l y .
e A ( S ) and 15,al> c A ( S ) .
Hence I a , B l =
A(S).
and ( F l ) , or (M7)* i n ( F 2 ) then y i e l d t h a t each non-void subset
Theorem 3.1.4,la)
o f A i s S-autonomous
.
I t i s easy t o see t h a t each q u o t i e n t o f a s e m i - l i n e a r s t r u c t u r e i s again semi-
linear.
This observation implies, together w i t h ( M 6 ) , t h a t L(S) i s monotonically
closed, i . e .
i f n e L ( S ) then u
E
L(S) f o r a l l u
t h e question o f whether L ( S ) contains a
E
V ( S ) with
least element,
which may serve as a r e p r e s e n t a t i o n o f L ( S ) .
II
.c< u.
This leads t o
i .e. a f i n e s t p a r t i t i o n II*
I f t h i s i s t h e case, we o b t a i n t h e
second decomposi t i m > p r i r r c i p l e :
L e t S be a s t r u c t u r e on A f o r which L(S) contains a f i n e s t par-
Theorem 3.4.4: tition
TI*.
I n t h i s case each a ) If S/II* i s l i n e a r , then S / a i s l i n e a r f o r each TI E L . C E A ( S ) i s e i t h e r SIB-autonomous f o r some B e II*o r t h e convex u n i o n o f classes o f II*( i . e . the s e t o f classes o f
T*
whose union i s C i s convex w . r . t .
the l i n e a r
order inducing A ( S / n * ) ) . b ) I f S/a* i s degenerate, then S / T i s degenerate f o r each TI r L ( S ) . I n t h i s A ( S ) i s e i t h e r SIB-autonomous f o r some B E II*,o r t h e union o f case each C classes o f Proof: linear.
TI*.
a ) It i s easy t o see t h a t each q u o t i e n t o f a l i n e a r s t r u c t u r e i s again Let 6 be a l i n e a r order on A/n* suchthat A ( < ) = A ( S / n * ) . I f a) i s not
true, there e x i s t C L e t w.1.o.g.
6
L1 6 L2.
a r e S-autonomous.
A ( S ) and L1,L2
Then B := [C
E.
T*.
f o l l o w s analogously.
nC
L],
#
$ and
L2Q
C #$.
and L1\B and L l n B
B l E V(S).
{Ll\B,Lln
It
L ( S ) (an associated l i n e a r o r d e r 2 i s obtained
from & by r e p l a c i n g L1 by L1\B and
b)
L1\C # 4 , L1
Because o f ( M 8 ) , u := (II*\{L~))U
can then be v e r i f i e d t h a t u contradicts u <
L TI*w i t h
u L1
L1nB w i t h
t h e o r d e r L1\B
-3
L l n B.
This
321
Substitution decomposition for discrete structures I n general, L(S) may n o t c o n t a i n a f i n e s t p a r t i t i o n h o l d s , we o b t a i n :
IT*.
B u t i f ( F l ) o r (F2)
L e t S be a s t r u c t u r e on A f u l f i l l i n g ( F l ) o r ( F 2 ) .
Theorem 3.4.5:
contains a f i n e s t p a r t i t i o n
Then L ( S )
and e i t h e r of t h e cases o f Theorem 3.4.4 a p p l i e s .
IT*
The p r o o f i s r a t h e r l a b o r i o u s , b u t s t r a i g h t f o r w a r d , so t h a t we g i v e o n l y
Proof:
t h e main i d e a s .
Claim:
I f r1,v2
E
L(S), then
r1 A r2E
L(S).
IT^
Lemma 3.2.1 and Theorem 3.2.2 show t h a t
A
r 2 E V ( S ) f o r t h e cases ( F l ) and
Because o f Theorem 3.1.3,
(FZ), r e s p e c t i v e l y .
we may assume t h a t
Assume f i r s t t h a t S/nl and S/.n2 a r e b o t h l i n e a r , and t h a t “1 and l i n e a r orders.
r1 #
Since, w.l.o.g.,
IT^
A
n2
# r 2 , there are
IT^ 62
A
IT^
=
IT
0
.
a r e associated
ao,Bo E
A
with
Then, w.l.o.g., a l s o [a0]IT 2 “2 [aJni # [BJri, i = 1,2, and [ao]~,~l [bJn,. [ B ~ I T ( o~ t h e r w i s e t a k e t h e dual o f G ~ ) . Each 4. induces a p a r t i a l o r d e r 4 . on B if
1
IT^<^
o r i f a = B, i = 1,2.
[B]ri
by c o m p a t a b i l i t y o f <1 and <2 on A. B <*
(*I
{“f o cr1 a l l
Since
IT
1
A
n
a
r2 =
C2
, E~ IT
0
1
A
Then one can show t h e f o l l o w i n g
f3
A such t h a t
IT^
#
[B]IT~,
i = 1,2.
,
B d e f i n e s a l i n e a r o r d e r on
otherwise
A.
We s h a l l show t h a t A($) c A ( S ) , which proves t h e
L1 i = 1,2. It follows from s
Let
[ ~ , BEJ A ( $ ) .
L e t B~ :=
0
Thus a l l i n t e r v a l s [a,@] o f A(<) from ( F l ) o r (F2).
a r e members o f A ( S ) .
A(&) G A(S) then follows
The cases i n which one o r b o t h o f S/nl,
S / I T ~ a r e degenerate,
f o l l o w analogously.
So under ( F l ) , which covers t h e f i n i t e case, t h e theorem f o l l o w s f r o m t h e c l a i m and Theorem 3.4.3.
Under ( F 2 ) one can show w i t h Z o r n ’ s lemma t h a t L(S) must have
minimal elements, which proves t h e theorem by t h e c l a i m .
Here a g a i n t h e c o n s t r u c -
t i o n o f an a p p r o p r i a t e l i n e a r o r d e r t o show s e m i - l i n e a r i t y i s r a t h e r l a b o r i o u s .
R.H. Mohring and F.J. Radermacher
328
Based on the two decomposition p r i n c i p l e s , we w i l l d i s t i n g u i s h the f o l l o w i n g types o f s t r u c t u r e s , which subsequently l e a d t o t h e d e f i n i t i o n o f the composition tree.
D e f i n i t i o n : Let S be a s t r u c t u r e on A .
Then e x a c t l y one o f t h e f o l l o w i n g cases
applies:
1.
S has a maximal d i s j o i n t decomposition
P ( s i n c e S/G* i s
prime
if
G*
G*.
Then S i s s a i d t o be o f t h e
eV(S)).
2. S i s n o t of the type P, b u t L ( S ) has a f i n e s t p a r t i t i o n l i n e a r o r degenerate. respectively
3.
D,
.
I n a l l o t h e r cases, S i s s a i d t o be o f the type
Definition:
n* such t h a t S/n* i s
Then S i s s a i d t o be o f t h e t y p e L o r t h e t y p e
L e t S be a s t r u c t u r e on A .
-.
Define the labeled t r e e
B(S) as f o l l o w s .
Each node o f B ( S ) i s an S-autonomous s e t .
1.
The r o o t o f B ( S ) i s A.
2.
A node C o f B ( S ) w i t h ICI 3 2 i s l a b e l e d w i t h t h e type of t h e s u b s t r u c t u r e
S I C o f S.
Nodes C w i t h ICI = 1 a r e n o t labeled.
3. A node C w i t h l a b e l P,L o r D i s assigned t h e classes o f t h e associated part i t i o n $ cr n; o f SIC as i t s immediate successors. I n t h e case o f type L, these successors a r e ordered "from l e f t t o r i g h t " according t o an associated l i n e a r order on$.
A l l o t h e r nodes have no successors.
4. I f t h i s procedure ends f o r each sequence A
= Co
C
C1 c...= C,
C
... o f
nodes
a f t e r f i n i t e l y many steps, then B ( S ) i s d e f i n e d t o be t h e r e s u l t i n g t r e e o f f i n i t e depth.
Otherwise, B ( S ) c o n s i s t s o f t h e r o o t A w i t h l a b e l
-.
B ( S ) i s c a l l e d t h e composition t r e e o f S. For examples, c f . Section I V . S i s called f i n i t e l y decomposable i f no node o f B ( S ) i s l a b e l e d w i t h a. This i s o f course t h e case i f ( F l ) o r (F2) holds f o r each node. For f i n i t e l y decomposable s t r u c t u r e s , B ( S ) contains a l l t h e i n f o r m a t i o n about A ( S ) i n an e a s i l y a c c e s s i b l e way:
Theorem 3.4.6:
L e t S be a f i n i t e l y decomposable s t r u c t u r e on A.
i s S-autonomous i f f one o f t h e f o l l o w i n g cases a p p l i e s : 1.
B i s a node o f 6(S).
A subset B o f A
Substitution decomposition for discrete structures
2.
B i s t h e u n i o n o f immediate successors o f a node o f t h e t y p e L which f o r m a convex s e t w . r . t .
3.
329
the associated l i n e a r order.
6 i s t h e u n i o n o f immediate successors o f a node o f t h e t y p e D.
Proof: -
Theorem 3.4.1 and 3 . 4 . 4 . r
T h i s r e s u l t on t h e r e p r e s e n t a t i o n o f A ( S ) i s b a s i c f o r i n v e s t i g a t i n g t h e a l g o r i t h m i c compl e x i t y o f t h e decomposition i n 111.6.
Furthermore, i t g i v e s many i n s i g h t s
i n t o t h e s t r u c t u r a l p r o p e r t i e s o f t h e s u b s t i t u t i o n decomposition,
some o f which
a r e s u m a r i z e d below. 1.
If S has a c o m p o s i t i o n s e r i e s (and f u l f i l l s (M9), which i s necessary f o r t h e
t r e e c o n s t r u c t i o n ) , t h e n S i s f i n i t e l y decomposable, b u t n o t v i c e versa.
I n this
sense B(S) i s a s t r o n g e r i n s t r u m e n t f o r s t r u c t u r a l a n a l y s i s t h a n c o m p o s i t i o n series.
If, i n a d d i t i o n , (M12) h o l d s t o o , t h e n t h e Jordan-Holder theorem may be
d e r i v e d from t h e p r o p e r t i e s o f B ( S ) .
2. I f S i s f i n i t e l y decomposable, t h e n A ( S ) i s s y m m e t r i c a l l y c l o s e d i f f B ( S ) c o n t a i n s no node o f t h e t y p e L. Moreover, t h e symmetric c l o s u r e A ( S ) S Y o f A ( S ) w.r.t.
symmetric d i f f e r e n c e s i s o b t a i n e d f r o m
B(S) by r e p l a c i n g a l l l a b e l s L by D.
F o r p a r t i a l o r d e r s S t h i s extends a l s o t o t h e symmetric c l o s u r e o f S i t s e l f , i . e . t o t h e a s s o c i a t e d c o m p a r a b i l i t y graph, c f . Theorem 4.1.4. 3. I f , i n t h e f i n i t e case, we c o n s i d e r A ( S ) U { g } as a s u b l a t t i c e o f t h e l a t t i c e P(A) o f a l l subsets o f A, t h e n t h e r e s u l t s on B(S) show t h a t A ( S ) i s composed o f t h r e e t y p e s o f s u b l a t t i c e s , w h i c h correspond t o t h e systems A(S/B) f o r t h e t h r e e d i f f e r e n t t y p e s o f nodes. Chein e t a l . [27],
T h i s i s an i n t e r e s t i n g c o n n e c t i o n w i t h a r e s u l t o f
which c h a r a c t e r i z e s t h e l a t t i c e A ( S ) i n t h e case t h a t ( A l ) - ( A 4 )
hold, i n e s s e n t i a l l y t h e same way.
Their r e s t r i c t i o n t o t h e symmetrically closed
case i s m o t i v a t e d by t h e i r t r e a t m e n t o f graphs and s e t systems, which f u l f i l (A4) i n general, c f . S e c t i o n I . 4.
F o r t h e f i n i t e case, i t i s shown i n [149]
by e x p l o i t i n g o n l y t h e p r o p e r t i e s
o f autonomous s e t s t h a t t o each c o m p o s i t i o n t r e e B ( S ) o f an a r b i t r a r y s t r u c t u r e S
t h e r e i s a r e l a t i o n a l system R = (Ri)ieI
B(S) = B(R).
such t h a t A ( S ) = A(R) ( =
i21 A(Ri))
and
So i n t h e f i n i t e case, t h e p r o p e r t i e s o f a r b i t r a r y .composition t r e e s
can a l r e a d y be o b t a i n e d by s t u d y i n g c o m p o s i t i o n t r e e s o f r e l a t i o n a l systems. T h i s shows a n o t h e r d i f f e r e n c e w i t h t h e i n f i n i t e case, i n which r e l a t i o n s and s e t systems behave q u i t e d i f f e r e n t l y .
R. H. Mohrittg and t;.J. Radermacher
330 111.5
CONNECTIONS WITH THE S P L I T DECOMPOSITION
The decomposition p r i n c i p l e s used here t o o b t a i n B(S) a r e s t r o n g l y r e l a t e d t o t h e s p l i t decomposition i n t h e f i n i t e case.
I n f a c t , degenerate and l i n e a r s t r u c t u r e s
correspond t o the more general b r i t t l e and s e m i - b r i t t l e s t r u c t u r e s i n [38].
The
c e n t r a l r e s u l t f o r t h e s p l i t decomposition i s t h a t each s t r u c t u r e has a unique u n r e f i n e d decomposition i n t o prime, b r i t t l e and s e m i - b r i t t l e s t r u c t u r e s , which may be v i s u a l i z e d i n an ( u n d i r e c t e d r a t h e r than d i r e c t e d ) composition t r e e .
The
analogous r e s u l t f o r t h e s u b s t i t u t i o n decomposition i s obtained by r e p l a c i n g each node B of B(S) w i t h
IBI
b
2 by ( S ~ B ) / T ~ where ,
sg
i s the p a r t i t i o n o f B i n t o i t s
imnediate successors, and a p p l y i n g t h e theorems l e a d i n g t o t h e c o n s t r u c t i o n o f B(G).
In s p i t e c f these c l o s e connections between t h e s p l i t decomposition and t h e subs t i t u t i o n decomposition w . r . t . series i s d i f f e r e n t .
composition trees, t h e s i t u a t i o n w . r . t .
composition
I f we t h i n k o f a composition s e r i e s f o r t h e split-decomposi-
t i o n as a sequence o f s p l i t s , where one s u b s t r u c t u r e i s split-indecomposable ( t h e " f a c t o r " ) and t h e o t h e r s u b s t r u c t u r e i s the n e x t s t r u c t u r e i n the composition series (the "quotient"),
then t h e i n v a r i a n c e o f t h e l e n g t h i s v a l i d .
However,
the invariance o f t h e f a c t o r s , which would correspond t o a unique (up t o isomori s l o s t i n general, as i s shown
phism) prime decomposition i n the sense o f [38], by the examples f o r b i n a r y r e l a t i o n s i n [35].
111.6
ON THE ALGORITHMIC COMPLEXITY OF DECOMPOSITION
With regard t o the l a r g e scope o f a p p l i c a t i o n o f t h e s u b s t i t u t i o n decomposition ( c f . Section 1 1 ) , e f f i c i e n t algorithms f o r determining the decomposition p o s s i b i l i t i e s o f a ( f i n i t e ) s t r u c t u r e S p l a y an important r o l e .
Because o f Theorem 3.4.6,
t h e composition t r e e B(S) o f a s t r u c t u r e S seems t o be an a p p r o p r i a t e instrument t o represent the system A ( S ) and thus, i n t h e f i n i t e case, a l l t h e decomposition p o s s i b i l i t i e s o f S.
The f o l l o w i n g theorem shows t h a t & ( S ) i s indeed an approp-
r i a t e data s t r u c t u r e f o r t h e r e p r e s e n t a t i o n o f A ( S ) , since i t contains o n l y l i n e a r l y ( i n j A l ) many nodes, whereas I A ( S ) I may be exponential i n
Theorem 3.6.1:
L e t S be a s t r u c t u r e on a f i n i t e s e t A, and l e t
number o f nodes o f B ( S ) .
Then
I&(S)l<
2[AI
r e l a t i o n s , s e t s y s t e m s and Boolean f u n c t i o n s . Proof: _ _
I n d u c t i o n on
IA( .I
-
1.
IAl. IB(S)I denote the
This bound i s t i g h t f o r
33 I
Substitution decomposition for discrete structures
I n o r d e r t o o b t a i n some i n s i g h t s i n t o a l g o r i t h m s f o r d e t e r m i n i n g t h e decomposition p o s s i b i l i t i e s o f a s t r u c t u r e , we c o n s i d e r t h e f o l l o w i n g a l g o r i t h m i c t a s k s . Task 1:
Task 2:
A s t r u c t u r e S on A. o u t p u t : "Prime", i f S i s p r i m e A n o n - t r i v i a l S-autonomous s e t , o t h e r w i s e Input:
Input:
A s t r u c t u r e S on A, a subset B o f A.
output:
The S-autonomous c l o s u r e B* o f B, which i s d e f i n e d as t h e l e a s t S-autonomous s e t c o n t a i n i n g B. 3.1.4,l.a,
Task 3:
(Because o f Theorem
B* e x i s t s i n t h e f i n i t e case).
Input:
A s t r u c t u r e S on A.
output:
The c o m p o s i t i o n t r e e B ( S ) .
Apparently, these t h r e e t a s k s a r e o f i n c r e a s i n g d i f f i c u l t y , o r viewed as o r a c l e s , they a r e o f increasing a b i l i t y .
I n p a r t i c u l a r , Task 1 does what one would con-
s i d e r as a minimal requirement i n o r d e r t o c a r r y o u t a decomposition, w h i l e Task 3 determines a l l decompositions o f S i n t h e f o r m o f B ( S ) .
I t i s t h e r e f o r e perhaps
s u r p r i s i n g t h a t a l l t h r e e t a s k s t u r n o u t t o be e s s e n t i a l l y e q u i v a l e n t i n t h e sense t h a t (up t o c e r t a i n s t r u c t u r a l o p e r a t i o n s ) t h e y a r e T u r i n g r e d u c i b l e [59] each o t h e r .
To f o r m u l a t e t h e r e s u l t , l e t Pi(n)
( i = 1,2,3)
to
denote t h e worst-case
c o m p l e x i t y o f a l g o r i t h m Pi f o r t a s k i, vhen a p p l i e d t o s t r u c t u r e s on A = { l , ...,nl. Furthermore, l e t Ql(n) denote t h e c o m p l e x i t y o f t e s t i n g a g i v e n s e t f o r S-autonomy, l e t Q 2 ( n ) denote t h e c o m p l e x i t y o f c o n s t r u c t i n g a s u b s t r u c t u r e SIB, and l e t Q 3 ( n ) denote t h e c o m p l e x i t y o f c o n s t r u c t i n g a q u o t i e n t S / r B ,
where
rB := { B , { a ) l a
E:
A\BZ
a ) F o r each a l g o r i t h m P1 t h e r e i s an a l g o r i t h m P3 w i t h P3(n) = O(n2Pl(n) t n2Ql(n) + n 2 Q 2 ( n ) t nQ3(n) t n 4 )
Theorem 3.6.2:
b)
F o r each a l g o r i t h m P2 t h e r e i s an a l g o r i t h m P3 w i t h P3(n) = O(n3P2(n))
c)
F o r each a l g o r i t h m P3 t h e r e a r e a l g o r i t h m s P1 and Pz w i t h Pl(n)
= 0(1)
+
P3(n),
P2(n) = o(n2)
+
p3(n).
T h i s shows t h a t f o r an e f f i c i e n t d e t e r m i n a t i o n o f B ( S ) , i t s u f f i c e s t o f i n d s u f f i c i e n t methods f o r c o n s t r u c t i n g t h e autonomous c l o s u r e o f a g i v e n s e t B. a c t u a l l y what w i l l be done f o r r e l a t i o n s and c l u t t e r s i n S e c t i o n
IV.
This i s
R.H. Mohring and F.J. Radermacher
332
IV. ALGEBRAIC AND ALGORITHMIC ASPECTS OF THE SUBSTITUTION DECOMPOSITION FOR RELATIONS, SET SYSTEMS, AND BOOLEAN FUNCTIONS I n t h i s Section we r e i n t e r p r e t t h e r e s u l t s obtained i n the general model f o r t h e special s t r u c t u r e s discussed i n Section I .
I n p a r t i c u l a r , we c h a r a c t e r i z e the
r e s p e c t i v e l i n e a r and degenerate s t r u c t u r e s , discuss t h e complexity o f decomposit i o n and i n t e r p r e t t h e i n v a r i a n c e s mentioned i n Section I i n connection w i t h t h e We w i l l s t a r t w i t h r e l a t i o n s , since they c o n s t i t u t e a c l a s s f o r
composition t r e e .
which a l l c o n d i t i o n s h o l d i n t h e strong sense, and afterwards t r e a t s e t systems and Boolean f u n c t i o n s , which have weaker p r o p e r t i e s .
IV.l
RELATIONS
R e l a t i o n s f u l f i l c o n d i t i o n s (Ml)
-
(M5) w i t h t h e s u r j e c t i v e mappings d e r i v e d from
the s u b s t i t u t i o n o p e r a t i o n i n 1.4 as t h e s u r j e c t i v e homomorphisms, and w i t h t h e s t i p u l a t e d i d e n t i f i c a t i o n of r e l a t i o n s up t o r e f l e x i v e t u p l e s ( a ,...,a), which i s necessary f o r (M5). I t i s easy t o see t h a t (M6) and the strong versions (M7)* and (M8)* o f (M7) and
(M8) hold, as do the a d d i t i o n a l c o n d i t i o n s (M9)
-
(M12).
So r e l a t i o n s c o n s t i t u t e
an example f o r a c l a s s o f s t r u c t u r e s which f u l f i l a l l p r o p e r t i e s discussed so f a r i n t h e i r strongest v e r s i o n .
I n p a r t i c u l a r , we have t h e i n v a r i a n c e of t h e
l a s t f a c t o r i n composition s e r i e s , t h e unique r e c o n s t r u c t i o n p r o p e r t y ( M l l ) ,
the
embedding o f V(R) as a complete s u b l a t t i c e i n t o Z(A), and t h e e x i s t e n c e o f a f i m est p a r t i t i o n
i n L(R).
TI*
With regard t o the c h a r a c t e r i z a t i o n o f degenerate and l i n e a r r e l a t i o n s , we f i r s t o b t a i n the f o l l o w i n g r e s u l t concerning t h e v a l i d i t y o f (A4), which, c o n t r a r y t o s e t systems o r Boolean f u n c t i o n s , does n o t h o l d f o r a r b i t r a r y r e l a t i o n s .
Theorem 4.1.1:
L e t R be a k-ary r e l a t i o n on A, k
>,
2.
a)
A ( R ) i s symmetrically closed i f k 5 3.
b)
I f k = 2, then A(R) i s symmetrically closed i f R i s symmetric.
Proof: D3
Let B1,B2
a)
:= B2\B1
B E DIU D3.
u B2)
A ( R ) overlap.
are R-autonomous.
assume t h a t (a1,. (B1
E
. . ,ak) E
D2 := B1n B2 and
Then D1 := B1\B2,
I n order t o show t h a t B1 A
R and w.1 .o.g.
a l E D1
U
We must show t h a t (Bs02 ,...,a k ) E R.
o r i f B and al are i n t h e same Di
.
B2
=
DIU
D3
E
D3, a2 E A \ ( D 1 u D3).
A(R), Let
This i s t r i v i a l i f a 3 e A\
So assume
a1 E
D1 and B E. D3.
Then
Substitution decomposition for discrete structures
333
i f a 3 E D1, we show by successive exploitation of t h e autonomy of D l Y B1 and B2 t h a t the following sequence of k-tuples belongs t o R (only the f i r s t t h r e e components vary) : ("1 sa23~3sa49... , a k )
=i>
(a1,a2,al ,a4,..
. , a k ) =>
(a1,B,al , a q y . . . 'ak)
Because of Theorem 4.1.1 , l i n e a r r e l a t i o n s can obviously only occur i n the case k = 2. In f a c t , i t turns out t h a t l i n e a r r e l a t i o n s correspond e s s e n t i a l l y t o l i n e a r orders.
Theorem 4.1.2: A binary, r e f l e x i v e r e l a t i o n R on A with I A l >, 3 i s l i n e a r i f f i t i s a l i n e a r order. Moreover, i f \< i s a l i n e a r order on A such t h a t A($) = A ( R ) , then R i s equal t o 6 o r i t s inverse 5 . Proof: Let R be l i n e a r , i . e . there e x i s t s a l i n e a r order B on A with A ( < ) = A ( R ) . I n the following, a l l i n t e r v a l s [ c L , ~r]e f e r t o 4. By assumption, they a r e Rautonomous. Claim 1 : Given a E A, t h e r e i s y E A\Ia3 with (a,y) E R o r (y,a) E R. Since I A l 3 and R i s l i n e a r , t h e r e e x i s t s (a1,a2) ~l R with a l # a2. Let w.1.0.g. a l 6 a*. If a 4 a1 then [.,a1] E A ( R ) and thus ( a l , a 2 ) e R implies t h a t ( a , a 2 ) E R. The cases a1 B ~1 6 a2 and a2 6 follow s i m i l a r l y . Claim 2: Given a,B E A with a # 8 , then ( ~ 1 ~ E 8 ) R o r ( 8 , a ) E R. Let w.1.o.g. 01 4 8 . Because of Claim 1 , there e x i s t s Y E A such t h a t w.1.o.g. (a,Y)E R. If CL 4 Y and 8 4 Y, then [8,Y] E A ( R ) and a I [B,Y]. Hence (a,Y)E R y i e l d s t h a t ( a , ~E) R. The o t h e r cases follow s i m i l a r l y . Claim 3: Let
R i s transitive.
( a , 8 ) E R,
( 8 , Y ) E R and, w.l.o.g.,
from the f a c t t h a t similarly with
[Y,B]
[a,B] E
a 6
8.
A ( R ) and (8,Y) E R.
and (a,@).
If Y & If Y h
b,B], one [a,B],
obtains ( ~ , Y ) G R one concludes
R.H. Miihriiig and bl J. Rudermacher
334 C l a i m 4:
R i s asymmetric.
S i n c e A(R) = A(Rc) ( c f . 1.4), C l a i m 2 h o l d s a l s o f o r kc. Claims 2
-
4 show t h a t R i s a l i n e a r o r d e r .
This proves Claim 4.
The r e s t o f t h e a s s e r t i o n t h e n means
t h a t a l i n e a r o r d e r i s d e t e r m i n e d up t o d u a l i t y by i t s system o f i n t e r v a l s .
This
i s s t r a i g h t f o r w a r d l y shown..
Theorem 4.1.3:
i s empty ( i . e . ,
Proof:
A k - a r y r e l a t i o n (k
2) R on A w i t h I A l 3 3 i s degenerate i f f R R = a ) o r complete ( i . e . , R = A k ) up t o r e f l e x i v e t u p l e s ( a ,..., a). E R and t h a t ai # a . f o r some i,j e. { l , ...,k l . J A c o n t a i n a t l e a s t two d i f f e r e n t elements. We must show t h a t
Assume t h a t (a,,...,a k )
L e t { @l,...,Bk)
5;
(B1 ,..., B k ) r R i f R i s degenerate. =
@,
>/
T h i s i s t r i v i a l i f {al
s i n c e each { a . , B . 1 i s R-autonomous, 1
1
i = 1,.
.
,k.
,..., ak3n {bl ,...,'k'
A l l o t h e r cases t h e n
f o l l o w w i t h t h e same argument i f we show t h a t , f o r each p e r m u t a t i o n p o f
0 ,...,k ) , a l s o ( c x ~ ( ~ ) , . .
) e R . T h i s can be a c h i e v e d by making use o f a -aP(k) t h i r d element i f ] { a , , ...,a k l [ = 2 ( n o t e t h a t / A ( & 3 ) and t h e f a c t t h a t each two-
element subset o f A i s R-autonomous. The o p p o s i t e d i r e c t i o n i s obvious.. I t i s i n f o r m a t i v e t o e x t e n d t h i s c h a r a c t e r i z a t i o n o f degenerate r e l a t i o n s t o t h e
composition tree, too.
T h i s i s done by t h e d e s t i n c t i o n o f nodes o f t h e t y p e D
i n t o nodes of t y p e s Do and
D1, depending on whether t h e c o r r e s p o n d i n g q u o t i e n t
s t r u c t u r e i s (up t o r e f l e x i v e t u p l e s ) empty o r complete.
Furthermore, f o r nodes
of t h e t y p e P w i t h o n l y two successors, we a l s o use t h e t y p e s
Do, 0, o r L
i f the
a s s o c i a t e d q u o t i e n t has t h e c o r r e s p o n d i n g p r o p e r t y o f Theorem 4.1.2 o r 4.1.3. Then t h e i n v a r i a n c e s o f A(R) and V ( R ) d e s c r i b e d i n S e c t i o n 1.5 may be i n t e r p r e t e d as f o l l o w s
Theorem 4 . .4: r e l a t i o n o f R.
a ) L e t R be a r e l a t i o n on A and l e t RC denote t h e complementary Then B(Rc) i s o b t a i n e d f r o m B(R) by exchanging l a b e l s Do and D1.
L a b e l s I. and P remain unchanged. b)
L e t o be a p a r t i a l o r d e r on A and G ( o ) be i t s c o m p a r a b i l i t y graph.
B(G(O)) i s o b t a i n e d f r o m S ( 0 ) by r e p l a c i n g a l l l a b e l s L by D1. remain unchanged.
In p a r t i c u l a r ,
Then
L a b e l s Do and P
0 i s p r i m e i f f G ( 0 ) i s p r i m e ( c f . Theorem 1 . 5 . 1 ) .
There a r e s e v e r a l consequences o f t h e t r e e r e p r e s e n t a t i o n f o r p a r t i a l o r d e r s and t h e i r c o m p a r a b i l i t y graphs.
One o f them concerns t h e c h a r a c t e r i z a t i o n o f s e r i e s -
335
Substitution decomposition for discrete struttires
para1 l e l p a r t i a1 o r d e r s and t h e i r comparabi 1i t y graphs (known as complement reducible graphs PI]). They a r e exactly those p a r t i a l orders o r graphs whose composition t r e e has no node with label P. Based on t h i s representation one can a l s o derive the well-known characterization of these s t r u c t u r e s by means of forbidden substructures in the sense of [31], [44], 0601, c f . POq, p03]. Another application concerns the characterization of threshold graphs [29], which turn out t o correspond to those complement-reducible graphs G , f o r which each node in B ( G ) has a t most one non-singleton immediate successor; c f . [102], 11103). Furthermore, f o r comparability graphs G , the composition t r e e provides a means t o determine a l l p a r t i a l orders o such t h a t G = G(o), i . e . a l l t r a n s i t i v e o r i e n t a tions of G. Given such a p a r t i a l order 8 , a l l other such 0' a r e obtained from 0 by appropriately permuting the successors of a node of the type D, in B ( e ) and by possibly inverting some olB f o r which B i s a node of the type P i n S ( O ) , c f . a l s o D411.
~
A generalization of the composition t r e e t o a r b i t r a r y i n f i n i t e graphs and p a r t i a l orders i s given in p65l. A d i r e c t consequence of t h i s i n f i n i t e t r e e construction ( a l s o given i n [165]) i s t h a t p a r t i a l orders with the same comparability graph have t h e same dimension. This r e s u l t was shown e a r l i e r in [2] ( f o r f i n i t e and c e r t a i n i n f i n i t e p a r t i a l orders c f . a l s o [67], [lo71 , [158]) b u t t h e t r e e cons t r u c t i o n seems t o provide an e a s i e r proof.
With regard t o the computational complexity of decomposihg r e l a t i o n s , t h e d e f i n i t i o n of autonomy leads straightforwardly t o the following algorithm f o r determining the autonomous closure B* of a given s e t B y c f . P04].
Algorithm 4.1.5: Let R be a obtained a s follows: 1 . P u t C : = B. 2. If t h e r e a r e a ,,..., a k e i , j E El ,... , k l , and B E C by C {al ,...,c( 1 and k 3. Otherwise, B* = C.
k-ary r e l a t i o n on A and l e t B s A.
Then B* i s
A w i t h ( a l ,..., a k ) R, a 6 C, a. & C f o r some i J C w i t h ( a l ,..., a i - l , B , a i + l ,..., a k ) & R, then replace apply 2 . again.
Obviously, t h i s algorithm has complexity O ( n 2 m 2 ) -s O ( n 2 k + 2 ) , where n = I A l and m = I R I . This, together with Theorem 3.6.2, shows t h a t the composition t r e e B ( R ) of an a r b i t r a r y r e l a t i o n R on A can be constructed i n polynomial ( i n I A [ ) bounded time. Of course, more e f f i c i e n t algorithms may be possible, as i s the case f o r graphs or p a r t i a l orders, c f . the bibliographical notes below.
R.H. Mohring and FJ. Rademcher
336 Example 4.1.6:
L e t R be t h e b i n a r y r e l a t i o n on A = I1,
d i r e c t e d graph o f F i g u r e 4.1. i n F i g u r e 4.1. position o f T*
Note t h a t a* = K1,2,31,{4),{5,611
R l { l , ...,6 1
={{7XI8,9,101,111}~
Theorem 3.4.4.
..., 111
represented by the
The associated composition t r e e B(R) i s a l s o given
i s t h e maximal d i s j o i n t decom-
i n the sense o f Theorem 3.4.2.
Similarly,
i s t h e f i n e s t p a r t i t i o n o f L(Rl{7,
...,111)
i n t h e sense o f
Furthermore, t h e composition t r e e o f t h e complementary r e l a t i o n
R C o f R i s obtained from g(R) by exchanging a l l l a b e l s Do by D1 and vice-versa ( c f . Theorem 4.1 . 4 ) .
F i g u r e 4.1:
A r e l a t i o n and i t s composition t r e e
We conclude t h i s summary chapter on t h e decomposition o f r e l a t i o n s w i t h some a d d i t i o n a l h i n t s on the l i t e r a t u r e and r e l a t e d work. The decomposition o f r e l a t i o n s by means o f composition s e r i e s and t h e c h a r a c t e r i z a t i o n o f t h e i r congruence p a r t i t i o n l a t t i c e s was f i r s t i n v e s t i g a t e d i n D22] and
PI
*
The c h a r a c t e r i z a t i o n o f degenerate r e l a t i o n s occurs i n r e l a t i o n s i n [102].
p8],
that o f linear
For b i n a r y r e l a t i o n s , they f o l l o w i n t h e f i n i t e case a l s o
from r e s u l t s on the s p l i t decomposition f o r b i n a r y r e l a t i o n s i n [35J.
331
Substitution decomposition for discrete structures
The c o m p o s i t i o n t r e e occurs f o r a r b i t r a r y graphs f i r s t i n p2], O(n4)-algorithm f o r i t s construction.
t o g e t h e r w i t h an
An O ( n 3 ) - a l g o r i t h m based on methods f o r
c o m p a r a b i l i t y graph r e c o g n i t i o n was g i v e n i n [68J.
Independently, s i m i l a r r e s u l t s
f o r graphs and p a r t i a l o r d e r s t o g e t h e r w i t h an O(n3) a l g o r i t h m were o b t a i n e d i n [24]. F o r CPM-networks, e a r l i e r a l g o r i t h m s t o f i n d "most" o f t h e decomposition p o s s i b i l i t i e s a r e g i v e n i n p33], D 3 q , [ 1 4 g . I t i s shown i n [87] t h a t t h e methods o f u33],
L134-J a l s o l e a d t o an O(n3) a l g o r i t h m by t h e use o f dominance
trees. S p e c i a l r e s u l t s on t h e decomposition o f s e r i e s p a r a l l e l p a r t i a l o r d e r s and complement r e d u c i b l e graphs a r e found i n [23],
Pl], p2],
[31],
Q60].
They c o n t a i n ,
among o t h e r t h i n g s , a b i n a r y t r e e r e p r e s e n t a t i o n which can be f o u n d i n l i n e a r t i m e . For ( b i c o n n e c t e d ) b i n a r y r e l a t i o n s t h e r e a r e O ( n 4 ) - a l g o r i t h m s developed f o r t h e s p l i t decomposition, which reduce t o O ( n 3 ) - a l g o r i t h m s f o r t h e s u b s t i t u t i o n decomposition f o r a r b i t r a r y b i n a r y r e l a t i o n s , c f . L 3 q . F i n a l l y , a r e c e n t a p p l i c a t i o n of t h e s p l i t decomposition t o d i s t r i b u t i v e l a t t i c e s i s g i v e n i n [54].
IV
2.
SET SYSTEMS
Set systems f u l f i l c o n d i t i o n s ( M l )
-
(M5) w i t h t h e s u r j e c t i v e mappings d e r i v e d
f r o m t h e s u b s t i t u t i o n o p e r a t i o n i n 1.3 as t h e s u r j e c t i v e homomorphisms. I t i s easy t o v e r i f y t h a t (M6), (M7) and (M8)*,
(M9), (M11) and (M12) h o l d , b u t
n o t (M7)* and (M10) (even n o t f o r c l u t t e r s , c f . Example 4.2.2
below).
So f o r s e t systems, some c o n d i t i o n s h o l d i n g f o r r e l a t i o n s a r e l o s t , i n g e n e r a l .
As a consequence, t h e r e i s no embedding o f V ( T ) i n t o Z ( A ) as a complete s u b l a t t i c e ( i t i s o n l y a sub-semilattice w . r . t . existence o f a f i n e s t p a r t i t i o n
1~*
t h e meet
A,
c f . Theorem 3.2.2),
and t h e
i n L ( T ) cannot be guaranteed, i n g e n e r a l .
B u t we s t i l l have, o f course, t h e i n v a r i a n c e o f t h e l a s t f a c t o r and t h e unique reconstruction property. I n t h i s c o n t e x t , one would l i k e t o know whether t h e r e e x i s t subclasses o f s e t
systems which f u l f i l (as r e l a t i o n s do) t h e s t r o n g c o n d i t i o n s (M7)* and (M10).
It
i s easy t o see t h a t s e t systems having a c o m p o s i t i o n s e r i e s , s e t systems w i t h f i n i t e members o n l y ( c f . p6])
and conformal c l u t t e r s c o n s t i t u t e such c l a s s e s .
Also, though n o t t h a t easy t o o b t a i n , maximal and minimal c l u t t e r s f u l f i l these properties.
With r e g a r d t o t h e c o m p o s i t i o n of such s e t systems, one o b t a i n s t h e
f o l 1owing theorem.
R. H. Mohriiig aiid ?I J. Rndermacher
3 38
The c l a s s o f s e t systems f u l f i l l i n g (M7)* and (M10) i s a sub-
Theorem 4.2.1:
c a t e g o r y o f t h e c a t e g o r y o f a l l s e t systems which i s c l o s e d w . r . t
composition.
The p r i n c i p l e s used t o p r o v e t h i s theorem a r e s i m i l a r t o t h o s e used f o r Theorem 1.3.5,
b u t even more i n v o l v e d , l e a d i n g t o a p r o o f o f c o n s i d e r a b l e l e n g t h , so t h a t
we w i l l o m i t t r e a t m e n t here. Next, we g i v e two examples, which show t h a t (M7)* and (M10) may n o t h o l d , i n genera 1 .
Example 4.2.2:
a ) L e t (An),,&
be a f a m i l y o f p a i r w i s e d i s j o i n t c o p i e s o f IN,
say An = Iamn\m EN}, and l e t A := m(N
T = [a
m(n) ,n
In
E
IN1 o f (An)m
Then A ( T ) c o n s i s t s o f a l l An, A(T) f u l f i l s
tions
n1 :=
L e t A1 = i a n l n
L e t T b e t h e system o f t r a n s v e r s a l s
i n which m(n) i s bounded. n rOU, and a l l u n i o n s and subsets t h e r e o f .
Thus
However, V(T) i s n o t a l a t t i c e , s i n c e t h e congruence p a r t i -
(M10).
{Ia2k-l,n,a2k,n11k
n G N } have no j o i n i n V ( T ) . b)
An.
E.
OU)
E (T,
IN, n E ( N I and v
T~
and A2 = I b n l n
= IAnln
n 2 :=
I{al,n),Ia2k,n
,a2k+l,nllkrCN,
eOU1 6 V ( T ) ) .
be d i s j o i n t c o p i e s o f
A
u
hold.
N e v e r t h e l e s s V(T) i s a ( i n c o m p l e t e ) s u b l a t t i c e o f Z(A).
LN
and l e t
L e t T be t h e system o f t r a n s v e r s a l s o f ({an,bnl)na which c o n t a i n := A1 A*. o n l y f i n i t e l y many a, o r b., I t i s easy t o v e r i f y t h a t n e i t h e r (M7)* n o r (M10)
Other examples d e m o n s t r a t i n g s t i l l o t h e r e f f e c t s i n t h e i n f i n i t e case can be found i n [16]
and [102].
Since s e t systems a r e s y m m e t r i c a l l y c l o s e d (Theorem 1.3.4), s e t systems.
Degenerate s e t systems a r e c h a r a c t e r i z e d i n Theorem 4.2.3.
end, n o t e t h a t an T1 S T2
u,n
E
t h e r e a r e no l i n e a r
ideal
I (T, z T2
E
To t h i s
i s a s e t system 1 c_ P(A) such t h a t I ) i m p l i e s T1 e I and T1 ,T2 f I i m p l i e s T1 u T2 E I ( a dual i d e a l ) o f P(A)
T~ E 1 ) .
Theorem 4.2.3:
A s e t system T on A ( w h i c h c o v e r s A) i s degenerate i f f one o f t h e
following conditions applies: 1.
T = {IaiJnE A } .
2.
Tu 10) i s an i d e a l o f P ( A ) .
3.
There e x i s t an i d e a l I and a p r o p e r dual i d e a l F o f P(A) such t h a t T =
-In F.
Substitution decomposition for discrete structures
339
( 1 = P(A) i s p o s s i b l e )
Furthermore, 3. h o l d s i f f T l n T2 # $ f o r a l l T1,T2 E T.
Proof:
L e t T be degenerate and d i f f e r e n t f r o m ICaIla e A } .
We t h e n p r o v e t h a t T
has t h e f o l l o w i n g p r o p e r t i e s : T , T i n T2
# 4
=’
a)
T1,Tp
b)
T1 ,Tp e T => T i
c)
T1,T2 E T , T1 s T s T2 3 T E T ( i . e . T i s G-convex)
d)
T1 ,T2,T3 E T, T1 n T2 = $, Q # T
U T2
T i n Tp
T
E T
c T3
P r o p e r t y a ) f o l l o w s f r o m Ex(T2,T2,T1) To show b ) , assume f i r s t t h a t ITII empty s e t s D1,D2. D1
u D2
E
A ( T ) and
D2uD3
A(T).
E
U (T2n
Now l e t T1,T2 be s i n g l e t o n s .
= T1
1.
T
E
n T2 E
T. T, s i n c e T2
Since T i s degenerate,
T2.
Since T # I I a I l a r A},
t h e r e i s To
I f To # T1
u T2,
I T o [ > 1. a p p l i e s t o t h e d i s j o i n t T-autonomous s e t s T1,T2 and T o \ (T,U E
T.
Thus T1 can be e n l a r g e d on T2 t o TI
c ) f o l l o w s f r o m Ex(T1,T,T2)
A ( T ) = P(A).
Lemma 1.3.2 t h e n i m p l i e s t h a t T1 can be
D3) = TIU
Because o f t h e above, T o U T2 E T.
To U T2
E
Then T1 decomposes i n t o two d i s j o i n t non-
D3 := T2\T1 # $.
L e t w.1.o.g.
e n l a r g e d on D3 t o T1
>
*
E
T with
Lemma 1.3.2
T2), s i n c e
U T2
= T.
To p r o v e d ) , we f i r s t show t h e weaker p r o p e r t y ( * ) s t a t i n g t h a t i f T1, T2
E
t h a t T 1 n T2 = 9, t h e n each non-empty subset T o f T1 o r T2 belongs t o 7.
For
T -C T1,
t h i s f o l l o w s f r o m E x ( T 2 , T U T2,T1)
= T.
T such
Now l e t T3 be a r b i t r a r y and
# T &T3. IfT c o n t a i n s a s u b s e t o f T1 o r T2, T E T because o f c ) and ( * ) . So l e t T (7 (Tl U T2) = 9 . Because o f (*), we may assume t h a t T l n T3 # $. Then a ) y i e l d s t h a t T1 n T3 0 T. Hence (Tin T3) U T E T because o f c ) . B u t then
$
T E T because of ( * ) and t h e f a c t t h a t [(T1 Properties a )
-
n T3) U T] n T2
= $.
d) o b v i o u s l y y i e l d t h a t one o f t h e cases 1. -3. must a p p l y .
To
show t h e o p p o s i t e d i r e c t i o n , s i m p l y observe t h a t f o r T1,T2 E T and B -C A, T1 n T2 c Ex(T1,B,T2)
C o r o l l a r y 4.2.4:
a)
s T1 (J T2, which proves Ex(T1 ,B,T2)
2. and 3.
I f A i s f i n i t e , then 2. and 3. reduce t o t h e cases T =
P ( A ) \ { $ } and T = I T E P ( A ) I B 0 c T I f o r some $ # B,
b)
E T i n cases
I f T is a c l u t t e r , t h e n
C_
A, r e s p e c t i v e l y .
2. and 3. reduce t o t h e case
T = {A}.
R.H. Mohringand F.J. Radermacher
340
S i m i l a r l y t o r e l a t i o n s , f o r s e t systems, too, i t i s i n f o r m a t i v e t o i n t r o d u c e i n t o t h e composition t r e e a d i s t i n c t i o n o f nodes o f t h e type D i n t o those o f type Do,
D1 and D2, depending on whether the corresponding q u o t i e n t f u l f i l s l.,2.,
o r 3.
( F o r c l u t t e r s , o f course, we o n l y have types Do and DI).
i n Theorem 4.2.3.
c l a s s i f i c a t i o n i s a l s o extended t o nodes of the type
This
P w i t h o n l y two successors.
The i n v a r i a n c e r e s u l t s o f 1.3 l e a d t o t h e f o l l o w i n g i n v a r i a n c e s f o r t h e composit i o n tree o f clutters.
Theorem 4.2.5:
L e t T be a c l u t t e r on A and l e t b[T]
tence we assume) and LLIT]
Then B ( b [ T ] ) i s obtained from
t h e a n t i b l o c k e r o f T.
B ( T ) by exchanging l a b l e s Do and D,.
be t h e b l o c k e r (whose e x i s -
I f T i s conformal, then B ( a [ T ] ) i s a l s o
obtained t h i s way, and B ( T ) = B(G(T)).
( I n t h a t case, even B ( a [ T ] ) = B ( b [ T ] ) and
thus A ( a [ T ] ) = A ( b [ T ] ) , although u[T] # b[T]
i n general.)
With regard t o t h e computational complexity o f decomposing s e t systems, t h e s i t u a t i o n i s d i f f e r e n t from t h a t f o r r e l a t i o n s , s i n c e t h e d e f i n i t i o n o f autonomy does n o t provide e v i d e n t c r i t e r i a f o r how t o c o n s t r u c t t h e autonomous c l o s u r e . For c l u t t e r s , however, i t i s p o s s i b l e t o reduce t h e d e t e r m i n a t i o n o f the autonomous c l o s u r e e s s e n t i a l l y t o the c o n s t r u c t i o n o f separators i n c e r t a i n subsystems.
To t h i s end, observe f i r s t t h a t f o r unconnected c l u t t e r s C ( i . e . c l u t t e r s o f t h e type Do) the associated f i n e s t p a r t i t i o n
TI*
o f L(C) i s j u s t the p a r t i t i o n o f the
graph G ( C ) ( c f . Section 1.5) i n t o i t s connected components, which can be found i n polynomial time by standard graph methods. So we w i l l r e s t r i c t ourselves i n t h e f o l l o w i n g t o connected c l u t t e r s .
In accordance w i t h the p r o p e r t i e s o f separators o f matroids n 5 9 ] , n632 , we d e f i n e a separator o f a c l u t t e r C on A t o be a s e t B E A such t h a t Ex(T ,B,T2) cz C f o r a l l T1 ,Tp E C .
i s o f t h e type D1.
Obviously, a c l u t t e r C has n o n - t r i v i a l separators i f f i t
The corresponding f i n e s t p a r t i t i o n
TI*
i n t h e sense o f
Theorem 3.4.4 then t u r n s o u t t o be t h e p a r t i t i o n o f C i n t o i t s minimal non-empty separators.
I t f o l l o w s from t h e p r o p e r t i e s o f
T*
t h a t f o r each s e t B G A t h e r e
e x i s t s a smallest B c o n t a i n i n g separator, t h e separator c l o s u r e BA o f B. course, B* =
Of
@IT*.
An e q u i v a l e n t p r o p e r t y t o being a separator o f a c l u t t e r C on A i s t h a t a l l c&-
cuits o f
C ( i . e . t h e c - m i n i m a l dependent sets o f t h e independence system associatej
w i t h C ) a r e contained i n B o r A\B.
So t h e knowledge o f t h e c i r c u i t s o f C (or,
the knowledge o f whether two elements o f A a r e contained i n a common c i r c u i t )
Substitution decomposition .for discrete structures
p r o v i d e s a way t o f i n d
34 1
and t h e s e p a r a t o r c l o s u r e B A .
i~*
L e t C be a connected c l u t t e r on A and Bc A.
A l g o r i t h m 4.2.6:
Then B* i s
o b t a i n e d as f o l l o w s :
c
: = B.
1.
Put
2.
Determine t h e s e p a r a t o r c l o s u r e CA o f C i n t h e c l u t t e r C,
on AC : =
u
T.
:= {T E C l T n C
# 6)
I f CA f C, r e p l a c e C b y CA and a p p l y 2.Lagain.
TECc
3.
If CA = C, t h e n B* = C .
Proof:
cf.
[lOq, Dog.
Observe t h a t C E A ( C ) i f f C i s a s e p a r a t o r i n Cc.
So f o r connected c l u t t e r s , d e t e r m i n a t i o n o f t h e autonomous c l o s u r e i s shown t o be T u r i n g - r e d u c i b l e t o d e t e r m i n i n g t h e s e p a r a t o r c l o s u r e o r , e q u i v a l e n t l y , t o determi n i n g whether two elements a r e c o n t a i n e d i n a common c i r c u i t . T h i s l e a d s t o polynomial ( i n [ A \ ) t i m e decomposition a l g o r i t h m s f o r those c l a s s e s o f c l u t t e r s i n which t h e r e a r e p o l y n o m i a l methods t o d e c i d e whether two elements a r e c o n t a i n e d i n a common c i r c u i t . One such c l a s s i s g i v e n by t h e c l a s s o f conformal c l u t t e r s C, which may be c h a r a c t e r i z e d by t h e p r o p e r t y t h a t a l l c i r c u i t s o f C c o n s i s t o f e x a c t l y two elements.
( I n t h i s case t h e above a l g o r i t h m i s e s s e n t i a l l y e q u i v a l e n t t o A l g o r i t h
Another such c l a s s i s , f o r each k 4.1.5 a p p l i e d t o G(C).) c l u t t e r s C w i t h IT1 6 k f o r a l l T E C.
E (N,
the class o f
Furthermore, i t can be shown t h a t two elements a,@ E A a r e c o n t a i n e d i n a c o m o n c i r c u i t o f C i f f t h e r e e x i s t T1, T2 (T1 f l T2)
u Ial,ci21
E
C with al E T1\T2,
i s c o n t a i n e d i n some T E C.
a2
c T 2 \ T 1 such t h a t
Since t h i s p r o p e r t y can be
checked i n O(n-m3) time, where m := I C I , t h e above methods l e a d t o a l g o r i t h m s o f c o m p l e x i t y O(n4m3) f o r d e t e r m i n i n g t h e autonomous c l o s u r e f o r a r b i t r a r y c l u t t e r s . T h i s upper c o m p l e x i t y bound seems t o be r e l a t i v e l y t i g h t i n m, s i n c e one can show t h a t each o r a c l e a l g o r i t h m [71],
[89]
f o r c l u t t e r decomposition a l r e a d y r e q u i r e s
O(m) ( t h u s i n g e n e r a l e x p o n e n t i a l l y ( i n I A I ) many) c a l l s on t h e o r a c l e i n o r d e r t o decide whether a g i v e n c l u t t e r C on A i s decomposable o r n o t , even i f t h e o r a c l e can d i s t i n g u i s h whether a g i v e n s e t B i s a c l u t t e r s e t , a s u b s e t o f a c l u t t e r s e t , a s u p e r s e t o f a c l u t t e r s e t , a c i r c u i t , o r a dependent s e t which i s
342
R. H. Molirbig and F.J. Radermaclier
n o t a superset o f a c l u t t e r s e t and n o t a c i r c u i t , c f . CIOZ],
Example 4.2.7:
L e t C denote t h e c l u t t e r on A = { l ,
m a t r i x o f F i g u r e 4.2.
4.2.
DO4-J
..., 101 g i v e n
by t h e i n c i d e n c e
The a s s o c i a t e d c o m p o s i t i o n t r e e i s a l s o g i v e n i n F i g u r e
C i s n o t conformal, and t h e system A ( G ( C ) ) i n t h i s case c o n t a i n s more
elements t h a t A ( C ) .
The c o m p o s i t i o n t r e e o f b[C]
can be o b t a i n e d f r o m B(C) by
exchanging a l l l a b e l s Do by D l and v i c e - v e r s a .
C
1 1
2
3
4
5
6
7
8
9
10
1
1
0
1
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
1
1
0
1
1
0
0
0
0
0
1
0
1
1
1
0
0
0
0
0
1
1
0
1
0
1
0
0
0
0
1
0
1
1
0
1
0
0
0
0
1
0
0
0
1
1
F i g u r e 4.2:
A c l u t t e r and i t s c o m p o s i t i o n t r e e
We conclude t h i s s u b s e c t i o n w i t h some h i n t s t o t h e l i t e r a t u r e and r e l a t e d work. Decomposition o f c l u t t e r s by means o f c o m p o s i t i o n s e r i e s and t h e c h a r a c t e r i z a t i o n o f t h e i r congruence p a r t i t i o n l a t t i c e s was f i r s t i n v e s t i g a t e d i n DO6-J.
343
Substitution decomposition for discrete structures The c h a r a c t e r i z a t i o n o f degenerate c l u t t e r s appears a l s o i n systems, see [34]
,
[38]
DOa,
f o r f i n i t e set
.
The c o m p o s i t i o n t r e e was f i r s t a p p l i e d by Shapley t o c l u t t e r s i n t h e c o n t e x t o f t h e decomposition o f s i m p l e n-person games, c f .
P38-J.
For a g e n e r a l i z a t i o n t o
c o o p e r a t i v e games c f . c941.
[lo]
Billera
has developed a d i f f e r e n t decomposition a l g o r i t h m f o r c l u t t e r s which
i s based on p r o p e r t i e s o f t h e b l o c k e r .
H i s a l g o r i t h m i s p o l y n o m i a l l y bounded i f f
t h e b l o c k e r o f t h e g i v e n c l u t t e r and c e r t a i n s u b c l u t t e r s can be c o n s t r u c t e d i n polynomial t i m e .
But since, even i n t h e case IT1 6 2 f o r a l l T c C, f i n d i n g a
minimal b l o c k i n g s e t i s an NP-complete problem [83], t h i s a l g o r i t h m w i l l p r o b a b l y be e x p o n e n t i a l i n general. There a r e ( a t l e a s t ) two a p p l i c a t i o n s o f t h e s u b s t i t u t i o n decomposition t o matroids.
One i s o b t a i n e d by c o n s i d e r i n g t h e system o f t h e independent s e t s o f
t h e m a t r o i d , w h i l e t h e o t h e r , more i m p o r t a n t one i s based on t h e p a t h c l u t t e r s e t system d e r i v e d f r o m t h e system o f c i r c u i t s o f a (non-separable) m a t r o i d , c f .
pg.
I n t h i s i n t e r p r e t a t i o n , t h e m a t r o i d s u b s t i t u t i o n decomposition t u r n s o u t t o be e q u i v a l e n t t o t h e s p l i t decomposition o f t h e system o f c i r c u i t s o f t h e m a t r o i d , c f . n i ’ ] , [MI.
T h i s second a p p l i c a t i o n t o m a t r o i d s has many connections w i t h
m a t r o i d c o n n e c t i v i t y (e.g., t e d m a t r o i d s 1171).
t h e indecomposable m a t r o i d s a r e e x a c t l y t h e 3-connec-
I n p a r t i c u l a r , t h e s p l i t v e r s i o n g e n e r a l i z e s t h e decompositim
o f u n d i r e c t e d graphs i n t o 2-connected components, c f . Pg],
f u r t h e r d e t a i l s and r e f e r e n c e s .
p57], p61] f o r
Due t o t h e connections w i t h m a t r o i d c o n n e c t i v i t y ,
t h e r e a l s o e x i s t p o l y n c m i a l t i m e decomposition a l g o r i t h m s f o r t h i s case, c f . c341
El,
-
IV. 3.
BOOLEAN FUNCTIONS
Boolean f u n c t i o n s f u l f i l c o n d i t i o n s ( M l )
-
(M4) w i t h t h e s u r j e c t i v e homomorphisms
o b t a i n e d f r o m t h e s u b s t i t u t i o n decomposition and w i t h t h e s t i p u l a t e d i d e n t i f i c a t i o n o f Boolean f u n c t i o n s up t o complementation o f v a r i a b l e s which i s necessary f o r (M4). Then t h e congruence p a r t i t i o n s o f a Boolean f u n c t i o n F a r e e x a c t l y ( s i n c e we o n l y deal w i t h t h e f i n i t e case) t h e p a r t i t i o n s i n t o F-autonomous s e t s . The s u b s t r u c t u r e s ( i n t h e a l g e b r a i c sense) o f a Boolean f u n c t i o n F a r e t h e subf u n c t i o n s induced by F-autonomous s e t s 8, which a r e o b t a i n e d by l e t t i n g a l l v a r i a b l e s x . , j pI B t a k e a p p r o p r i a t e c o n s t a n t values. I n t h i s sense s u b s t r u c t u r e s a r e J ( c f . 1.2) o n l y determined up t o complementation, so t h a t (M5) seems o n l y t o be
R.H. Mohring and F.J. Radermacher
344
s a t i s f i e d i n a weaker v e r s i o n than required.
I n f a c t , t h e use o f (M5) i s r e s t r i c -
t e d t o cases i n which a q u o t i e n t F ' o f F i s given and where B i s r e l a t e d t o t h e classes o f the corresponding congruence p a r t i t i o n ( c f . i n p a r t i c u l a r t h e d e f i n i t i o n o f the factors).
I n t h a t case, t h e substructures a r e u n i q u e l y determined
( c f . 1.2), so t h a t t h e r e s u l t s o f t h e general theory ( i n p a r t i c u l a r t h e uniqueness o f f a c t o r s i n composition s e r i e s ) a r e preserved.
(M6) - (M9) have already been shown i n Section 1.2.
(M11) f o l l o w s from [40,
Theorem 4.43, w h i l e (M12) i s obvious from what has been s a i d above.
So, a l t o g e t h e r , Boolean f u n c t i o n s f u l f i l a l l c o n d i t i o n s t h a t a r e necessary t o guarantee t h e v a l i d i t y o f a l l r e s u l t s o f t h e model f o r t h e f i n i t e case. O f course, a g e n e r a l i z a t i o n t o t h e i n f i n i t e case may be p o s s i b l e .
I n t h a t case phenomena
s i m i l a r t o those f o r s e t systems w i l l occur, since a monotonic Boolean f u n c t i o n F ( b u t n o t a r b i t r a r y Boolean f u n c t i o n s , as may be concluded from t h e c h a r a c t e r i z a t i o n o f t h e r e s p e c t i v e degenerate s t r u c t u r e s ) can, w . r . t .
decomposition, equiva-
l e n t l y be described by t h e system T(F) = I B I F ( x ( B ) ) = 1, where x . ( B ) = 1 i f f j eB} o f i t s t r u t h sets.
J
( c f . a l s o 1.5).
Since Boolean f u n c t i o n s a r e s y m e t r i c a l l y closed ( c f . 1.2), t h e r e a r e no " l i n e a r " Boolean f u n c t i o n s .
Theorem 4.3.1:
Degenerate Boolean f u n c t i o n s have t h e f o l l o w i n g c h a r a c t e r i z a t i o n
A Boolean f u n c t i o n F ( xl,...yxn)
degenerate i f f F(xl
,...,xn)
= xi1*
without inessential variables i s
... * xnEn, where
E'
x
j stands f o r Boolean a d d i t i o n (+), Boolean m u l t i p l i c a t i o n
(
0
)
x.
and * J o r r i n g sum a d d i t i o n
denotes x j o r
(0,. Proof:
E i t h e r by i n d u c t i o n o r by r e d u c t i o n t o [40,
case, note t h a t the s e t s {1,2},{2,3},
...,{ n - l , n i
Theorem 4 . q .
I n the l a t t e r
a r e F-autonomous..
As f o r r e l a t i o n s and s e t systems, we i n t r o d u c e t h e d i s t i n c t i o n o f Boolean functions of the type
D i n t o those o f the type Do, D1 and D2, depending on whether the
associated q u o t i e n t i s a Boolean sum, a Boolean product, o r a r i n g sum, respectively. The i n v a r i a n c e r e s u l t s o f 1.2 then l e a d t o t h e f o l l o w i n g i n v a r i a n c e s f o r t h e comp o s i t i o n t r e e o f Boolean f u n c t i o n s .
Theorem 4.3.2:
L e t F be a Boolean f u n c t i o n .
345
Substitution decomposition for discrete structures
a)
I f F* denotes t h e dual f u n c t i o n o f F, t h e n B(F*) i s o b t a i n e d f r o m B ( F ) by exchanging l a b e l s Do and D1.
b)
I f F i s monotonic and CF i s t h e a s s o c i a t e c l u t t e r o f p r i m e i m p l i c a n t s o f F, t h e n B(F) = B(CF) and B ( F * ) = B ( b p J ) .
A l s o f o r t h e decomposition o f Boolean f u n c t i o n s , d i f f e r e n t a l g o r i t h m s have been developed due t o t h e s i g n i f i c a n c e o f decomposition f o r s w i t c h i n g design, c f . f o r i n s t a n c e [41],
[43],
p39],
p40],
[155].
These a l g o r i t h m s a r e e i t h e r based on
t h e e v a l u a t i o n o f Ashenhurst's decomposition c h a r t s o r use a d i f f e r e n t i a l c a l c u l u s f o r d e t e r m i n i n g F-autonomous s e t s .
I n a l l cases t h e y have an e x p o n e n t i a l ( i n t h e
number o f v a r i a b l e s ) worst-case c o m p l e x i t y , a l t h o u g h i n some cases a good average performance was observed i n e m p i r i c a l s t u d i e s , c f . p 4 0 ] . F o r monotonic Boolean f u n c t i o n s t h e r e s u l t s on c l u t t e r decomposition i n I V . 2 l e a d t o more e f f i c i e n t a l g o r i t h m s .
On t h e o t h e r hand, t h e n e g a t i v e r e s u l t on t h e
c o m p l e x i t y o f o r a c l e decomposition a l g o r i t h m s mentioned i n IV.2,
shows t h a t
a l r e a d y f o r monotonic Boolean f u n c t i o n s , u n i v e r s a l polynomial t i m e decomposition algorithms are very u n l i k e l y t o e x i s t .
L e t F(xl,
Example 4.3.3:
.
F ( x ~ , . . ,xl0)
...,xl0)
be g i v e n by i t s normal d i s j u n c t i v e form:
= x i j i 2 j i . 3 ~ 4+ X1x2X3x4 t X1X2x3x4 t x1x2x3x4
+
x6x728xg
+
The c o m p o s i t i o n t r e e o f F i s g i v e n i n F i g u r e 4.3. the following,
e q u i v a l e n t r e p r e s e n t a t i o n F(xl,
(x6x7@x8)(x9
+
+
Xgxlo
+ Xgx10. Based on t h i s t r e e , one o b t a i n s
...,xlo)
( c f . Theorem 4.3.1),
= (xl
@ x2
8 x3)x4 +
where t h e node B = {6,
corresponds t o t h e prime Boolean t h r e s h o l d f u n c t i o n G(yl ,y2,y3) y2y3.
x5
X6x8x9 t x7x8x9
t x6X7ji8x10 t X6X8x10 t x7X8x10
Xlo)
+
= y1y2
x5
+
...,101
+ y1y3 +
R. H. Moltritig atid F.J. Radertnacher
346
D..
F i g u r e 4.3:
The composition t r e e o f F from Example 4.3.3
ACKNOWLEDGEMENT We would l i k e t o thank t h e l a t e R. Kaerkes,and a l s o W.
Oberschelp
and M.M.
R i c h t e r f o r constant support d u r i n g t h e many years o f research t h a t have gone i n t o t h i s paper, as w e l l as W.H.
Cunningham f o r i n t e n s i v e d i s c u s s i o n s and many
suggestions, among o t h e r t h i n g s concerning t h e connections w i t h t h e s p l i t decomposition. We would a l s o l i k e t o thank L.J.
B i l l e r a and R.E.
Bixby f o r some discussions and
m a t e r i a l concerning c l u t t e r decomposition and connections between s u b s t i t u t i o n decomposition and c o m b i n a t o r i a l o p t i m i z a t i o n , and A.J. t h e (max,+)-algebra,
Hoffman f o r h i s h i n t s on
which i n f a c t m o t i v a t e d t h e research f i n a l l y l e a d i n g t o t h e
general v e r s i o n o f Theorem 2.3.1. P a r t i c u l a r thanks a r e a l s o due t o t h e r e f e r e e s f o r t h e many c o n s t r u c t i v e h i n t s and t h e i r enormous work i n improving and s t r e a m l i n i n g t h e p r e s e n t a t i o n o f t h i s paper. L a s t b u t n o t l e a s t , we would l i k e t o thank t h e o r g a n i z e r s o f t h e Bad-Honnef conference on " A l g e b r a i c S t r u c t u r e s i n Operations Research" f o r s t i m u l a t i n g t h e
347
Substitution decomposition for discrete structures
p r e s e n t a t i o n o f t h i s survey on p r e s e n t l y a v a i l a b l e i n s i g h t s i n t o t h e s u b s t i t u t i o n decomposition and P.L. Hammer f o r h i s personal encouragement.
REFERENCES A i g n e r , M. and P r i n s , G., U n i q u e l y p a r t i a l l y o r d e r a b l e graphs, J . London Math. SOC. 2 (1971) 260-266. A r d i t t i , J.C. and Jung, H.A., The dimension o f f i n i t e and i n f i n i t e comparab i l i t y graphs, J . London Math. SOC. 21 (1980) 31-38. Ashbacher, M., A homomorphism theorem f o r f i n i t e graphs, Proc. Amer. Math, SOC. 54 (1976) 468-470. Ashenhurst, R.L., The decomposition o f s w i t c h i n g f u n c t i o n s , i n : Proceedings o f t h e i n t e r n a t i o n a l symposium on t h e t h e o r y o f s w i t c h i n g ( P a r t I), (Harvard U n i v e r s i t y Press, Cambridge, 1959). Baker, K.A., Fishburn, P.C. and Roberts, F.S., 2, Networks 2 (1971) 11-28.
P a r t i a l o r d e r s o f dimension
Balas, E..and Zemel, E., Graph s u b s t i t u t i o n and s e t packing p o l y t o p e s , Networks 7 (1977) 267-284. Barlow, R.E.,and Proschan, F., S t a t i s t i c a l Theory o f R e l i a b i l i t y and L i f e T e s t i n g ( P r o b a b i l i t y Models), ( H o l t , R i n e h a r t and Winston, New York, 1975). Berge, C.,
Graphs and Hypergraphs ( N o r t h Holland, Amsterdam,
1973).
B i l l e r a , L.J., C l u t t e r decomposition and monotonic Boolean f u n c t i o n s , Annals o f t h e New York Academy o f Science 175 (1970) 41-48. B i l l e r a , L.J., On t h e c o m p o s i t i o n and decomposition o f c l u t t e r s , J . Comb. Th. B 11 (1971) 234-245. B i l l e r a , L.J. and Bixby, R.E., Decomposition t h e o r y f o r a c l a s s o f combinat o r i a l o p t i m i z a t i o n problems, i n : O p t i m i z a t i o n Methods f o r Resource A l l o c a t i o n , Proc. Nato Conf. E l s i n o r e (1971) ( E n g l i s h U n i v e r s i t y Press, London, 1974) 427-434. B i l l s t e i n , N . and Radermacher, F.J., Res. 27 (1977) 274-294. B i r k h o f f , G., 1973).
Time-cost o p t i m i z a t i o n , Methods o f Oper.
L a t t i c e Theory, Amer. Math. SOC. C o l l . Pub1 .,(Providence,
Birnbaum, Z.W. and Esary, J.D., Modules o f c o h e r e n t b i n a r y systems, S I A M J. A p p l i e d Math. 13 (1965) 444-462. Bixby, R.E., On t h e l e n g t h - w i d t h i n e q u a l i t y f o r compound c l u t t e r s , J. Comb Th. 11 (1971) 246-248. Bixhy, R.E., Cnrnposition and decomposition o f m a t r o i d s and r e l a t e d t o p i c s , Thesis, C o r n e l l U n i v e r s i t y , 1972. Bixby, R.E.,
A composition f o r m a t r o i d s , J . Comb. Th. B 18 (1975) 59-72.
348
p8]
R.H. Mohring and EJ. Radennacher Bixby, R.E.,
A composition f o r p e r f e c t graphs, p r e p r i n t , 1982.
@9] Bixby, R . E . ,and Cunningham, W.H. , Matroids, graphs, and 3 - c o n n e c t i v i t y , Bnndy, J.A. and Murty, U.S.R. (eds): Graph Theory and r e l a t e d t o p i c s , (Acad. Press, New York, 1978) 94-123.
in:
Po]
Blass, A., 19-24.
Pl]
Blass, A. and Harary, F., P r o p e r t i e s o f almost a l l graphs and complexes, J. Graph Th. 3 (1979) 225-240.
12q
Bollobas, B.,
B33
Brylawski, T., A c o m b i n a t o r i a l model f o r s e r i e s - p a r a l l e l networks, Trans. Amer. Math. SOC. 154 (1971) 1-22.
@4]
Buer, H. and Mohring, R.H., A f a s t a l g o r i t h m f o r t h e decomposition o f graphs and posets, Vath. Oper. Res. 8 (1983) 170-:84.
Graphs w i t h unique maximal clumpings, J. Graph Th. 2 (1978)
Extremal Graph Theory, (Academic Press, London, 1978).
Burkard, R.E. and Zimmermann, U., Combinatorial o p t i m i z a t i o n i n l i n e a r l y ordered semimodules: a survey, Report 80-12, .Math. I n s t . , Univ. KEln, (1980); i n : Korte, B. (ed.): Modern Applied Hathematics: O p t i m i z a t i o n and Operations Research ( N o r t h Holland, Amsterdam, 1982) 391-436. p6]
Butterworth, R.W., A s e t t h e o r e t i c treatment o f coherent systems, S I A M J . Appl. Math. 22 (1972) 590-598.
pi’]
Chein, M., Habib, M. and Maurer, M.C., 37 (1981) 35-50.
p8]
Chvatal, V, On c e r t a i n polytopes associated w i t h graphs, J . Comb. Th. ( B ) 18 (1975) 138-154.
F9-J
Chvatal, V . and Hamner, P.L., Aggregation o f i n e q u a l i t i e s i n I n t e g e r Prog r a m i n g , Annals o f D i s c r . Math. 1 (1977) 145-162.
PO] Pl]
Cohn, P.M.,
Universal Algebra,
P a r t i t i v e Hypergraphs, Discr. Math.
(Harper and Row, New York, 1965).
C o r n e i l , D.G., Lerchs, H. and Stewart Burlingham, L., graphs, D i s c r . Appl. Math. 3 (1981) 163-174.
Complement r e d u c i b l e
[32]
Cowan, D.D., James, L.O. and Stanton, R.G., Graph decomposition f o r undirect e d graphs, i n : Hoffman, F. and Levow, R.B. (eds.), 3 r d South-Eastern Conf. Combinatorics, Graph Theory, and Computing, U t i l i t a s Math. (Winnipeg, 1972) 281 -290.
p3]
Cunninghame-Green, R.A., 1979).
G4-j
Cunnincharn, W.H., A combinatorial decomposition theory, Thesis, U n i v e r s i t y o f Waterloo, 1973.
[35]
Cunningham, W.H., Decomposition o f d i r e c t e d graphs, S I A M J . Algebraic and D i s c r e t e Methods 3 (1982) 214-228.
p6]
Cunningham, W.H. , Polyhedra f o r composed independence systems, Annals o f D i s c r e t e Math. 16 (1982) 57-67.
Minimax Algebra, Springer Lecture Notes ( B e r l i n ,
349
Substitution decomposition for discrete structures
~ 7 1Cunningham, W.H., Decomposition o f submodular f u n c t i o n s , Report No. 8221s-OK, I n s t . f u r Okon. und Oper. Res.,(University o f Bonn, 1982). Cunningham, W.H. and Edmonds, J., J. Math. 32 (1980) 734-765. Cunningham, W.H. p a r a t i o n , 1982.
A combinatorial decomposition theory, Can.
and Edmonds, J., Decomposition o f l i n e a r systems, i n p r e -
C u r t i s , H.A., A New Approach t o t h e Design o f Switching C i r c u i t s , Nostrand, Princeton, 1962). Davio, M., Deschamps, J.P. and Thayse, A., tMcGraw-Hill, New York, 1978).
(Van
D i s c r e t e and Switching Functions,
De F l o r i a n i , L., A method f o r o r g a n i z i n g a p r o j e c t network i n t o a t r e e s t r u c t u r e , Methods o f Oper. Res. 35 (1979) 119-122. Deschamps, J.P., B i n a r y simple decomposition o f d i s c r e t e f u n c t i o n s , D i g i t a l Processes 1 (1 975) 123-1 40. D u f f i n , K.J., Topology o f s e r i e s - p a r a l l e l networks, Journal o f Math. A n a l y s i s and A p p l i c a t i o n s 10 (1965) 303-318. D u f f i n , R.J. and Pate, R.S., An a b s t r a c t t h e o r y o f t h e Jordan-Holder compos i t i o n s e r i e s , Duke Math. J. 10 (1943) 743-750. Dushnik, B. and M i l l e r , E., 600-61 0.
P a r t i a l l y ordered sets, h e r . J. Math. 63 (1941)
Edmonds, J. and Fulkerson, D.R., 299-306.
Bottleneck extrema, J. Comb. Th. 8 (1970),
Elmaghraby, S.E., A c t i v i t y Networks: P r o j e c t Planning and Control by Network Models, (J. Wiley & Sons, New Yor k, 1977). Erne, M. , S t r u k t u r - und Anzahlformeln Manuscripta Math. 11 (1974) 221-259.
f k r Topologien a u f endlichen Mengen,
Esary, J.D., Proschan, F. and Walkup, D.W., A s s o c i a t i o n o f random v a r i a b l e s w i t h a p p l i c a t i o n s , Annals o f Math. S t a t i s t i c s 38 (1967) 1466-1474. Fagin, R.,
P r o b a b i l i t i e s on f i n i t e models, J. Symbolic Logic 41 (1976) 50-58.
Ford, L.R. and Fulkerson, D.R., Princeton, New Jersey, 1962).
Flows i n Networks, ( P r i n c e t o n Univ. Press,
F u j i s h i g e , S. , Canonical decompositions o f symmetric submodular systems, D i s c r e t e Appl. Math. 5 (1983) 175-190. F u j i s h i g e , S., A decomposition o f d i s t r i b u t i v e l a t t i c e s , Report No. 83256-0R, I n s t i t u t f u r Okonometrie und Operations Research, ( U n i v e r s i t y o f Bonn, 1983). Fulkerson, D.R., Networks, frames, and b l o c k i n g systems, i n : Dantzig und V e i n o t t (eds.): Mathematics o f t h e Decision Sciences, P a r t 1, Amer. Math. SOC. ( 1 968) 303-334. Fulkerson, D.R., Blocking and a n t i - b l o c k i n g p a i r s o f polyhedra, Math. Progr. 1 (1971) 168-194.
R. H. Mohring and F J. Raderrnacher
Fulkerson, D . R . , PniiblockinS polyhedra, J . Comb. T h . B 1 2 (1972) 50-71. G a l l a i , T . , T r a n s i t i v o r i e n t i e r b a r e Graphen, Acta Math. Acad. S c i e n t . Hung. Tom. 18 (1967) 25-66. Garey, M . R . and Johnson, D.S., Computers and I n t r a c t i b i l i t y : A Guide t o the Theory of NP-completeness,(Freeman, San Francisco, 1979). Gilmore, P . C . and Hoffman, A.J., A c h a r a c t e r i z a t i o n of comparability graphs and of interval graphs, Can. J . Math. 16 (1964) 539-548. Ginzburg, A . , Algebraic Theory of Automata,(Academic Press, New York, 1968). Glinz, N . and MGhring, R.H., Reduction theorems for networks with general sequencing r e l a t i o n s , Methods of Oper. Res. 27 (1977) 124-162. Goldie, A.W., The scope of the Jordan-Hb'lder theorem in a b s t r a c t algebra, Proc. London Math. SOC. 2 (1952) 349-368. Golumbic, M . , Algorithmic Graph Theory and Perfect Graphs, (Academic Press, New York, 1980). Gorman, W.H., The s t r u c t u r e of u t i l i t y functions, Rev. Econ. Stud. 35 (1968) 367390. GrB'tzer , G . , Universal Algebra, (Springer Verl ag , New York, 1979). Gysin, R . , Dimension t r a n s i t i v o r i e n t i e r b a r e r Graphen, Acta Math. Acad. S c i . Hung. 29 (1977) 313-319. Habib, M. a n d Maurer, M.C., On the X-join decomposition f o r undirected graphs, J . Appl . Discr. Math. 3 (1979) 198-207. Hansel, G . , Sur l e nombre des fonctionsboolgennes monotones de n v a r i a b l e s , C . R . Acad. S c i . P a r i s 262 (1966) 1088-1090. Hashimoto, J . , Ideal theory f o r l a t t i c e s , Math. Japanicae 2 (1952) 149-186. Hausmann,D. and Korte, B . , Lower bounds on t h e worst-case complexity of some oracle algorithms, Discrete Mathematics 24 (1978) 261-276. Hemminger, R . L . , 408-41 8.
The group of an X-join of graphs, J . Comb. T h . 5 (1968)
Herrlich, H. and Strecker, G . E . , 1973).
Category Theory, (Allyn & Bacon, Boston,
Hiragushi, T . , On the dimension of p a r t i a l l y ordered s e t s , S c i . Rep. Kanazawa Univ. 1 (1951) 77-94. Hapfinger, i. and Steinhardt, U., Two New Procedures f o r the Evaluation of F i n i t e Acyclic A c t i v i t y Networks with Stochastic Durations of A c t i v i t i e s ; Discussion paper, I n s t i t u t f . Wirtschaftstheorie und Operations Research of the University Karlsruhe, No. 54, 1975.
Hu, S.T.,
I?71
Threshold Logic ( U n i v . of California Press, Berkeley, 1965).
Kaerkes.I R . . . Netzalan Theorv. Methods of Ooer. Res. 27 (1977) 1-65
Substitution decomposition for discrete structurcs
P81
35 1
Kaerkes, R . , lrlangel und Fehlerquelien ckr herkomnlichen rietzplantechnik und neue Konzepte zu i h r e r Behandlung, i n : Iienn, R. e t a1 ( e d s . ) : Q u a n t i t a t i v e Wirtschafts- und Unternehmensforschung, (Springer, Belin, 1980). Kaerkes, R . and Leipholz, B., Generalized network functions i n flow networks, Methods of Oper. Res. 27 (1977) 225-273. Kaerkes, R . and Mohring, R . H . , Vorlesungen uber Ordnungen und Netzplantheorie Schriften zur Informatik und Angewandten Mathematik 45,(RWTH Aachen, 1978). Kaerkes, R . and Radermacher, F.J., P r o f i l e s , network functions and f a c t o r i z a t i o n , Methods of Oper. Res. 27 (1977) 66-87. Karp, R . M . , Functional decomposition and switching c i r c u i t design, J . SOC. I n d u s t . Appl. M a t h . 11 (1963) 291-335. Karp, R.M., Reducibility among combinatorial problems, in: Miller, R.E. and Thatcher, J.W. ( e d s ) , Complexity of Computer Computations, (Plenum Press, New York (1972) 85-103. Klein Haneveld, W.K., Distributions with known marginals and d u a l i t y of mathematical programming w i t h applications t o PERT, P r e p r i n t 90 (0R-8203), I n t e r f a c u l t e i t der Actuariele Wetenschappen en Econometrie, University of Groni ngen, 1982. Kleitman, D.J., On Dedekind's problem: The number of monotone Boolean funct i o n s , Proc. Amer. Math. SOC. 21 (1969) 677-682. Kleitman, D.J. and Rothschild, B . L . , Asymptotic enumeration of p a r t i a l orders on a f i n i t e set, Trans. Amer. Math. SOC. 205 (1975) 205-220. Knodel, W . , Ein O(n3)-Algorithmus zur Dekomposition eines Digraphen, Arbeitspapier, I n s t . f & Informatik, Univ. S t u t t g a r t , 1979. Korte, B . , Matroids and independence systems, i n : Korte, B.(ed.), Modern Applied Mathematics, Optimization and Oper. Res. (North Holland, Amsterdam, 1982) 517-553. Korte, B. and Schrader, R . , A survey on oracle techniques, Report No. 81186O R , Inst. f u r Okon. und Oper. Res., Univ. of Bonn; t o appear in: Proceedings Mathematical Foundations o f Computer Science. Strbsk6 Pleso, Czechoslovakia 1981; Lecture Notes in Computer Science (Springer-Verlag, B e r l i n ) . Lawler, E . L . , Combinatorial Optimization: Networks and Matroids, (Holt, Rinehart and Winston, New York, 1976). Lawler, E . L . , Graphical algorithms and their complexity, Math. Centre Tracts 81 (1976) 3-32. Lawler, E.L., Sequencing j o b s t o minimize t o t a l weighted completion time subject t o precedence c o n s t r a i n t s , Ann. Discrete Math. 2 (1978) 75-90. Lawler, E . L . , Lenstra, J.K. and Rinnooy Kan, A . H . G . , Recent developments i n deterministic sequencing and scheduling, i n : Dempster, M.A.H. e t a l . ( e d s ) , Deterministic and Stochastic Scheduling, (Reidel, Dordrecht, 1982) 35-73.
p4]
Megiddo, N. Tensor decomposition o f cooperative games, SIAM J . Appl. Math. 29 (1975) 388-405
R.H. Mohring and F.J. Radermacher M e i l i j s o n , I . and Nadas, A., Convex m a j o r i z a t i o n w i t h an a p p l i c a t i o n t o the l e n g t h o f c r i t i c a l paths, J. Appl. Prob. 16 (1979) 671-677. Meyer, H., Strukturaussagen i n Entscheidungsnetzwerken, Math. Syst. i n : Economics 66, (Anton Hain, K i n i g s s t e i n (Ts), 1981). Moder, J.J. and P h i l l i p s , C.R., (Reinhold, New York, 1964).
P r o j e c t management w i t h CPM and PERT,
f 6 h r i n g y R.H., Untersuchungen zur Homomorphietheorie von Relationalsystemen, Thesis, Techn. Univ. o f Aachen, 1975. Mijhring, R.H., On t h e d i s t r i b u t i o n o f l o c a l l y undecomposable r e l a t i o n s and independence systems, Methods o f Oper. Res. 42 (1981) 33-48. Mijhring R.H., M i n i m i z i n g Costs o f Resource Requirements s u b j e c t t o a f i x e d Completion Time i n P r o j e c t Networks, t o appear i n : Oper. Res., extended a b s t r a c t i n Methods o f Oper. Res. 32 (1979) 181-182. Mohring, R.H.,
Almost a l l c o m p a r a b i l i t y graphs a r e UPO, t o appear i n Discr. Path.
Mohring, R.H., Dekomposition d i s k r e t e r S t r u k t u r e n m i t Anwendungen i n der kombinatorischen Optimierung, H a b i l i t a t i o n , Techn. Univ. o f Aachen, 1982. Mijhring, R.H., On t h e c h a r a c t e r i z a t i o n o f s e r i e s - p a r a l l e l networks, complement r e d u c i b l e graphs, and t h r e s h o l d graphs, p r e p r i n t 1982 (paper presented a t t h e 7. Symp. on Oper. Res., S t . Gallen, 1982). Miihring, R.H., On t h e computational complexity o f t h e s u b s t i t u t i o n decompos i t i o n f o r r e l a t i o n s , s e t systems and Boolean f u n c t i o n s , p r e p r i n t 1982 (paper presented a t t h e X I . I n t . Symp. on Math. Progr., Bonn, 1982). Mohring, R.H., preparation.
Congruence p a r t i t i o n l a t t i c e o f d i s c r e t e s t r u c t u r e s ,
Mijhring, R.H. and Radermacher, F.J., Oper. Res. 27 (1977) 88-112.
in
P r o f i l e s and homomorphisms, Methods o f
M h r i n g , R.H. and Radermacher, F.J., Dimension, r e v e r s i b i l i t y and chainequivalence of posets, and connections w i t h homomorphisms, I n : Beckmann, PI., Eichhorn, U. and K r e l l e , W. (eds.), Mathematische Systeme i n d e r Okonomie, (Athenaum, K o n i g s t e i n ( T s ) , 1982) 415-432. Monma, C.L. and Sidney, J.B., Sequencing w i t h s e r i e s - p a r a l l e l precedence c o n s t r a i n t s , Math. o f Oper. Res. 4 (1979) 215-224. kadas, A . , P r o b a b i l i s t i c PERT, IBM J . Res. Develop. 23 (1979) 339-347. Natvig, B . , Improved bounds f o r t h e a v a i l a b i l i t y and u n a v a i l a b i l i t y i n a f i x e d time i n t e r v a l f o r systems o f maintained, interdependent components, Adv. Appl. Prob. 12 (1980) 200-221. Neggers, J . , Counting f i n i t e posets, Acta Math. Acad. Scient. Hung., Tom. 31 (1978) 233-258. D l 2 1 Neumann, K., Operations Research Verfahren, Band 111, ( C a r l Hanser Velag, Munchen, 1975).
Substitution decomposition for discrete structures
353
p13)
Neumann, K. and Steinhardt, U., 1979).
[114]
Oberschelp, W., M o n o t o n i c i t y f o r s t r u c t u r e numbers i n t h e o r i e s w i t h o u t i d e n t i t y , i n : Foata,D. (ed.), Combinatoire e t Representation du Groupe Symgtrique, Lect. Notes i n Math. 579 (Springer, 1977) 297-308.
[llq
Oberschelp, W., P r o p e r t i e s o f almost a l l parametric r e l a t i o n s , paper presented a t t h e Oberwolfach workshop "Kombinatori k geordneter Mengen", Oberwolfach, May 1980.
p 1 a
Decomposition o f p r o j e c t networks, ManageParikh, S.C. and Jewell, W.S., ment Sci , 11 (1965) 8397-8402.
n17]
Parthasarathy, K.R., New York, 1967).
n 1 q
P f a l t z , J.L.,
GERT Networks, (Springer Verlag, B e r l i n ,
.
P r o b a b i l i t y Measures on M e t r i c Spaces (Academic Press,
Graph s t r u c t u r e s , J. ACM 19 (1972) 411-422.
Pfanzagl, J . , Theory o f Measurement, (Physica Verlag, Wurzburg, 1971). c203
P r i t s k e r , A.A.B., M o d e l l i n g and Analysis u s i n g Q-GERT Networks, (John Wiley & Sons, New York, 1977).
u21]
Radermacher, F.J. ( 1 974) 167-1 76.
[I
Radermacher, F. J , Theorie der r e d u z i erenden Ordnungshomomorphi m e n , Thesis, Techn. Univ. o f Aachen, 1974.
221
, Reduktion
von Netzplanen, Proceedings i n Oper. Res. 3
.
[I231
Radermacher, F.J., Invarianzaussagen f u r s t o c h a s t i s c h e Netzplane, Methods o f Oper. Res. 22 (1976) 136-148.
p24]
Radermacher, F.J., (1977) 163-224.
[125]
Radermacher, F.J., Kapazitatsoptimierung i n Netzplanen, Math. Syst. i n Econ. 40, (Anton Hain, Meisenheim, 1978).
n2a
Radermacher, F.J., E i n F a k t o r i s i e r u n g s s a t z f u r d i e Mobiusfunktion, paper presented a t t h e "23. Kurztagung uber Allgemeine Algebra", K a i s e r s l a u t e r n , November 1981 1.
u271
Radermacher, F.J. and Spelde, H.G., Reduktion von Flussnetzplanen, Proceedings i n Oper. Res. 3 (1974) 177-186.
[12q
R i c h t e r , G.,
0291
Riordan, J., An I n t r o d u c t i o n t o Combinatorial Analysis, New York, 1958).
-
F l o a t s i n p r o j e c t networks, Methods o f Oper. Res. 27
K a t e g o r i e l l e Algebra (Akademie Verlag, B e r l i n , 1979).
(J. Wiley & Sons,
F3q
On t h e Foundations o f Combinatorial Theory I , Rota, G.C., l i c h k e i t s t h e o r i e 2 (1964) 340-368.
p31]
Roy, B., Alg'ebre moderne e t t h 6 o r i e des graphes,Vol. 1969, 1970).
Q32]
Sabidussi, G.,
Z. Wahrschein-
I,II,(Dunod,
Paris,
Graph d e r i v a t i v e s , Math. Z e i t s c h r i f t 76 (1961) 385-401.
351
R. H. Mohring and F.J. R u d e m e h e r
[133]
Schlee, W., E i n Algorithmus f u r T e i l n e t z p l a n e , Z e i t s c h r i f t f . Oper. Res. ( B ) 17 (1973) 167-172.
c343
Schwarze, J . , An a l g o r i t h m f o r h i e r a r c h i c a l r e d u c t i o n and decomposition o f a d i r e c t e d graph, Computing 25 (1980) 47-57.
[13g
Seeling, R. and Spelde, H.G., Verfahren zur d i r e k t e n Berechnung von stochastischen Netzplanen, Proceedings i n O.R. 5, (Physica Verlag, Wurzburg, 1975) 221 -230).
[136]
Shannon, C.E., A symbolic a n a l y s i s o f r e l a y s w i t c h i n g c i r c u i t s , Trans. h e r . I n s t . Elec. Engrs. 57 (1938) 713-723.
[13g
Shapley, L.S., S o l u t i o n s o f compound simple games, i n : Advances i n Game Theory, Annals o f Math. Study No. 52 ( P r i n c e t o n Univ. Press, Princeton, 1964) 267-305.
p38]
Shapley, L.S., On Comnittees, i n : Zwicky, F. and Wilson, A.G. ( e d s . ) , New Methods o f Thought and Procedure (Springer-Verlag, B e r l i n , New York, 1967) 246-270. Shen, V.Y. and McKellar, A.C., An a l g o r i t h m f o r t h e d i s j u n c t i v e decompos i t i o n o f s w i t c h i n g f u n c t i o n s , I E E E Trans. Computers C - 19 (1970) 239-248. Shen, V . Y . , McKellar, A.C. and Weiner, P., A f a s t a l g o r i t h m f o r t h e d i s j u n c t i v e decomposition o f s w i t c h i n g f u n c t i o n s , I E E E Trans. Computers C 20 (1971) 304-309. Shevrin, L . N . and F i l i p p o v , N.D., P a r t i a l l y ordered s e t s and t h e i r comparab i l i t y graphs, Siber. Math. J . 11 (1970) 497-509. Shogan, A.W., Bounding d i s t r i b u t i o n s f o r s t o c h a s t i c PERT-network, 7 (1977) 359-381.
Networks
Shogan, A.W., A decomposition a l g o r i t h m f o r network r e l i a b i l i t y a n a l y s i s , Networks 8 (1978) 231-252. Sidney, J.B., Decomposition a l g o r i t h m s f o r single-machine sequencing w i t h precedence r e l a t i o n s and d e f e r r a l costs, Oper. Res. 23 (1975) 283-298. S i e l k e n j r . , R.L., and H a r t l e y , H . O . , A new s t a t i s t i c a l approach t o p r o j e c t scheduling, i n : Tsokos, Ch.P. and T h r a l l , R.M. (eds.), Decision Information, (Academic Press, New York, 1979) 153-184. Spelde, H.G., Stochastische Netzplane und i h r e Anwendung i m Baubetrieb, Thesis, Techn. Univ. o f Aachen (1976). Spelde, H.G., Bounds f o r t h e d i s t r i b u t i o n f u n c t i o n o f network v a r i a b l e s , Methods o f Oper. Res. 27 (1977) 113-123. Stoyan, D., Qua1it a t i v e Eigenschaften und Abschatzungen s t o c h a s t i s c h e r Modelle, (R. Oldenbourg Verlag, Munchen, 1977). Strassner, K., Zur S t r u k t u r t h e o r i e e n d l i c h e r n i c h t d e t e r m i n i s t i s c h e r Automaten I , Zum Verband der I-Kongruenzen von endlichen Relationalsystemen, J , I n f o r m a t i o n Processing and Cybernetics (EIK) 17 (1981) 113-120. Strassner, K., Zur S t r u k t u r t h e o r i e e n d l i c h e r n i c h t d e t e r m i n i s t i s c h e r Automaten 11, Gesteuerte Sumnen und Stedrbaume n i c h t d e t e r m i n i s t i s c h e r
Substitution decornposition for discrete stncctures
355
Automaten, J. I n f o r m a t i o n Processing and Cybernetics (EIK) 17 (1982) 51 1-522. u51]
Strassner, K. , Zur S t r u k t u r t h e o r i e e n d l i c h e r n i c h t d e t e r m i n i s t i s c h e r Automaten 111, Zum Steuerrelationenverband eines n i c h t d e t e r m i n i s t i s c h e n Automaten, submitted t o J. I n f o r m a t i o n Processing and Cybernetics (EIK).
[152]
Suchowizki , S.I. and Radtschik, I.A., Mathematische Methoden d e r Netzplantechnik, (B.G. Teubner, L e i p z i g , 1969).
[15q
Sumner, D.P., Graphs undecomposable w i t h r e s p e c t t o t h e X-join, Math. 6 (1973) 281-298.
[154]
Szasz, G., 1962) .
[155]
Thayse, A., A f a s t a l g o r i t h m f o r t h e proper decomposition o f Boolean funct i o n s , P h i l i p s Res. Rep. 23 (1972) 140-150.
[156)
Thumb, G., Grundlagen und P r a x i s der Netzplantechnik, Moderne I n d u s t r i e , Munchen, 1975).
[157]
Trakhtenbrot, B.A., Towards a t h e o r y o f non-repeating c o n t a c t schemes, T r u d i Mat. I n s t . Akad. Nauk. SSSR 51 (1958) 226-269 (Russian).
D58)
T r o t t e r jr., W.T. Moore jr., J.I. and Sumner, D.P., The dimension o f a c o m p a r a b i l i t y graph, Proc. Amer. Math. SOC. 60 (1976) 35-38.
059)
T u t t e , W., 1-48.
p60]
Valdes, J . , Tarjan, R.E. and Lawler, E.L., The r e c o g n i t i o n o f series-paral l e l digraphs, Proc. 11th. Annual ACM Symp. on Theory o f Comp., ACM (1979) 1-12.
p61]
Walsh, T.R.S., Counting u n l a b e l l e d three-connected and homeomorphically i r r e d u c i b l e two-connected graphs, J. Comb. Th. B 32 (1982) 12-32.
[162]
Weiss, G., S t o c h a s t i c bounds on d i s t r i b u t i o n s o f optimal value f u n c t i o n s w i t h a p p l i c a t i o n t o PERT networks, flows and r e l i a b i l i t y , Techn. Report, L e h r s t u h l f u r I n f o r m a t i k I V , Techn. Univ. o f Aachen (1982).
[163
Welsh, D.J.A.,
[164]
Whitehouse, G.E., Systems Analysis and Design Using Networks Techniques ( P r e n t i c e - H a l l , Englewood C l i f f s , 1973).
[165]
W i l l e , R., Lexicographic decomposition o f ordered s e t s (graphs), P r e p r i n t No 705, Fachbereich Mathematik, Techn. Univ. o f Darmstadt, (1983).
p66]
Zimmermann, U., L i n e a r and Combinatorial O p t i m i z a t i o n i n Ordered Algebraic S t r u c t u r e s , Anals o f D i s c r . Math. 10,(North Holland, Amsterdam, 1981).
Discr.
Einfuhrung i n d i e Verbandstheorie, ( A k a d h i a i Kiado', Budapest,
(Band 1,Verlag
Lectures on Matroids, J. Res. Nat. Bur. Standards 69 B (1965)
M a t r o i d Theory (Academic Press, London, 1976).
This Page Intentionally Left Blank
Annals of Discrete Mathematics 19 (1984) 357-362 0 Elsevier Science Publishers B.V. (North-Holland)
357
ON MAX-SEPARABLE OPTIMIZATION PROBLEMS
K. Zimnermann M a t e m a t i c k o - F y z i k a l n i Fakul t a U n i v e r z i t y K a r l o v y Katedra k y b e r n e t i k y I n f o r m a t i k y a Operacniho Vyzkumu 118 00 Praha
The aim o f t h i s c o n t r i b u t i o n i s t o g e n e r a l i z e some [4] c o n c e r n i n g c e r t a i n c l a s s r e s u l t s f r o m [2], [3], o f o p t i m i z a t i o n problems, i n which b o t h t h e o b j e c t i v e f u n c t i o n and t h e c o n s t r a i n t f u n c t i o n s a r e o f t h e x ) = “max-separable” type, i . e . o f t h e form g ( x max g j ( x j ) , where g . E -+ ~ t land j E i s l’totalyy j* 14JSn ordered s e t .
...,
INTRODUCTION The aim o f t h i s c o n t r i b u t i o n i s t o g e n e r a l i z e some r e s u l t s from [2],
[3],
[4]
c o n c e r n i n g c e r t a i n c l a s s o f o p t i m i z a t i o n problems which can be c a l l e d maxs e p a r a b l e o p t i m i z a t i o n problems. We s h a l l assume t h a t I f L i s a subset o f
E E,
i s a t o t a l l y o r d e r e d s e t w i t h a t o t a l l y o r d e r i n g r e l a t i o n 6. t h e n element i a L w i t h t h e p r o p e r t y x 6
w i l l be c a l l e d maximal element o f L.
i
x
for all x e L
We s h a l l denote t h i s element max x, s o t h a t xeL = max x.
XeL f o r j = 1 ,...,n w i l l be c a l l e d an n - v e c t o r o v e r J t h e s e t o f a l l n - v e c t o r s o v e r E w i l l b e denoted E ~ .
X = (xl
,...,xn)
A mapping f :
E
n
with
X.LE
E
E
and
w i t h the property
f ( X ) = max f . ( x . ) J J jaN
tl
X = (xl,...,x
n
)
E E
n
where N = {l,...,n}, V j e N, w i l l be c a l l e d max-separable f u n c t i o n f j : E-E n over E . I n t h e sequel, we s h a l l s t u d y o p t i m i z a t i o n problems o v e r sn w i t h maxseparable o b j e c t i v e f u n c t i o n and max-separable c o n s t r a i n t f u n c t i o n s . The maximiz a t i o n o r m i n i m i z a t i o n o f a f u n c t i o n i n t h e s e problems i s meant w i t h r e s p e c t t o
K. Zimmermann
358 the r e l a t i o n
+
introduced on
i.e.
E,
min f ( Y ) = f ( i ) 4 f ( X ) 4 max f ( Y ) = f ( X ) YeM YeM f o r a l l X E M f o r a r b i t r a r y M, M
f o r which max and min e x i s t s .
CE",
We s h a l l d e f i n e r e l a t i o n s 3 , 6 , > on the b a s i s o f 4 as usual: a a R - 6 4 a r~
6 andaP B
B -a<
<
8 < a
s > 6 >-
MAX-SEPARABLE OPTIMIZATION PROBLEMS Max-separable o p t i m i z a t i o n problems are d e f i n e d as o p t i m i z a t i o n problems over
E~
o f t h e f o l l o w i n g form: f ( X ) :max f j ( x j ) d min o r max jaN subject t o
where S1
u S2 = S
5
C1,
max r .1J . ( x .J) jeN
a b
V
iES1
max r i j ( x j ) jrN
\< bi
W
i E S 2
N = 11 ,..., n l , bi,
..., ml,
k . , K . a r e g i v e n elements o f J J
E.
We s h a l l now formulate general c o n d i t i o n s under which e x p l i c i t formulae f o r optimal s o l u t i o n s of (P) can be found.
This extends t h e r e s u l t s g i v e n i n [3]
We s h a l l confine ourselves t o m i n i m i z a t i o n problems o f t h e form (P).
[4].
w.., L~ and M as f o l l o w s :
L e t us d e f i n e t h e s e t s Vij, E
I
kj 6 xj
€ E
1
kj
Vij
= 1X. t
w l. J.
= {x.
J
L.
1
1
= 1j E N
M = fx
Lma 1
E
E
n
1J
6
r . . xj) 1J
Kj,
x j < Kj, rij
3
i E S1, j r N,
bi}
x . ) 4 bil J
w i E S2,
j E N,
V . . # 01 W i € S 1 , 1J
I
max r i j ( x . ) J j EN
%
bi w i E S , ,
[ 3 i 0 t S1 such t h a t Li
0
=
0J
-
k . 4 x j 6 K . W j e N}. J J
M =
0,
and
On ma-separable optimization problems
Proof
0, i o e S1 and l e t X Therefore
L e t Li
=
f o r a l l j E N?
E E ~ .
359
Then r . . ( x . ) < bi and k . 6 x . 1oJ J 0 J J
<
max r (xj) < b. jeN io j l0
0. Q.E.D.
and t h u s M =
Theorem 1
If X
Proof
,... ,xn)
X = (xl
and k . < x . 4 K . J J J
w
E M ->
i cS1 3 j(i)
N such t h a t x j f i )
E.
Vij(.)
M, t h e n
E
max r i j ( k j ) jeN
so t h a t W i e S 1 3 j ( i )
E
= r lj(i) .
(
~ 3 (bi ~w i ~E S1, ,
~
N such t h a t xj(.)
E Vij(i).
b e f u l f i l l e d and l e t i be an
L e t t h e r i g h t hand s i d e o f t h e - > - r e l a t i o n a r b i t r a r y i n d e x o f S1. rij(i)(xj(i)
E
j e N.
Then xj(i)
S .
Vij(i)
>
bi and thus max r ( x . ) jaN i j J
>/
f o r some j(i) rij(i)(xj(i))
>, bi
N so t h a t and X
E
M, Q.E.D.
THE EXPLICIT SOLUTION OF MAX-SEPARABLE OPTIMIZATION PROBLEMS We s h a l l assume now t h a t problem (P) s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s .
(1)
A l l nonempty Vij,
I. J
(2)
=
{x. J
1
k. J
W . . are subintervals o f 1J
4 x. &
J
KjI;
F o r f i x e d j t h e s e t s Vij
form a chain ( w . r . t .
set inclusion).
( 3 ) The f u n c t i o n s f . a t t a i n t h e i r minimum on any s u b i n t e r v a l o f I
j.
J
If S 2 # 0 and ( 1 ) i s s a t i s f i e d , t h e n f o r a l l j c N: R.
J
5
ix. J
4 E
1
r..(x.) 1J J
6 bi,
x . 6 K., W i & S 2 } = (l W . . J J ieS2 'J
k. J
and t h u s f o r each j e i t h e r R . = 0 and hence t h e s e t o f f e a s i b l e s o l u t i o n s o f (P) J i s empty o r t h e r e e x i s t k ! , K'. e E such t h a t J J
R. = {x. J
J
E
E
1
k'. J
4
X.
J
c K!]. J
Therefore ifR . # 0, W j E N, t h e n t h e o r i g i n a l o p t i m i z a t i o n problem can be J reduced t o t h e problem of t h e form (P) w i t h these new bounds k ' K j and w i t h
j'
K. J
K. Zimmermann
360 S1 = S , S2 =
0.
Therefore we can c o n f i n e ourselves i n the sequel t o problems ( P )
w i t h S1 = S and S 2 = 0.
We s h a l l show t h a t under t h e c o n d i t i o n s ( l ) , ( 2 ) , ( 3 )
e x p l i c i t fonnulae f o r t h e components of an optimal s o l u t i o n o f ( P ) can be given. Theoran 2
L e t us suppose t h a t t h e s e t o f f e a s i b l e s o l u t i o n s o f ( P ) i s nonempty,
S1 = S, S 2 = 0 and c o n d i t i o n s ( 1 )
xj(i)
-
( 3 ) are s a t i s f i e d .
L e t us d e f i n e elements
and sets S ( j ) , H ( j ) as f o l l o w s : f. . J(1)
(X.J ( i ) )
min
min
= j e ~. x. r ~
j
i
L e t X = ( x l,...,xn)
x
Then
fj(xj),
ij
be d e f i n e d according t o t h e f o l l o w i n g r e l a t i o n s :
i s an optimal s o l u t i o n o f t h e m i n i m i z a t i o n Droblem ( P ) .
I t f o l l o w s immediately from Lemma 1 t h a t Li # 0 t/ i t S so t h a t f o r each i C S there e x i s t s j ( i ) e N such t h a t f . . . j = m i n min f j ( y j ) and ~ ( 1 ) ~ ( 1 ) jcLi y.rv
Proof
(x.
i e S(J(i)).
Then according t o ( 2 ) ^x
j(i)
E
J ij H(j(i)) c V.. 1 J ( i ) so t h a t rij(.)(Xj(.))
3 bi and thus . .(X .) 5 r i j ( i ) ( i j ( i ) ) max r lJ
3 bi,
v iE S
jeN
and
X
i s a f e a s i b l e s o l u t i o n of ( P ) .
I t remains t o prove t h a t
f ( X ) 5 f ( X ) = max f . ( i . ) = f ( j j c ~J J P P f o r a l l f e a s i b l e s o l u t i o n s of (P). solution.
If
=
0,
IfS(p) # 0 and f p ( x p ) =
f(X).
f (x )
P
P
L e t X = (x ly...,xn)
fp(ip).
then again f ( X ) = max [ f . ( x . ) ] + j , ~ J J
I t remains t o i n v e s t i g a t e t h e case t h a t c
be an a r b i t r a r y such
we have
f ( x ) = max f j ( x j ) P P jcN
= f(X) holds.
f (x ) P
P
+
f
P
( iP)
# 0 and a t t h e same time
We s h a l l show i n t h i s case t h a t t h e r e
On max-separable optimization problems
e x i s t s an i n d e x
If fp(xp)
<
1=
l ( p ) E. N such t h a t f - ( x - ) >/ f ( x ). t t p p
fp(Xp),
then x
e x i s t s an i n d e x i ( p )
min Y$"i
fp(yp) =
(PIP
36 1
P
+
L e t us n o t e t h a t a c c o r d i n g t o ( 2 ) t h e r e
such t h a t H(p) =
E
min min f . ( y . ) . J J jaL i( p) Y j e V i ( PI j
(k
) = and t h u s f %PIP P P Since X i s a f e a s i b l e s o l u t i o n o f
( P ) t h e r e e x i s t s according t o Theorem 1 an i n d e x
e = l(i(D))
such t h a t xt
V. 1(P)t
and we o b t a i n :
so t h a t
I n a l l p o s s i b l e cases we o b t a i n e d f ( X ) > f ( X ) and t h e r e f o r e X i s an o p t i m a l s o l u t i o n o f t h e m i n i m i z a t i o n problem ( P ) , Q.E.D.
0
Remark 1 The " l i n e a r " o p t i m i z a t i o n problem C ' c o n d i t i o n s (A @ X)i r e l a t i o n s 6.
+,
i = 1,
: bi,
= considered i n
[3]
... ,my , [4]
X e
E
X+
i s a s p e c i a l t y p e o f max-separable o p t i -
m i z a t i o n problem w i t h f . ( x . ) = c . @ x . and r . . ( x . )
J
Remark 2
J
J
The c o n d i t i o n s (l), (2),
max o r m i n under t h e
n , where : stands f o r one o f t h e
J
1J
J
= aij
0 xj.
(3) are f u l f i l l e d f o r instance i f
E
i s a subset
o f r e a l numbers, f . a r e c o n t i n u o u s and monotone, and rij a r e c o n t i n u o u s and
J
monotone f o r a l l i E S2, j
E
N and c o n t i n u o u s and s t r i c t l y i n c r e a s i n g f o r a l l T h i s case was
i E S1, j e Li o r s t r i c t l y decreasing f o r a l l i a S1, j r Li. i n v e s t i g a t e d i n a s l i g h t l y more g e n e r a l f o r m i n
[q.
L e t u s remark t h a t i t can
b e shown t h a t ( I ) , ( Z ) , (3) r e m a i n f u l f i l l e d f o r t h e f u n c t i o n s rij(xj) = min ( r i j ( x j ) , a i j ) ,
Fij(xj)
= max ( r . . ( x . ) , a . . ) 1J
J
1J
=
where r . . s a t i s f y t h e assump1J
t i o n s mentioned above and a . . a r e g i v e n numbers. 13
REFERENCES
p]
Cuninghame-Green, R.A., Minimax Algebra, L e c t u r e Notes i n Economics and Mathematical Systems, ( S p r i n g e r Verlag, 1979) 166.
[Z]
Zimmermann, K., The e x p l i c i t s o l u t i o n o f max-separable o p t i m i z a t i o n problems, Ekonornicko-matematickj obzor, 4 (1982).
362
K. Zimmermann
131
Zimnermann, U., On some extremal o p t i m i z a t i o n problems, Ekonomickomatematick5 obzor, 4 ( 1 9 7 9 ) .
[4]
Zimmermann, U., L i n e a r and c o m b i n a t o r i a l o D t i m i z a t i o n i n ordered a l g e b r a i c s t r u c t u r e s , Annals o f D i s c r e t e Mathematics, 10 (North-Holland, 1981).
Annals of Discrete Mathematics 19 (1984) 363- 382 0 Elsevier Science Publishers B.V. (North-Holland)
36 3
MINIMIZATION OF COMBINED OBJECTIVE FUNCTIONS ON INTEGRAL SUBMODULAR FLOWS
U. Zimmermann Mathematisches I n s t i t u t U n i v e r s i t a t zu KBln 0-5 K o l n 41 Weyertal 86
We develop a n e g a t i v e c i r c u i t method and an augmenting p a t h method f o r t h e m i n i m i z a t i o n o f o b j e c t i v e f u n c t i o n s on i n t e g r a l submodular f l o w s . The c l a s s o f o b j e c t i v e f u n c t i o n s c o n s i d e r e d admits c e r t a i n combinations o f t h e usual l i n e a r o b j e c t i v e f u n c t i o n and a f i x e d - c o s t o b j e c t i v e f u n c t i o n .
1.
INTRODUCTION
I n sequence of a paper o f EDMONDS and GILES [1976] v e s t i g a t e d by s e v e r a l a u t h o r s . t i e s a r e due t o FUJISHIGE [1978], [1980].
submodular f l o w s have been i n -
R e l a t e d models w i t h s i m i l a r c o m b i n a t o r i a l p r o p e r HASSIN ( n 9 7 8 ] ,
[1981])
F o r i n t e g r a l submodular f l o w s , FRANK [1982]
m i n i m i z i n g (maximizing) l i n e a r o b j e c t i v e f u n c t i o n s .
and LAWLER and MARTEL
developed a n a l g o r i t h m f o r T h i s method i s q u a s i p o l y -
nomial and, i n t h e case o f 0-1-valued submodular f l o w s , p o l y n o m i a l .
A polynomial
method f o r t h e i n t e g r a l case i s supposed t o be p o s s i b l e u s i n g s c a l i n g t e c h n i q u e s . The m i n i m i z a t i o n o f c e r t a i n o b j e c t i v e f u n c t i o n s on R-valued submodular f l o w s , where R i s a t o t a l l y o r d e r e d group ( r i n g ) i s discussed i n ZIMMERMANN p982b-J.
In
p a r t i c u l a r , a n e g a t i v e c i r c u i t method i s based on a theorem showing t h a t t h e d i f f e r e n c e o f two submodular f l o w s i s a c i r c u l a t i o n i n a c e r t a i n a u x i l i a r y graph. I n t h e f o l l o w i n g , we use t h a t ' d i f f e r e n c e ' theorem i n o r d e r t o m i n i m i z e f u r t h e r d i f f e r e n t o b j e c t i v e f u n c t i o n s on i n t e g r a l submodular f l o w s . t i o n s g e n e r a l i z e f u n c t i o n s i n FRIESDORF and HAMACHER ([198la],
These o b j e c t i v e f u n c n981b]).
They
p r o v e t h e v a l i d i t y o f a n e g a t i v e c i r c u i t method as w e l l as o f a s h o r t e s t augmentirg p a t h method f o r m i n i m i z i n g t h e s e f u n c t i o n s on network f l o w s o f maximum f l o w v a l u e . We develop b o t h methods i n t h e g e n e r a l case.
I n p a r t i c u l a r , a s h o r t e s t augmenting
p a t h method f o r m i n i m i z i n g (maximizing) l i n e a r o b j e c t i v e f u n c t i o n s on submodular flows i s given. S e c i i c r , ? c n n t a i n s t h e necessary c o m b i n a t o r i a l r e s u l t s ; i n p a r t i c u l a r , a c o n c i s e statement of t h e ' d i f f e r e n c e ' theorem.
I n s e c t i o n 3, we i n t r o d u c e o b j e c t i v e
f u n c t i o n s combining a l i n e a r o b j e c t i v e and a f i x e d - c o s t o b j e c t i v e on submodular
U.Z i m m e m n n
364 flows.
Some examples a r e given i n a theorem.
The c l a s s o f o b j e c t i v e f u n c t i o n s
considered i n general i s d e f i n e d by s t a t i n g c e r t a i n important p r o p e r t i e s . The negative c i r c u i t method i s developed i n s e c t i o n 4 and t h e augmenting p a t h method i s developed i n s e c t i o n 5.
2.
SUBMODULAR FLOWS
EDMONDS and GILES p976] discuss a r i c h combinatorial s t r u c t u r e i n c l u d i n g network flows, polymatroid i n t e r s e c t i o n s and d i r e c t e d c u t s .
I n a previous paper p982b]
we g e n e r a l i z e t h e i r concept i n a c e r t a i n a l g e b r a i c sense and develop a negative c i r c u i t method f o r t h e d e t e r m i n a t i o n o f submodular f l o w s m i n i m i z i n g c e r t a i n That approach i s based on an a u x i l i a r y
f u n c t i o n s on r m g u a l u e d submodular f l o w s .
graph which FRANK p982) uses f o r t h e development of a polynomial a l g o r i t h m f o r maximizing a r e a l - v a l u e d l i n e a r o b j e c t i v e f u n c t i o n on 0-1-valued submodular f l o w s .
I n t h e f o l l o w i n g we l i s t d e f i n i t i o n s and r e s u l t s on i n t e g e r - v a l u e d submodular flows. denote a digraph w i t h v e r t e x s e t V and a r c s e t E.
L e t G = (V,E)
V
A family F S 2
i s called a crossing family i f [S
n T # 0, SU T #
= [S flT,
SU T E F]
(2.1)
Two members S , T o f F w i t h S 6 T, T SS, S flT # 0 and 9 V a r e c a l l e d c r o s s i n g members o f F. A f u n c t i o n h: F + Z i s c a l l e d submodular (on F) i f (2.2) h(S) + h(T) a h ( S n T) + h ( S U T )
f o r a l l S, T E F. S
T
f o r a l l crossing members of 6(s).
Let
5
:= V\S.
F.
The s e t of a l l a r c s l e a v i n g S gV i s denoted by
Then 6 f S ) c o n t a i n s t h e arcs e n t e r i n g S.
f o r A C E we d e f i n e x ( A ) := Zed\
For x
E ZE and
x(e).
E
A vector x E Z s a t i s f y i n g x ( ~ ( S ) )- x ( & ( S ) ) 4 h(S) i s c a l l e d an ( i n t e g e r - v a l u e d ) submodular f l o w . and upper bounds on t h e arcs, i . e . flow x i s called feasible, i f
e
II C
(Z&Jr-m))E,
(5
E
F)
(2.3)
With r e s p e c t t o given lower and c
E
(m{-l)E, a submodular
6 x .c< c.
L e t x be a f e a s i b l e , submodular f l o w .
A member S E: F i s c a l l e d t i g h t ( w i t h
respect t o x ) i f (2.3) holds w i t h e q u a l i t y .
Our previous n o t i o n ' s t r i c t ' i s
replaced by ' t i g h t ' ( c f . ZIMMERMANN c982b])
s i n c e ' s t r i c t ' seems t o be misunder-
standable when used f o r a n o n - s t r i c t i n e q u a l i t y .
NOW, a : 2'
+
ZZ
, defined
by
Combined objective functions on integral submolar flows
-
o ( S ) := X ( 6 ( S ) )
365
X(6(S))
( w i t h r e s p e c t t o x ) i s a modular f u n c t i o n , i . e . a(s)
f o r a l l S, T
For v
GV.
+ U(T)
n T)
=
t
o(s u
T)
(2.4)
V, l e t P ( v ) denote t h e i n t e r s e c t i o n o f a l l t i g h t s e t s
E.
S w i t h v E 5.
The a u x i l i a r y graph Gx = (V,Ex)
Ex : = E+U
E-u Eo,
contains three types o f arcs ( w i t h respect t o x),
d e f i n e d by
Et := I u v E- := { v u Eo := {uv
I I I
x ( u v ) < ~ ( u v ) , uv E E }
"forward arcs",
~ ( u v )< ~ ( u v ) , uv E E }
"backward a r c s " ,
v h P(u); u,v
E
V,U #
"red arcs".
V l
The backward a r c corresponding t o e i s denoted by Z , i . e . i f e = uv t h e n E = vu. The n o t i o n i s m a i n l y drawn from network f l o w t h e o r y . those i n FRANK [1982]
The r e d a r c s c o i n c i d e w i t h
We i n t r o d u c e p o s i t i v e , upper
w i t h reversed d i r e c t i o n .
bounds d on Ex, d e f i n e d by d ( v u ) : = ~ ( u v )- ~ ( u v )resp. d ( u v ) = c ( u v )
-
x ( u v ) on
backward r e s p . f o r w a r d arcs, and by d ( u v ) := m i n {h(S) A vector
on r e d a r c s .
EX
AX E
Z+
-
o(S)
I
S
E
F, u
E
S, v 4 Sl
i s called a circulation i f
AX(dx(i))
-
AX(dx(V)) = 0
(V c v )
where 6x(S) denotes t h e s e t o f a l l a r c s f r o m Ex l e a v i n g S c V.
6+ ( S ) and 6 0 ( S ) A nonnegative c i r c u l a t i o n i s c a l l e d ( + ) - f e a s i b l e [feasible] i f i t s a t i s f i e s t h e upper bounds on E,- [Ex]. A c i r c u l a t i o n A X i n GX E d e f i n e s a v e c t o r x ' E Z i n G by
are defined similarly.
x'(UV) := ( X @ AX)(uV) := X(UV)
t
AX(lrV)
-
AX(VU)
f o r a l l uv e E (we i n t e r p r e t Ax(uv) by 0 i f an a r c does n o t o c c u r i n Gx). F o r a g i v e n f e a s i b l e , submodular f l o w x ' , we d e f i n e
-
~'(uv) ~(uv) Ax(uv) :=
x(vu)
-
~'(vu)
[o f o r a l l uv Now, f o r Ax
E
i f ~ ' ( u v )> ~ ( u v ) , uv
E
E+,
i f x ( v u ) > ~ ' ( v u ) ,uv r
E-,
otherwise
Ex.
t
EX
Z+
, we c a l l Ax conformal
if
Ax(UV) AX(VU) = 0
U.Zimmermann
366
f o r a l l p a i r s o f forward/backward arcs which a r e d e r i v e d from t h e same a r c i n E C l e a r l y , Ax i n ( 2 . 5 ) i s conformal.
I n a previous paper we prove t h e f o l l o w i n g theorem 2.4).
theorem i n a more general form (p982b],
Theorem 2.1 L e t x, x ' be f e a s i b l e , submodular f l o w s .
Then t h e r e e x i s t s a conformal, ( ? ) f e a s i b l e c i r c u l a t i o n Ax i n Gx such t h a t x ' = x @ Ax. It i s well-known t h a t such a c i r c u l a t i o n can be decomposed i n p o s i t i v e c i r c u i t
flows, i . e . AX = Xi
where each Ci
Ax(Ci,ni)
i s a ( d i r e c t e d ) c i r c u i t i n Gx and t h e c i r c u l a t i o n A X ( C ~ , ~has ~)
constant value ni > 0 on t h e arcs o f Ci
b u t vanishes on a l l o t h e r arcs.
The
number o f c i r c u i t s i n t h a t decomposition can be bounded by t h e number o f p o s i t i v e valued arcs i n Ax.
E + U E- i s an a r c o f some c i r c u i t i n t h a t decomposiTherefore, t h e decomposition i s c a l l e d conformal, i . e . i f
I f uv
t i o n then A X ( U V )> 0.
Q
t h e forward (backward) a r c e occurs i n some c i r c u i t then t h e corresponding backward (forward) a r c
6
does n o t occur i n any c i r c u i t o f t h a t decomposition.
p a r t i c u l a r , e E. C i m p l i e s t h a t property.
6 6 C.
In
I n t h i s paper we o n l y consider c i r c u i t s w i t h
C l e a r l y , i f x and x ' a r e f e a s i b l e , submodular f l o w s w i t h x ( e ) =
x ' ( e ) f o r some e E E then no c i r c u i t i n t h e conformal decomposition o f t h e ' d i f f e r e n c e ' c i r c u l a t i o n AX can c o n t a i n e o r Z . D i f f e r e n t from network f l o w theory i t may happen t h a t x
8 Ax
i s not a feasible,
submodular flow, even i f Ax i s a f e a s i b l e p o s i t i v e c i r c u i t f l o w i n Gx.
The
f o l l o w i n g p r o p e r t y o f a c i r c u i t excludes such a behaviour. L e t ~ x ( C , r i ) be a f e a s i b l e , p o s i t i v e c i r c u i t f l o w i n Gx. C does n o t c o n t a i n consecutive r e d arcs.
W.1.0.g.
we assume t h a t
Now, we consider another graph Gc
corresponding t o C i n which t h e r e d a r c s a r e t h e v e r t i c e s and i n which two vert i c e s uv and r s are l i n k e d by an a r c (uv,rs)
i f f us i s a r e d a r c i n Gx.
We c a l l
C admissible i f GC does n o t c o n t a i n a d i r e c t e d c i r c u i t .
I n o u r above mentioned paper (p982b],
theorem 2.6) we prove t h e f o l l o w i n g theorem
Theorem -2.2 L e t x be a f e a s i b l e , submodular f l o w .
I f Ax(C,n)
i s a f e a s i b l e c i r c u i t f l o w on an
admissible c i r c u i t C i n Gx, then X @ A X i s a f e a s i b l e , submodular f l o w .
367
Combined objecrive functions on integral submolar flows C l e a r l y , a c i r c u i t f l o w Ax(C,u) capacity, i . e . 0 6
p Q
i s feasible i f
u does n o t exceed t h e minimum a r c
d(C) f o r
I
d(C) = min Ed(uv)
uv c C l .
I f d(C) i s a t t a i n e d on a backward a r c e o f C w i t h p o s i t i v e w e i g h t b ( 5 ) ( c f .
s e c t i o n 3 ) and d(C) > 1 t h e n we d e f i n e a reduced c a p a c i t y d(C) : = d(C) o t h e r w i s e d(C) : = d(C). sections
3.
-
1;
The reduced c a p a c i t y w i l l be used i n t h e f o l l o w i n g
.
COMBINED OBJECTIVE FUNCTIONS
I n a r e c e n t paper FRIESDORF and HAMACHER 1 9 8 l b ] discussed t h e m i n i m i z a t i o n o f
I n t h e f o l l o w i n g we g e n e r a l i z e
c e r t a i n o b j e c t i v e f u n c t i o n s on network f l o w s . these f u n c t i o n s i n some a l g e b r a i c c o n t e x t .
Although t h e o b j e c t i v e f u n c t i o n s
c o n s i d e r e d a r e d i f f e r e n t f r o m t h o s e i n ZIMMERMANN [1982b]
t h e discussion f o l l o w s
quite similar lines. L e t (R,t,c)
be a t o t a l l y ordered, commutative and d i v i s i b l e group w i t h n e u t r a l
element 0.
R i s assumed t o be n o n t r i v i a l , i . e . R # {Ol.
Then, R i s a t o t a l l y
o r d e r e d vectorspace o v e r t h e f i e l d Q o f t h e r a t i o n a l numbers. The e x t e r n a l comn n p o s i t i o n i s denoted i n t h e usual m u l t i p l i c a t i v e form, i . e . (Fii,a) + m a := x, where x i s t h e s o l u t i o n o f t h e e q u a t i o n n.a = m.x and n - a := a t a t . ..+a
( n times),
!E Q and f o r a l l a E R. L e t R, := { a E R I a + 01. F o r a d e t a i l e d for all ! m d i s c u s s i o n o f t h e a l g e b r a i c s t r u c t u r e s appearing h e r e and i n t h e f o l l o w i n g sect i o n s we r e f e r t o ZIMMERMANN c 9 8 1 1 .
We remark t h a t , due t o c o m m u t a t i v i t y ,
d i v i s i b i l i t y o f R can be assumed w i t h o u t l o s s o f g e n e r a l i t y . L e t W := I x
E.
t
72
I
L 6 x 6 u } where .t and u a r e t h e l o w e r and upper bounds o f t h e
u n d e r l y i n g submodular f l o w problem ( c f . s e c t i o n 2 ) . F o r g i v e n w e i g h t v e c t o r s E E a E R and b R, we c o n s i d e r t h e l i n e a r o b j e c t i v e f u n c t i o n f: W + R d e f i n e d by
f(xf
= CerE x(e).a(e)
and t h e f i x - c o s t o b j e c t i v e f u n c t i o n g: W
+
R, d e f i n e d by
c
g(x) =
b(e).
x ( e ) > f i (E 1 With r e s p e c t t o a f e a s i b l e , submodular f l o w x we i n t r o d u c e w e i g h t s i n Gx: a(e)
i f e e E,, i f e r E-, otherwise,
U.Zimmermann
368
b x ( e ) :=
E,
. b(e)
if e
-
b(e)
i f e E E-
o
otherwise
l
E
A
x ( e ) = e(e),
A
x ( e ) = e(e)
.
Then, f o r a conformal, ( + ) - f e a s i b l e , c i r c u l a t i o n
t h e weights i n Gx r e f l e c t the
AX
change o f the o b j e c t i v e f u n c t i o n values i f x i s replaced by x fx(Ax) := ZecEx gx(AX) :=
+ 1,
Ax(e).ax(e)
0 Ax.
Let
,
bx(e).
1
Ax( e)>O
Proposition 3.1 L e t x be a f e a s i b l e , submodular f l o w and l e t Ax be a conformal, c i r c u l a t i o n i n Gx w i t h conformal decomposition
AX^
(1)
f ( x @ Ax) = f ( x )
+
fx(Ax) =
f(X)
(2)
s ( x @AX) 6 g ( x )
+
!iIx(AX)
g ( x ) + ZiieI g x ( A x i ) .
Q
(+)-feasible
:= ~ x ( C ~ , r tf ~ o r) i E I. Then
fx(Axi),
t EieI
I f Ax = Ax(C,p) w i t h 0 4 p 6 i ( C ) then
g ( x 8 Ax) = g ( x ) + SX(AX).
(3)
( 1 ) i s obvious. ( 2 ) and ( 3 ) f o l l o w from the f a c t t h a t x and Ax are i n t e g r a l . With r e s p e c t t o t h e l e f t i n e q u a l i t y , t h e c o n t r i b u t i o n t o t h e change i n
Proof.
t h e value o f g i s c a l c u l a t e d e x a c t l y f o r a l l forward arcs as w e l l as f o r t h e backward arcs E(e)
t
e with
x ( e ) = L(e)
t
I f f o r some backward a r c
1.
e with
x(e)
1 we have x ( e ) = AX(^) then g(x
0 Ax)
<
g ( x ) + gx(A&
provided t h a t b ( e ) > 0. With respect t o t h e r i g h t i n e q u a l i t y , i f t h e conformal decomposition contains more than one c i r c u i t using the same forward a r c e w i t h x ( e ) = n(e), we have g(X) + gx(AX) < g(X) + Z i e ~ S,(Axi)* Contributions f o r other arcs are calculated exactly. 0c
p 6
I f Ax = Ax(C,p) w i t h
d(C) then s t r i c t i n e q u a l i t i e s can n o t occur since Ax(e) < X(e) f o r a l l
backward arcs i n C w i t h x ( e ) > e(e)
t
m
1.
The l i n e a r and t h e f i x e d - c o s t o b j e c t i v e f u n c t i o n a r e combined t o a s i n g l e object i v e f u n c t i o n F by a f u n c t i o n r: D ordered s e t .
Then F: W
+.
-+
T with D
T i s d e f i n e d by
5;
R2 where (T,<) i s a t o t a l l y
Combined objective functions on integral submolar flows
369
(x E W).
F ( x ) := r ( f ( x ) , g ( x ) )
I t i s assumed t h a t t h e domain D o f r i s l a r g e enough such t h a t F i s w e l l - d e f i n e d
P r o p o s i t i o n 3.1 shows t h a t e s t i m a t i o n s may appear i n which we e n l a r g e
on W.
t h e second component o f t h e arguemnt o f r. belong t o D.
Then t h e new argument i s assumed t o
Furthermore, r s h o u l d n o t be i n c r e a s e d by such an o p e r a t i o n .
Furthermore, we w i l l assume t h a t D i s convex. a, b a R2.
F o r d e f i n i n g c o n v e x i t y i n R2, l e t
Then [a,b]
:= {a
+
h(b-a)
I
0 6 h
6
1, A
E
QI.
L e t S E R2.
S i s c a l l e d convex w i t h r e s p e c t t o t h e vectorspace R i f [a,b]
f o r a l l a, b
E
S.
G S
Then,
' " cone ( S ) := { c hia Q i=1
I
i i a E S, h i E Q+; m EINI
i s t h e convex cone generated by S. We summarize t h e above assumptions on t h e f u n c t i o n r i n t h e f o l l o w i n g p r o p e r t y .
P r o p e r t y 3.2 (1)
D c R2 i s convex,
(2)
(cr,B)
(3)
r i s n o n i n c r e a s i n g i n i t s second argument.
E
D,
y E
R with
y >,
implies ( a , y )
E
D,
P r o p e r t y 3.2 c o n t a i n s some t e c h n i c a l assumptions which a r e e a s i l y v e r i f i e d f o r a given function.
The second p r o p e r t y assumed i s e s s e n t i a l ; i t i s i n s p i r e d by d i s -
cussions i n FRIESDORF and HAMACHER b98lbJ as w e l l as i n ZIMMERMANN p982bJ.
P r o p e r t y 3.3
D and S
Let a
C_
0.
I f r ( a ) 4 r ( b ) f o r a l l b E S then
r(a) 6 r(d)
for all
d E [a
+
cone (S-a)]
Q
n D.
T s a t i s f y i n g p r o p e r t y 3.2 and p r o p e r t y 3.3. In t h e development o f an augmenting p a t h method, p r o p e r t y 3.3 w i l l n o t s u f f i c e
We w i l l o n l y c o n s i d e r f u n c t i o n s r: D
+
and has t o be r e p l a c e d by t h e f o l l o w i n g s t r o n g e r p r o p e r t y .
ProDertv 3.4 Let a
E
D and S G D.
I f r ( a ) c. r ( b ) f o r a l l b E S t h e n
U. Zimmermann
310
r ( c ) 6 r ( d ) for all d and f o r a l l c E D. Furthermore, r ( a ) d = c
t Zks
<
[c
E
t
coneQ(S-a)]n D
r ( p ) f o r some p E S leads t o r ( c )
xb(b-a) i f X p
<
r(d) for
0.
>
In our above mentioned paper, s i m i l a r p r o p e r t i e s were assumed f o r t h e o b j e c t i v e function i t s e l f ; i n general, F w i l l not have such p r o p e r t i e s . Similarly t o t h e approach i n t h a t paper, one may formulate a general s t r a t e g y f o r t h e minimization of F which applies t o optimization prob ems d i f f e r e n t from submodular flow problems. Two necessary conditions f o r property 3 3 a r e r(a) for all a , b
E
D, h
E
r(b)
6
6
r a ) 4 r ( a t h(b-a))
+ h(b-a)
Q t with a
r(a)
=>
E
D and
r ( b ) , r ( a ) -s r ( c ) => r ( a ) 4 r ( b + c - a )
f o r a l l a , b, c e D with b+c-a f i c i e n t f o r property 3.3, t o o .
Since D i s convex, (3.1) and (3.2) a r e suf-
D.
E
(3-2)
Similarly, property 3.4 i s equivalent t o the following conditions:
for a l l a , b, d
E
==,
r(d)
r(a)
=>
r ( d ) < r(d t x(b-a))
D, i E
r(a)
r(d + x(b-a)),
r ( a ) s. r ( b ) <
r(b)
6
(3.3)
Q+ with a + h(b-a) E D and
r(b), r(a)
6
r(c)
=>
r(d)
6
r(dt(b-a)+(c-a))
r(b), r(a)
6
r(c)
=>
r(d)
<
r(dt(b-a)t(c-a))
(3.4) r(a)
c
f o r a l l a , b, c , d E D w i t h d + ( b - a ) + ( c - a ) E 0.
r: D
+
Some examples f o r functions
T a r e given i n the following theorem.
Theorem 3 . 5 Let R be a t o t a l l y ordered, commutative and d i v i s i b l e group. functions s a t i s f y property 3.2. (1)
Let r :
RxR
-F
The following
R be defined by
r ( a ) := ual + 8
-
Ya 2
f o r g i v e n fi E Q , Y E Q+ and P E R. I f R i s even a ring then allowed, too. I n both cases r s a t i s f i e s property 3.4. (2)
Let R be a f i e l d and l e t R: D
+
R be defined by
C(
E.
R, y E R,
is
Combined objective functions on integral submolar flows
f o r given
CI, B,
6 6 R , y c: R,
R2
D := {a
I
Proof.
on
aal
Then, r s a t i s f i e s property 3.3.
37 1
t
B > 0, Ya2
+ 6
> 01.
I f a = 0 then r s a t i s f i e s p r o p e r t y 3.4.
Property 3.2 i s e a s i l y v e r i f i e d .
Now we consider ( 1 ) .
r ( d ) 6 r(dtb-a)
i s equivalent t o
-
0 4 a(bl-al)
y(b2-a2).
The same equivalence holds w i t h s t r i c t i n e q u a l i t i e s . a r e independent o f d. Now we consider ( 2 ) .
Both derived i n e q u a l i t i e s Thus, (3.3) and (3.4) a r e i m p l i e d .
r ( a ) 6 r ( b ) i s equivalent t o a ( v a 2 t 6 ) ( bl-al ) a Y (aaltB) ( b2-a2).
Thus, (3.1) and (3.2) are implied.
L e t a = 0.
Then, r ( d ) 6 r(d+b-a) i s equiv-
alent t o
0
>/
vB(b2-a2)
and t h e same equivalence holds w i t h s t r i c t i n e q u a l i t i e s . t i e s are independent o f d.
Both d e r i v e d i n e q u a l i -
Thus (3.3) and (3.4) a r e implied.
8
We remark t h a t t h e domain D o r r i n Theorem 3.2(2) i m p l i e s a bound on f. example, i f a > 0 then f(x)
For
' -@/a
f o r a l l x E W must be s a t i s f i e d i n view o f a proper d e f i n i t i o n o f F.
That bound
r e s t r i c t s t h e p o s s i b l e choice o f t h e c o e f f i c i e n t v e c t o r d e f i n i n g f. I f we assume t h a t R i s Archimedean then R i s a dense subgroup o f t h e a d d i t i v e group of r e a l numbers.
Let
R: D + W w i t h D E. R2 s a t i s f y (3.2).
e q u i v a l e n t t o quasi-concavity,
Then, (3.1) i s
i.e. (3.1)'
r ( a ) 4 r ( b ) => r ( a ) 6 r ( d ) f o r a l l a, b E: D and f o r a l l d s t r i c t quasiconcavity, i.e.
[a,b].
A necessary c o n d i t i o n f o r (3.3) i s If r satisfies
(3.1)' w i t h s t r i c t i n e q u a l i t i e s .
(3.4) and ( 3 . 1 ) ' then (3.3) i s e q u i v a l e n t t o r ( a ) < r ( b ) => r ( c ) < r(c+d-a) f o r a l l a, b, c
E
D and f o r a l l d c [a,b]
i n t e r v a l s are defined w i t h respect t o Q.
such t h a t ctd-a e D.
We remind t h a t
U.Zimmermann
372 4.
M I N I M I Z A T I O N USING NEGATIVE CIRCUITS
We consider the m i n i m i z a t i o n o f combined o b j e c t i v e f u n c t i o n s over P and Pe(a) where P denotes the s e t o f a l l f e a s i b l e submodular f l o w s and Pe(a):= { x x(e) = al f o r given a r c e E E and g i v e n cx e n + . We remind t h a t backward a r c corresponding t o e, i . e .
e:=vu i f e=uv.
e
E
P
I
denotes t h e
C l e a r l y , Pe(a) i s again a
submodular f l o w polyhedron which i s d e r i v e d from P by r e d e f i n i n g t h e lower and upper bounds on e, i . e . a ( e ) := u ( e ) : = a. Then, f o r x i n g a u x i l i a r y digraph G,(a)
i s d e r i v e d from G,
E
Pe(a), t h e correspond-
by d e l e t i n g e and
ience we assume t h a t P resp. Pe(a) are nonempty.
For F: W
F(x) := r ( f ( x ) , g ( x ) )
(x
+
s.
For conven-
T, d e f i n e d by
+w
we discuss min I F ( x ) min I F ( x )
I 1
x
E
Pl,
(4.1)
x B Pe(a)l.
(4.2)
I n t h i s s e c t i o n we assume t h a t r s a t i s f i e s p r o p e r t y 3.2 and p r o p e r t y 3.3. S t a r t i n g w i t h a f e a s i b l e s o l u t i o n x the proposed method c o n s i s t s i n t h e e l i m i n a t i o n o f negative c i r c u i t s i n Gx. F(x
A c i r c u i t C i n Gx i s c a l l e d negative, i f
8Ax)
<
F(x)
f o r a x = &x(C,l) and, i n t h e case o f ( 4 . 2 ) , e,
6
f C.
D i f f e r e n t from network
f l o w theory i t i s p o s s i b l e t h a t x 8 Ax i s n o t a f e a s i b l e s o l u t i o n .
Nevertheless,
i f a negative c i r c u i t e x i s t s , then we w i l l show t h a t we can improve F using
negative c i r c u i t s o f a s p e c i a l f o r m . Due t o the i n t e g r a l i t y o f t h e upper bounds i n Gx, Ax(C,l) flow f o r any c i r c u i t C i n Gx.
i s a feasible c i r c u i t
I f Gx contains a negative c i r c u i t , then i t con-
t a i n s a negative c i r c u i t a d m i t t i n g o n l y such r e d s h o r t - c u t t i n g a r c s which do n o t lead t o a negative c i r c u i t .
We c a l l such a c i r c u i t a s h o r t n e g a t i v e c i r c u i t .
I n p a r t i c u l a r , a n e g a t i v e c i r c u i t of s h o r t e s t l e n g t h i s a s h o r t negative c i r c u i t .
P r o p o s i t i o n 4.1 (1)
and F ( x ' )
(2)
I f C i s a s h o r t n e g a t i v e c i r c u i t i n Gx then x ' := x @ A x ( c , l ) E P
L e t x E P.
Let x
E
<
F(x).
Pe(a).
x ' := x@sx(C,l) Proof.
I f C i s a s h o r t n e g a t i v e c i r c u i t i n Gx w i t h e,
e Pe(a) and F ( x ' )
( 2 ) i s a consequence o f ( 1 ) .
<
e f C,
then
F(x).
Due t o theorem 2.2 i t s u f f i c e s t o show t h a t
373
Combined objective functions on integral submolar flows
C l e a r l y , a s h o r t n e g a t i v e c i r c u i t C does n o t c o n t a i n consecu-
C i s admissible.
We assume t h a t C i s n o t a d m i s s i b l e .
t i v e r e d arcs.
K ( c f . S e c t i o n 2).
Each a r c o f
Then, Gc c o n t a i n s a c i r c u i t
K together w i t h a s u i t a b l e p a r t o f C defines a
i E I , i n Gx. These c i r c u i t s c o v e r each non-red a r c f r o m C t h e same number o f times, say s t i m e s . L e t Axi := Ax(Ci,l), i I and l e t Ax = A X ( c , l ) . Now c i r c u i t Ci,
(i E I),
:= g ( X 8 A X i )
Bi
-
1-
1
b(E) x(€)=n(E) i’
g(x) =
ErC+
fl E,-
w i t h C i := Ci
E d i
-
( i E I),
,x(E)=e(E)+l
f o r i E I , and
imp1 i e s Sa =
C
a.,
ieI Then, F ( x
0 Axi)
>,
SB
=
2
i€1
B
i’
F ( x ) , i E I , which means r(f(x)+aiyg(x)+Bi)
f o r a l l i E I.
’
3 r(f(x).g(x))
P r o p e r t y 3.3 l e a d s t o r(f(x)+a,g(x)+8)
contrary t o F(x @Ax) < F(x).
a r ( f ( x ) ,g(x))
Thus C i s a d m i s s i b l e .
I
P r o p o s i t i o n 4.1 shows how t o improve F i f a n e g a t i v e c i r c u i t C i n Gx e x i s t s . Once a n e g a t i v e c i r c u i t i s determined we check f o r r e d s h o r t - c u t t i n g a r c s . Using such s h o r t - c u t t i n g s we f i n d a s h o r t n e g a t i v e c i r c u i t which y i e l d s a new f e a s i b l e s o l u t i o n improving F. On t h e o t h e r hand, i f Gx does n o t c o n t a i n n e g a t i v e c i r c u i t s t h e n we have t o p r o v e t h a t the current feasible solution i s optimal.
P r o p o s i t i o n 4.2 (1)
Let x
E
P.
I f Gx c o n t a i n s no n e g a t i v e c i r c u i t t h e n x i s an o p t i m a l s o l u t i o n
o f (4.1). (2)
Let x
E
Pe(,).
I f Gx c o n t a i n s no n e g a t i v e c i r c u i t C w i t h e ,
an o p t i m a l s o l u t i o n o f (4.2).
efC
then x i s
374
U. Zimmermann
Proof. 2.1, Let a
( 2 ) i s a consequence o f ( 1 ) .
t h e r e e x i s t s a conformal,
AX^
L e t x ' be a f e a s i b l e s o l u t i o n .
:= ~ x ~ ( C ~ , ni~E) I, , denote a conformal decomposition o f Ax.
:= f x ( A X i )
i 3.2 lead t o
and l e t Bi
we assume
TI. 1
Let
:= gx(AXi), f o r a l l i 6 I. P r o p o s i t i o n 3.1 and p r o p e r t y
F ( x ' ) = F ( x 8 A X ) b r ( f ( x ) + hi, g ( x ) W.1.o.g.
By theorem
( + ) - f e a s i b l e c i r c u l a t i o n AX i n Gx w i t h x ' = x @ AX
= 1 f o r a l l i E I.
+
ZB~)
Since Gx does n o t c o n t a i n a negative
circuit
+
r(f
Thus, p r o p e r t y 3.3 i m p l i e s
F(x
r ( f ( x ) + a i .g(x)+Bi) for a l l i6 I.
P r o p o s i t i o n 4.2 shows t h a t ' l o c a l o p t i m a l i t y ' c i r c u i t s i n Gx,
8
, i.e.
i s a s u f f i c i e n t condition f o r
t h e non-existence o f n e g a t i v e
global o p t i m a l i t y ' .
Together
w i t h p r o p o s i t i o n 4.1 we f i n d the f o l l o w i n g r e s u l t .
Theorem 4.3 (1)
Let x a P.
Then x i s an optimal s o l u t i o n o f (4.1) i f and o n l y i f Gx con-
t a i n s no negative c i r c u i t . (2)
L e t x c Pe(a).
Then x i s an optimal s o l u t i o n o f (4.2) i f and o n l y i f Gx
contains no n e g a t i v e c i r c u i t C w i t h e,
e $ C.
Sumnarizing, we conclude t h e v a l i d i t y o f t h e f o l l o w i n g method f o r s o l v i n g (4.1) resp. (4.2). Negative C i r c u i t Method 4.4 1.
F i n d x e P [resp.
2.
I f Gx contains no negative c i r c u i t p i t h e,
3.
F i n d a s h o r t negative c i r c u i t C i n Gx b i t h e,
x E Pe(a)].
e & C],
stop.
5 b q;
x : = x @Ax(C,1) and
go t o 2. An i n i t i a l f e a s i b l e f l o w i n step 1 can be determined using an 'augmenting' c i r c u i t method described i n ZIMMERMANN [1982b]
o r u s i n g t h e method o f FRANK 0982).
For many o b j e c t i v e f u n c t i o n s , we can e f f i c i e n t l y determine n e g a t i v e c i r c u i t s by i n t r o d u c i n g weights on t h e a r c s o f Gx.
I n p a r t i c u l a r , such an approach i s pos-
s i b l e f o r t h e o b j e c t i v e f u n c t i o n s i n theorem 3.5. F o r example, we consider r: R,
-+
R d e f i n e d by
Combined objective functions on integral submolar flows
375
“al + B r ( a ) : = ___ ya2 + 6 f o r given x
r
B , y, 6 e R,
a,
on a t o t a l l y o dered f i e l d .
P w i t h 0 < a f ( x ) + B =: u, 0 < y g ( x ) + 6 =: v.
L e t f,g
2,
0 on W and l e t
Then C i s a n e g a t i v e c i r c u i t i n
Gx i f f (4.3)
CECC S ( E
where S(E)
for all
E E
w i t h (4.3),
Ex.
:= a v a X ( E
The d e t e c t i o n o f c i r c u i t s w i t h n e g a t i v e ( l i n e a r ) w e i g h t , i . e .
i n a d i g r a p h i s a w e l l s o l v e d s t a n d a r d problem.
arguments as i n ZIMMERMANN n982b],
Then, u s i n g s i m i l a r
we f i n d t h a t s t e p s 2 and 3 a r e o f p o l y n o m i a l
C l e a r l y , i f F i s bounded from below, t h e n t h e n e g a t i v e c i r c u i t
complexity.
method i s f i n i t e .
A polynomial bound on t h e number o f s t e p s i s n o t known, and i n view o f t h e well-known p a r t i c u l a r case o f t h e minimum cost-max f l o w problem,
some k i n d of s c a l i n g argument may be necessary, as a l r e a d y mentioned i n FRANK
.
[1982]
I n s t e p 3 o f t h e n e g a t i v e c i r c u i t method we change x by a c i r c u i t f l o w Ax = Ax(C,ri)
with
11
= 1.
f l o w p r o v i d e d t h a t Ax(C,n)
By theorem 2.2,
x @ ax(C,u)
i s a f e a s i b l e , submodular
i s f e a s i b l e , i . e . provided t h a t
minimum a r c c a p a c i t y of C i n G
On t h e o t h e r hand, we can n o t be s u r e t h a t t h e
X’
o b j e c t i v e f u n c t i o n improves f o r
does n o t exceed t h e
11
11
# 1.
An example can be found i n FRIESDORF and
They propose t o assume m o n o t o n i c i t y o f r i n i t s f i r s t argument.
HAMACHER [198lb].
Then, even i n o u r more general s i t u a t i o n ,
11
can be chosen i n an o p t i m a l way.
We remind t h a t t h e reduced c a p a c i t y d ( C ) o f a c i r c u i t C i n G
X
i s e i t h e r equal t o
d(C) o r , i f d(C) > 1 and i f d(C) i s a t t a i n e d on a backward a r c e w i t h p o s i t i v e b ( e ) , i s equal t o d(C) - 1 .
P r o p o s i t i o n 4.5 Let x (1)
E
P [x
and l e t C be a n e g a t i v e c i r c u i t i n Gx [with e,
I f r i s n o n i n c r e a s i n g i n i t s f i r s t argument then and
(2)
Pe(a)]
E
17
:=
d(C)
ri
:=
d ( C ) otherwise.
Then F ( x @ AX(c,n)) \< F(X for all
o
6
u
4
q.
rl
:= 1 i f f x ( A x ( C , l ) )
6 0
rl
:= 1 i f f x ( A X ( c , l ) )
& O
otherwise.
I f r i s nondecreasing i n i t s f i r s t argument then and
54
i i ( ~ )p , i n t e g r a l .
@ Ax(c,u))
U.Zimmermann
316 By p r o p o s i t i o n 3.1,
Proof.
0
<
u c
b(C).
g ( x @ Ax(C,u)) i s o f constant value f o r a l l
Thus, f(x
8 AX(c,u))
i m p l i e s the o p t i r n a l i t y o f t h e choice o f
5.
. fx(Ax(c,l))
+ u
= f(x)
i n a l l cases.
TI
8
MINIMIZATION USING AUGMENTING C I R C U I T S
I n Section 4 we s o l v e t h e m i n i m i z a t i o n problems (4.1) and (4.2) u s i n g successive I n o r d e r t o b e g i n t h e procedure an a r b i t r a r y ,
elimination o f negative c i r c u i t s .
f e a s i b l e s o l u t i o n o f the r e s p e c t i v e problem i s necessary.
I n t h i s section, we
assume t h a t t h e s t a r t i n g s o l u t i o n x i s a l r e a d y optimal among a l l f e a s i b l e ,
E.
submodular flows i n Pe(a) w i t h a = x ( e ) f o r some e E ward a r c corresponding t o e. c i r c u i t s C w i t h e,
e 4 C.
Then, by Theorem 4.3,
e
denote t h e back-
x w i l l be c a l l e d a f l o w o f minimum c o s t ( w i t h respect
We want t o preserve t h a t p r o p e r t y when changing t h e c u r r e n t f l o w .
to F).
Clearly, i f e E C , decreased on e.
then x @ A x ( C , l )
mal c i r c u i t f l o w s .
e
E C,
it i s
We remind t h a t we consider o n l y c i r c u i t s l e a d i n g t o c o n f o r I n particular, e
P. An augmenting c i r c u i t
E
C implies
6 4
C.
A l l r e s u l t s can e a s i l y
e C, e & C.
be c a r r i e d over t o c i r c u i t s w i t h E.
i s increased on e and, i f
We w i l l o n l y discuss t h e case e eC. Then C i s c a l l e d an
augmenting c i r c u i t .
Let x
Let
does n o t c o n t a i n negative
G,
?
i n Gx i s c a l l e d o f minimum weight i f
F(x @ Ax(t,1)) 6 F(x @ Ax(c,1)) f o r a l l augmenting c i r c u i t s
c
i n G.,
If
2
i s an augmenting c i r c u i t o f minimum
weight then we can e a s i l y c o n s t r u c t an augmenting c i r c u i t C o f minimum weight a d m i t t i n g o n l y such r e d s h o r t - c u t t i n g arcs which do n o t l e a d t o an augmenting c i r c u i t o f minimum weight w i t h l e s s l e n g t h .
S i m i l a r l y t o short negative c i r c u i t s ,
we c a l l C a s h o r t augmenting c i r c u i t o f minimum weight. Unfortunately, f o r
F s a t i s f y i n g ' p r o p e r t y 3.2 and p r o p e r t y 3.3 i t may happen t h a t
augmentation o f a f e a s i b l e f l o w x o f minimum c o s t u s i n g a s h o r t augmenting c i r c u i t o f minimum weight leads t o a f e a s i b l e f l o w minimum c o s t . QSSlb].
X@AX
which i s n o t a f l o w o f
An example f o r network f l o w s i s g i v e n i n FRIESDORF and HAMACHER
Therefore, we assume i n t h i s s e c t i o n t h a t r s a t i s f i e s p r o p e r t y 3.2 and
t h e stronger p r o p e r t y 3.4.
P r o p o s i t i o n 5.1 Let x
6
P be o f minimum cost.
I f C i s a s h o r t augmenting c i r c u i t o f minimum
371
Combined objective functions on integral submolar flows
w e i g h t i n Gx t h e n x @ Ax(C,v)
i s a f e a s i b l e , submodular f l o w f o r a l l i n t e g r a l
u,
0 4 p 6 d(C). Proof.
i t s u f f i c e s t o prove t h a t C i s admissible.
Due t o Theorem 2.2,
Otherwise i = 1, k i n Gx c o v e r i n g each non-red a r c f r o m C e x a c t l y s times, s 6 k . I n p a r t i c u l a r , c1 ,. , .,c, a r e augmenting c i r c u i t s ar;d e, 6 Ci, s + l \< i 6 k. We d e f i n e
...,
as i n t h e p r o o f o f p r o p o s i t i o n 4.1 we f i n d c i r c u i t s Ci,
e
cx : =
fx(Ax(C,l)),
B := g x ( A x ( C , l ) )
a l l i. Then s ( a , b ) = zi
(ai,Bi)
and ai
:= fx(Ax(Ci,l)),
Bi
:= gx(Ax(Ciyl))
for
and
r(f(x)tai,g(x)+bi)
> r(f(x)+a,g(x)+b),
i = 1,2,...,~,
r ( f ( x ) + a i ,g(x)+Bi
+ r ( f ( x ) ,g(x)) ,
i = s+l,
...,k.
Since 0 i s convex, we f i n d bi
:= ( f ( x )
bi := ( f ( x )
>
Now
+
t
1 $atai),g(x)
1 1 + ;Ia,g(x) + 7B) e
and a : = ( f ( x ) r(bi)
+ y1i , g ( X )
r ( a ) f o r i = 1,2
0.
,... ,s
1
7Ei)
+
i = 1,2,...,s
d D
1 $B+Bi))
eD
..., k
i = Stl,
T h e r e f o r e (3.3) i m p l i e s
and r ( b i )
k
i l S ( b -a) = C (ai-a,Bi-B) i=1 i=1 C
3
t
r ( a ) f o r i = s+l, 1
'
...,k .
k
C (ai,Bi) i=s+l
= 0
t o g e t h e r w i t h p r o p e r t y (3.4) l e a d s t o t h e c o n t r a d i c t i o n r ( a ) > r ( a ) .
m
P r o p o s i t i o n 5.1 shows t h a t augmentation l e a d s t o a new f e a s i b l e s o l u t i o n .
For
c o n s e r v a t i o n o f t h e minimum c o s t - p r o p e r t y i t i s o f t e n necessary t o r e s t r i c t t h e choice o f p t o 1.
Let
i(c)
i f g x ( A x ( c , l ) ) \< 0
d(C) := otherwise f o r an augmenting c i r c u i t C i n Gx.
C l e a r l y , d(C) > 0.
Theorem 5.2 Let x
E
P be o f minimum c o s t .
w e i g h t i n Gx t h e n x @Ax(C,u) a l l integral p , 0 6 Proof.
LI
I f C i s a s h o r t augmenting c i r c u i t o f minimum i s a f e a s i b l e , submodular f l o w o f minimum c o s t f o r
6 d(C).
L e t x ' E Pe(X) w i t h A = x ( e ) t p .
By Theorem 2.1, t h e r e e x i s t s a
U. Zimmemann
378
conformal f l o w Ax i n Gx w i t h x ' = x @ Ax. a conformal decomposition o f
AX
with
L e t Axi
1
1
x
1
1 f o r a l l i. W.1.o.g.
rl. = 1
denote t h e augmenting c i r c u i t s i n t h e d e c o m p o s i t i o n ( u 6 k ) . E . = g ( A X . ) f o r a l l i.
,..., k be C1 ,..., Cu
i = 1,2
= Ax(C.,rl.),
Let
let ai
= fx(axi),
1
Then
+
r(f(x)+cci , 9 ( ~ ) + 6 ~ ) r ( f ( x ) , s ( x ) ) for all i
>
u.
P r o p e r t y (3.4) i m p l i e s r(f(x) +
1 ai,g(x) i> V
+
E
i> u
Bi)
3
r(f(x),g(x)).
The c o n s i d e r e d arguments b e l o n g t o D s i n c e e v e r y s u b s e t o f t h e c o n s i d e r e d c i r c u i t s leads t o a f e a s i b l e f l o w ( n o t n e c e s s a r i l y submodular). we can s h i f t t h e i n e q u a l i t y f r o m ( f ( x ) , g ( x ) )
to (f(x)
By p r o p e r t y 3.4
+ z
ai,g(x) idu
+ .z B i ) . l
Therefore
k
k
The c o r r e s p o n d i n g f l o w s have v a l u e
x
on e.
+
E ai,g(x) i6u
P r o p o s i t i o n 3.1 and p r o p e r t y 3.2
imply F(X') b r ( f ( x ) On t h e o t h e r hand, l e t f x ( A x ( C , l ) )
t
= a, gx(AX(C,l
Z Bi).
ia ) = B.
Then r ( f ( x ) t a i ,g( x)+Bi ) >, r ( f ( x ) +a .g( x)+B) f o r a 1 i<. u . A p p l y i n g p r o p e r t y 3.4 w i t h a = ( f ( x ) t a , g ( x ) + B ) , w i t h i = f(x)+Ui.g(x)+Bi), i 6 P, w i t h c = ( f ( x ) + u a , g ( x ) + u B ) and w i t h Xi
b
= 1, i \< u
1eads t o r(f(x)
+ c
ui,g(x) isu
+
I: E ~ b) r ( f ( x ) + u a , g ( x ) + u a ) . i6u
r(f(x)+uu,g(x)+vE) (5.1) and ( 5 . 2 ) show t h a t x @ Ax(C,u)
3
F ( x @ Ax(C,u)).
(5.2)
i s a f e a s i b l e , submodular f l o w o f minimum
cost. Theorem 5.2 shows hoe t o c o n s t r u c t f e a s i b l e , submodular f l o w s x o f minimum c o s t w i t h increasing x(e).
Flows x w i t h d e c r e a s i n g x ( e ) a r e determined i n t h e same
manner u s i n g c i r c u i t s C w i t h
C
r C .
Combined objective functions on integral submolar flows Augmenting c i r c u i t method f o r (4.2)
5.3 1.
L e t B 6 a. F i n d x E Pe(B) o f minimum c o s t .
2.
F i n d a s h o r t augmenting c i r c u i t C o f minimum weight i n Gx; X
3.
379
:= X@AX(c,,);
u
:= min (d(C),a-@
8 := Btp.
I f B = a stop. Otherwise go t o 2.
The augmenting c i r c u i t method can be m o d i f i e d f o r s o l v i n g (4.1). ience, we assume
:= a(e) >
We s t a r t w i t h x E P,(B)
o f minimum c o s t .
c i r c u i t then x i s o p t i m a l . either e E C or
6 a C.
For conven-
-m.
I f Gx does n o t c o n t a i n a n e g a t i v e
I f Gx c o n t a i n s a s h o r t n e g a t i v e c i r c u i t C then
W.1.o.g.
we assume e e C.
Let
L e t Ax = Ax(C,l),
c i r c u i t o f minimum weight i n Gx. F ( x @ A):
6
Ax' =
be a s h o r t augmenting Ax(t,1).
Then
F(x @ Ax) < F ( x ) .
we determine a s h o r t augmenting c i r c u i t C ' o f minimum weight.
I n Gx
Ax' = A x ( c ' , l )
and l e t
:= x @ A i .
F(X then x
If
0 Ax')
6 ~i i s optimal , s i n c e G
Let
3
F(X )
Ai does n o t
c o n t a i n negative c i r c u i t s .
p a r t i c u l a r , t h e r e i s no n e g a t i v e c i r c u i t C w i t h
6 e C as F(x 8 A i )
<
In
F(x).
Thus
we have proved t h e v a l i d i t y o f t h e f o l l o w i n g method f o r s o l v i n g (4.1).
5.4
Augmenting c i r c u i t method f o r (4.1)
--.
F i n d x ~l Pe(B) o f minimum c o s t .
1.
L e t B := a(e) >
2.
F i n d a s h o r t augmenting c i r c u i t C o f minimum weight i n Gx; i f C i s nonnegative, stop.
3.
F i n d an i n t e g r a l P , 0 < u 6 d ( C ) , d e f i n e d by F(x
@ Ax(c,p))
4 F(x @ A x ( c J ) )
f o r a l l 0 < A 6 d(C).
4. x
:= x @ A X ( C , ~ )and go t o 2
Both methods, 5.4 as w e l l as 5.3, cost.
can s t a r t w i t h an a r b i t r a r y f l o w o f minimum
The necessary m o d i f i c a t i o n s a r e obvious.
negative c i r c u i t method can be used.
I n order t o f i n d x
E
P,(B)
the
For determining s h o r t augmenting c i r c u i t s
o f minimum weight i t w i l l o f t e n be p o s s i b l e t o use standard a l g o r i t h m s a f t e r i n t r o d u c i n g weights on t h e a r c s o f Gx.
I n p a r t i c u l a r , such an approach i s known
f o r those o b j e c t i v e f u n c t i o n s i n Theorem 3.5 which s a t i s f y p r o p e r t y 3.4.
V. Zimmermann
380
Then, s i m i l a r t o reasoning i n Section 4, a l l steps i n methods 5.31'5.4 w i t h t h e exception o f t h e i n i t i a l step a r e p d y n o m i a l l y bounded. steps i s f i n i t e .
C l e a r l y , t h e number o f
D i f f e r i n g from t h e n e g a t i v e c i r c u i t method, where such a bound
is t h e d i f f e r e n c e between t h e i n i t i a l and t h e o p t i m a l o b j e c t i v e f u n c t i o n value, a bound i s t h e d i f f e r e n c e a - 6 .
Thus, i n t h e case o f f i n i t e c a p a c i t i e s , an a
p r i o r i bound i s known f o r t h e number o f the steps o f t h e augmenting p a t h method.
6.
CONCLUDING REMARKS
S i m i l a r methods can be developed along t h e same l i n e s f o r combining t h e l i n e a r o b j e c t i v e f u n c t i o n f w i t h o t h e r f i x - c o s t o b j e c t i v e f u n c t i o n s , f o r example w i t h
Some examples f o r combined o b j e c t i v e f u n c t i o n s on network f l o w s ( a . = 0) a r e considered i n FRIESDORF and HAMACHER p98lb-J. r := R:
+
They discuss r e a l v a l u e d f u n c t i o n s
R.and g i v e i n t e r p r e t a t i o n s o f f ( x ) / g ( x ) , f ( x )
-
g(x), f ( x )
+
g(x).
C l e a r l y , t h e i r examples can be formulated f o r any s p e c i a l c o m b i n a t o r i a l s t r u c t u r e
from the c l a s s o f submodular f l o w problems. I n p a r t i c u l a r , f o r t h e maximization o f a l i n e a r o b j e c t i v e f u n c t i o n on t h e s e t o f f e a s i b l e submodular flows, t h e augmenting c i r c u i t method 5.4
i s an e x t e n s i o n of
t h e s h o r t e s t augmenting p a t h methods i n network f l o w theory as w e l l as i n ( p o l y - ) m a t r o i d i n t e r s e c t i o n theory.
REFERENCES Edmonds, J . and G i l e s , R., A min-max r e l a t i o n f o r submodular f u n c t i o n s on graphs, Ann. D i s c r e t e Math. 1 (1977) 185-204. Frank, A., An a l g o r i t h m f o r submodular f u n c t i o n s on graphs, Ann. D i s c r e t e Math. 16 (1982) 97-120. F r i e s d o r f , H. and Hamacher, H., A n o t e on weighted minimal c o s t flows, Z e i t s c h r i f t f u r Operations Research 25 (1981a) 45-47. F r i e s d o r f , H. and Hamacher, H., Weighted rnin c o s t flows, European Journal o f Operational Research 11 (1982) 181-192. F u j i s h i g e , 5. , Algorithms f o r s o l v i n g t h e independent-flow problem, J . Operations Res. SOC. Japan 21 (1978) 189-204. Hassin, R.,
On network flows, Ph.D. Thesis, Yale U n i v e r s i t y (1978).
Hassin, R., Generalizations o f Hoffman's existence theorem f o r c i r c u l a t i o n s , Networks 11 (1981) 243-254.
Combined objective functions on integral submolar flows
381
[8]
Lawler, E.L. and M a r t e l , C.U. , Computing maximal " p o l y m a t r o i d a l " network flows, Math. o f Oper. Res. 7 (1982) 334-347.
[9]
Zimmermann, U. , L i n e a r and combinatorial o p t i m i z a t i o n i n ordered a l g e b r a i c s t r u c t u r e s , Ann. o f D i s c r e t e Math. 10 (1981).
[lo]
Zimmermann, U., M i n i m i z a t i o n o f some n o n l i n e a r f u n c t i o n s over p o l y m a t r o i d a l network flows, Ann. D i s c r e t e Math. 16 (1982a) 287-309.
[ll]
Zimmermann, U., M i n i m i z a t i o n on submodular flows, D i s c r e t e Appl. Math. 4 (1 982b) 303-323.
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