Linear and Combinatorial Optimization in Ordered Algebraic Structures (Annals of discrete mathematics, Volume 10)

ANNALS OF DISCRETE MATHEMATICS

mathematics Monaging Editor Peter L. HAMMER, University of Waterloo, Ont., Canada

Advisory Editors C. BERGE, UniversitC de Paris, France M.A. HARRISON, University of California, Berkeley, CA, U.S.A. V. KLEE, University of Washington, Seattle, WA, U.S.A. J.H. VAN LINT, California Institute of Technology, Pasadena, CA, U.S.A. (3.4’. ROTA, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.

ANNALS OF DISCRETE MATHEMATICS

10

LINEAR AND COMBINATORIAL OPT1MIZATION IN ORDERED A113 EBRAIC STRUCTURES

U. ZIMMERMANN Mathematisches Institut Universitat zu Koln

8 NORTHHOLLAND

PUBLISHING COMPANY - 198 1

All Rights mservui. No p ~ r of t this publimtion may b mprodurrd. s t o d in a mtrievalsystem. or transmitted, in any form or by any mtuns, electronic, mechanical. photmpying. recording or otherwise, without the prior permision of the copyright owner. Submission to this journal of a paper entails the author's irrevocableand exclusive authorization of the publisher to collect any sums or considerotions for copying or mpmduction payable by third parties (as mentioned in article I7pamgroph 2 of the Dutch Copyright Act of 1912 and in the Royal k r e e of June 20, 1974 (S.351) pumant to article 166 ofthe Dutch Copyright Act of 1912) and/or to act in or out of Court in connection themwith.

PREFACE

The object of this book is to provide an account of results and methods for linear and combinatorial optimization problems over ordered algebraic structures. In linear optimization the set of feasible solutions is described by a system of linear constraints; to a large extent such linear characterizations are known for the set of feasible solutions in combinatorial optimization, too. Minimization of a linear objective function subject to linear constraints is a classical example which belongs to the class of problems considered. In the last thirty years several optimization problems have been discussed which appear to be quite similar. The difference between these problems and classical linear or combinatorial optimization problems lies in a replacement of linear functions over real (integer) numbers by functions which are linear over certain ordered algebraic structures. This interpretation was not apparent from the beginning and many authors discussed and solved such problems without using the inherent similarity to linear and combinatorial optimization problems. Therefore results and methods which are wellknown in the theory of linear and combinatorial optimization have been reinvented from time to time in different ordered algebraic structures. In this book we describe algebraic formulations of such problems which make the relationship of similar problems more transparent. Then results and methods in different algebraic settings appear to be instances of general results and methods. Further specializing of general results and methods often leads to new results and methods for a particular problem. We do not intend to cover all optimization problems which can be treated from this point of view. We prefer to select classes of problems for which such an algebraic approach turns out to be quite useful and for which a common theory has been developed. Therefore nonlinear problems over ordered algebraic structures are not discussed; at the time being very few results on such problems are known. We assume that the reader of this book is familiar with concepts from classical linear and combinatorial optimization. On the other hand we develop the foundations of the theory of ordered algebraic structures covering all results which are used in the algebraic treatment of linear and combinatorial optimization problems. This first part of the book is self-contained; only some representation theorems are mentioned without explicit proof. These representation theorems are not directly applied in the investigation of the optimization problems considered, but provide a general view of V

vi

Preface

the scope of the underlying algebraic structure. In the second part of the book we develop theoretical concepts and solution methods for linear and combinatorial optimization problems over such ordered algebraic structures. Results from part one are explicitly and implicitly used throughout the discussion of these problems. The following is an outline of the content of the chapters in both parts. In chapter one we introduce basic concepts from the theory of ordered sets and lattices. Further we review basic definitions from graph theory and matroid theory. Chapter two covers basic material on ordered commutative semigroups and their relationship with ordered commutative groups. Lattice-ordered commutative groups are discussed in chapter three. Without proof we expose the embedding theorems of CONRAD, HARVEY and HOLLAND [ 19631 and of HAHN [ 19071 . Chapter four contains detailed characterizations of linearly ordered commutative divisor semigroups. In particular, we prove the decomposition theorems of CLIFFORD ( [1954] , 119581, [ 19591 ) and LUCOWSKI [ 1964 ] . The discussion of the weakly cancellative case is of particular importance for the applications in part two of the book. Many examples complete this chapter. In chapter five we introduce ordered semimodules generalizing ordered rings, fields, modules and vectorspaces as known from the theory of algebra. Basic definitions of a matrix calculus lead to the formulation of linear functions and linear constraints over ordered semimodules. Chapter six contains a discussion of linearly ordered semimodules over real numbers. We provide the necessary results for the development of a duality theory for algebraic linear programs. Chapter seven is an introduction to part two. Chapter eight covers algebraic path problems. We develop results on the stability of matrices and methods for the solution of matrix equations generalizing procedures from linear algebra. Several classes of problems are considered; the shortest path problem is covered as well as the determination of all path values. Chapter nine contains algebraic eigenvalue problems which are closely related to algebraic path problems. In chapter ten cxtremal linear programs are A weak duality theorem in ordered semimodules leads to explicit considered. solutions in the case of extremal inequalities. The case of equality constraints can be treated using a threshold method. In chapter eleven we develop a duality theory for algebraic linear programs. Such problems are solved by generalizations of the simplex method of linear programming. We discuss primal as well as primal dual procedures. As in classical linear optimization duality theory is a basic tool for solving combinatorial optimization problems over ordered algebraic structures. Chapter twelve covers solution methods for algebraic flow problems. We develop a generalization of the primal dual solution method of FORD and FULKERSON. Clearly primal methods can be derived from the simplex method in chapter twelve. We discuss several methods for the solution of algebraic transportation and assignment problems. Chapter thirteen contains solution methods for algebraic optimization problems in independence systems; in particular, we investigate algebraic matroid and 2-matroid intersection problems. Algebraic matching problems can be solved similarly; both

Preface

vii

classes of problems are mainly differing by their combinatorial structure whereas the generalization of the algebraic structure leads to the same difficulties. In writing this book I have benefited from the help of many people. In particular, I want to express my gratitude to Professor Dr. R.E. Burkard, who encouraged my work throughout many years. A great deal of the material on algebraic combinatorial optimization problems originated from collaborative work with him in the Mathematical Institute at the University of Cologne. He made it possible for me to visit the Department of Operations Research at Stanford University. Valuable discussions in Stanford initiated investigations of related lattice-ordered structures. I am much indebted to the Deutsche Forschungsgemeinschaft, Federal Republic of Germany, for the sponsorship of my stay in Stanford. The manuscript was beautifully typed by Mrs. E. Lorenz.

U.Z. Cologne

This Page Intentionally Left Blank

TABLE OF CONTENTS

PREFACE

V

PART I : ORDERED ALGEBRAIC STRUCTURES 1. Ordered Sets, Lattices and Matroids 2. Ordered Commutative Semigroups 3. Lattice-Ordered Commutative Groups 4. Linearly Ordered Commutative Divisor Semigroups 5. Ordered Semimodules 6. Linearly Ordered Semimodules over Real Numbers

1 30

41 51 85 97

PART I1 : LINEAR ALGEBRAIC OPTIMIZATION 7. Linear Algebraic Problems 8. Algebraic Path Problems 9. Eigenvalue Problems 10. Extremal Linear Programs 1 1. Algebraic Linear Programs 12. Algebraic Flow Problems 13. Algebraic Independent Set Problems

111 116 168 188 212 253 30 1

CONCLUSIONS

337

BIBLIOGRAPHY

339

AUTHOR INDEX

369

SUBJECT INDEX

373

ix

This Page Intentionally Left Blank

PART I

1.

ORDERED ALGEBRAIC STRUCTURES

O r d e r e d S e t s , L a t t i c e s and M a t r o i d s

In this chapter we introduce basic concepts from the theory of ordered sets and we consider certain types of lattices. Further we review basic definitions from graph theory and matroid theory as given in BERGE [ 1 9 7 3 1 and WELSH [ 1 9 7 6 ] . This chapter covers the necessary order-theoretic background for the -discussion o f the optimization problems in part 11. In particular, some representation theorems from lattice theory give a general view o f such structures. For further results and for some proofs in lattice theory we refer to GRATZER [19781.

The basic properties of the usual order relation numbers IR

5

of the real

are described by the following four axioms:

(RefZesivitp)

a z a a z b ,

b c a

*

a = b

( A n ti s y m m e t r y )

a'b,

c ' b

*

a c c

(Trans i t i o i t y )

or

b c a

(1.1)

a c b

(Linearity)

for all a , b , c E IR.

In general we consider a binary relation 5 o n a nonempty set H satisfying some o f these axioms. We always assume that the binary relation is transitive. Such a binary relation is called an o r d e r i n g . If an ordering is reflexive then it is called a

p r e o r d e r i n g or q u a s i o r d e r i n g ; if an ordering is antisymmetric then it is called a p s e u d o o r d e r i n g .

1

A

reflexive and anti-

Ordered Algebraic Smrcritrcs

symmetric orderinq is called a p n r t i a ? , o r d e r i n g ; if a part.ia1 ordering is linear then it is called a l i n e a r orderinq.

(or total)

system (H,f) with such an ordering o n H is

A

called a (pre-, quasi-, pseudo-, partially, linearly) ordered set. A

-

?reorder

on

14

satisfyinq

I S

called an e q u i v a 7 e n c e r e 7 n t * ’ o n . Then R : = { b € HI

LS

called the e q u i z a z e n c e c ? n s S of a. The set { R

a ~ c r f * ‘ + i oof~ H ,

i.e.

n Rb +

Ra

@

-

ard H is the union of all R

t

-

a}

a E H 1 is

Ra = Rb :

Vice versa a partition of H defines an equivalence relation cn H with a

-

b if and only if a and b are elements of the

same element ( b l o c k ) of the partitior..

T w 3 elements a , b in an ordered s e t H are called if

neither a

5

b nor b

5

a.

a

5

b but a

+

ixcomparable

b is denoted by a < b .

T h e ? r , t ; e r s e o r d u a l o r c i e r f n g of 5 is the ordering

a > -b

: -

L

defined by

b 5 a

fcr all a , b E H . A linearly ordered subset A of H is called

?ox?:r?nZ if there is no linearly ordered subset B o f H such that

A

*

R.

In order to obtain a general view o f ordered structures it is v e r y useful to introduce the concept o f isomorphism.

Ordered Sets, Luttices and Matroids

Let (HI()

V: H

+

and (H

, z ' ) denote

3

two ordered sets. A mapping

H' is cal ed an o r d e r i s o m o r p h i s m if

v

is bijective

and a c b for a

a,b E H. Then I? an

H' are called o r d e r - i s o m o r p h i c .

If (9 is an order isomorphism hetween H and V ( H ) but not necessarily surjective on H',then

UY

is called an O r d e r e m b e d d i n g

of H into HI. We say that H can be o r d e r - e m h e d d e d

into

H I .

For

example, each finite linearly ordered set with n elements is order-Isomorphic to the set {lI2,...,~n~linearly ordered by the usual order relation of the natural numbers and each finite linearly ordered set with less than n elements can be order-embedded in every finite linearly ordered set with at least n elements.

Ordered sets are often visualized by graphic representations. We assume that the reader is familiar with the concepts of a g r a p h and a directed graph ( d i g r a p h ) . We denote a graph G by a pair ( V , E ) where V = V ( G ) is the nonempty vertex set and E = E ( G ) is the set of e d g e s ; a digraph G is denoted similarly by ( V I A ) where V = V ( G ) I s the nonempty vertex set and A = A ( G ) the'set of a r c s . Arcs and edges are denoted as pairs (a,b) of vertices; then a,b are called e n d p o i n t s of (a,b); in particular, for an arc (a,b) the vertices a and b are called i n i t i a l and t e r m i n a l

e n d p o i n t s of (a,b).

...,en )

sequence of arcs (edges) (el,e2,

A finite

such that ei has one

common endpoint with ei-l and the other endpoint in common with ei+l for all i = 2,3,...,n-l

is called a c h a i n . The length

Ordered Algebmic Structures

4

of a chain is the number of its edges (arcs). V e r t i c e s o n a chain are called c o n n e c t e d

( b y the chain). A chain with arcs

e . = (ai,ai+l) for i = 1,2,. ..,n is called a p a t h from a to a n + l . a l and a

n+ 1

1

are called i n i t i a l and t e r m i n a l e n d p o i n t

of the path. The e n d p o i n t s o f a chain are defined similarly. A

chain (path) is called s i m p l e if its edges (arcs) are

mutually distinct.

A

simple chain (path) with coinciding end-

points is called a c y c l e

(circuit).

A

chain (path) not using

any vertex twice is called e l e r e n t a r y . A

s u b g r a p h o f G g e n e r a t e d by V'

5

V is the graph (V',E') con-

taining all edges (arcs) o f G which have both endpoints in V'. A g r a p h G is called c o n n e c t e d ted. The binary relation

if all vertices in G are connec-

-, defined o n V by a

-

b iff a and b

are connected, is an equivalence relation o n V. The elements

of the induced partition V 1 , V 2,... ,Vk of V generate the

c o n n e c t e d c o r p o n e n t s of

G.

A

graph is called s t r o n g l y c o n n e c t e d

if for all a , b E V there exist paths from a t o b and from b to a. The strongly connected subgraphs o f G are called s t r o n g l y

c o n n e c t e d c o m p o n e n t s and define another partition of the vertex set V.

F i g u r e 1.

Directed and undirected graph

An ordered set ( H , z ) is represented b y the (directed) graph with vertex set H and (arc set

A)

edge set E = {(a,b)( a 5 b).

Ordered Sets. Lattices and Matmids

5

In the case of a graph a < b is expressed by placing a below b in a drawing of the graph. Clearly, the above examples of graphs represent the same ordered'set. Let A denote a subset of an ordered set E and let b E H . Then b is called an U p p e r ( l o w e r ) bound of A if a

5 b (b 5 a) for

all a E A . An upper (lower) bound b of A i s called a l e a s t

u p p e r ( g r e a t e s t l o w e r ) bound or supremum ( i n f i m u m ) of A if b

5 c

(c

5 b) for all upper (lower) bounds

c of A.

An element a E A is called a maximum ( m i n i m u m ) o f A if a'

5 a

(a 5 a') for all a' E A . An element a E A is called maximal

( m i n i m a l ) if a 5 a' (a' 5 a) implies a' 5 a (a 5 a') for all a' E A . The corresponding sets are denoted by U(A) , L(A)

(upper, lower bounds),

sup A , inf A

(suprema, infima),

max A, min

A

(maxima, minima),

Max A, Min

A

(maximal, minimal elements).

The following example illustrates these denotations. On the set o f the vertices of the graph G = (V,A) in figure 2

0 3

Figure 2.

Digraph of an ordered set

an ordering i s defined by a

5 b iff (a,b) E A . This ordering

is only transitive. For A = 1 3 1 we find sup A = {l,Z], and Max A = A. For B = { 1 , 2 }

max A = @

we find sup B = max B = Max B = B.

Ordered Algebraic Structures

6

Suprema (infima) and maxima But if Max

is a finite subset of an ordered set H , then

A

A =

(minima) do not always exist.

a.

ordered set is called b o u n d e d if it has a maximum

AI:

tion: 1 or -1

and a minimum

(denotation: 0 o r - - I .

we find 1 E Sup H = max H = Max H

and

O E

(denotaClearly,

inf H = min H = Min H.

If H is pseudoordered then these sets contain exactly one element. A

partially ordered set

Let

called a r o o t s y s t e m if the

a 5 b} is linearly ordered for all a € H.

set { b E H I

(1.2)

(H,c) is

Proposition

(H,Z)

be a partially ordered set. Then H is a root system

if and only if n o pair of incomparable elements in H has a

common lower bound.

Proof. The set { d E H I c 5 d} for cEL({a,b))

is not linearly

ordered if a and b are incomparable. This immediately shows the proposed characterization of root systems.

A partially ordered set

and inf{a,b}

(H,L)

is called a l a t t i c e if sup{a,b}

exist for all a , b E H . For convenience we use the

denotations a

A

b:= inf{a,b},

in lattices.

A

a

sublattice

V

b:=

(H',z)

sup{a,b) of a lattice (H,z)

subset H' o f H which is closed with respect to m e e t j C i n v taken in H , i.e.

is a A

a , b E H' imply a v b , a h b E H ' .

not sufficient that (H',I) is a lattice.

A

and It is

lattice H is called

Ordered Sets. Lattices and Matroids

C o m p l e t e if sup that sup

A

for all B

7

and inf

A

exist for all

exists for all

A

C_ H if and only if inf B exists

A

A

H. We remark

5 a;

Two lattices H and A' are called l a t t i c e - i s o m o r p h i c iff there exists a bijection Ip: H + H' such that a

5 b iff Ip(a) 5 Ip(b)

for all a , b E H . This condition is equivalent to Ip(aAb) = Ip(a) A

Ip(b)

Ip(a v b ) = Ip(a) vIp(b) for all a , b E H . Then Ip is

and

called a lattice isomorphism. If Ip: H morphism then

@

+

cr)(H) is a lattice iso-

is called a l a t t i c e e m b e d d i n g of H in H'. We

say that H can be l a t t i c e - e m b e d d e d

into H ' .

In particular,

q(H) is a sublattice of HI.

Now let A be a root system and let H A be the linearly ordered set lR of real numbers for all A E A .

Let V = V(A) be the

following subset of the Cartesian product

x AEAHA: x =

belongs to V if and only if the set Sx = { A E A 1 x A

(.

* O }

. .xx.. . ) contains

no infinite ascending sequences. Further let M

XIY

:= { h E h l

xA * y A

for x , y E V . Then we define x hEM

XtY

.

As

and

x

lJ

= y p for all A -c

5 y if and only if x A 5 y A for all

in this definition of the binary relation

5 on V

we will often use the same symbol for different order relations; from the context it will always be clear which of the relations considered is meant specifically.

( 1 . 3 ) Proposition Let A be a roqt system. Then V = V(A) is a lattice. Proof. At first we show that V is a partially ordered set with respect to the above defined binary relation. Reflexivity

a

Ordered Algebraic Structures

follows from M

5

let x

5

If x

implies x A

z

and x

MX

A

for all 1-1 > A. y

= @.

= M implies antisymmetry. Now IY YrX y, y < z and let A E M x , z . Then x z A and x = z

xrx

= z6 =

yA

A

5

1-1

=

y p for all 1-1 > A

u u then x 5. y and

Otherwise let A < y with x

zA.

y6 for all 6 > y .

5

Then x

y and y

5

z

Y - z Y *yY

leads to

= x < yy. Secondly, for a , b E V we Y Y b, a v b in the following way.

the contradiction y define a

= z

1-1

*

< z

Y

Let M be a maximal linearly ordered subset of A. Then the p r o jeCtiOn V

M

vM : =

of V onto M, defined by ( v ~ v lv

=

A

o

for all A

M is linearly ordered. For v E M we call vM with vA:= v A for A E M and vM = 0 otherwise the p r o j e c t i o n Of

V.

A

5

if aM

b

M

For A E M we define

,

(a A b) A : = if aM > bM i f a

M

L b

I

M

,

M

.

(a v bIA : = ifa'
In this way a A b and a v b are well-defined on all maximal linear ordered sets M. We show this only for a A b . Let A E A . Then the set { p E A1 A 5

1-11 is contained in all maximal linearly ordered

sets M with A E M . If a

u

=

b

2

for all 1-1

1-1

A then

(a Ab)A = a A = b A

independently of the choice o f the particular M considered ( A EM). Otherwise let 1-1

2

We assume w.1.o.g.

A with a

a

1-1

< b

4

11

1-1

b

and a ?J

.

Y

= b

Y

for all y

2

p.

Then ( a ~ b =) a~A independently o f

the choice o f the particular M considered ( A E M ) . Next we show that a A b is the infimum o f a and b. Since a 5 b if and only if

9

Ordered Sets, Lattices and Manoids

aM < b

M

f o r a l l maximal l i n e a r l y o r d e r e d s u b s e t s M o f V ,

we

f i n d a h b E L ( { a , b ) ) . L e t c E L ( { a , b ) ) and l e t M b e a maximal l i n e a r l y o r d e r e d s u b s e t o f V. T h e n c M

(a h b )

M

M M M E { a ,b } w e f i n d cM 5 ( a h b )

5

.

aM and c M

Hence c

5

5

b

M

.

ahb.

AS

Simi-

l a r l y w e c a n show t h a t a v b i s t h e supremum o f a a n d b .

The l a t t i c e V ( h ) w i l l b e c a l l e d t h e r o o t l a t t i c e o f t h e r o o t system A . I n t h e f o l l o w i n g we c o n s i d e r l a t t i c e s f r o m a n o t h e r p o i n t o f view.

Since a h b , a v b a r e well-defined

functions A,V:

H X H + H

the

on a l a t t i c e H ,

a r e i n t e r n a l c o m p o s i t i o n s on H .

They s a t i s f y t h e f o l l o w i n g s y s t e m o f a x i o m s :

(1.5)'

ahb = bha

,

a v b = b v a

(Commutativity)

a h (bhc)= (ahb)h c

,

a v ( b v c ) = ( a v b ) Vc

(ASSOciatiUity)

ah(aVb) = a

,

av(ahb) = a

( A b s o r p t i o n1

for a l l a,b,cEH. As

an example, w e c o n s i d e r t h e l e f t a b s o r p t i o n i d e n t i t y .

know a

5

We

Thus a = a h ( a V b ) .

a hb.

On t h e o t h e r h a n d we may c o n s i d e r a n o n e m p t y s e t H w i t h t w o i n t e r n a l compositions

(1.6)

a z b ,

for a l l a,bEH. commutative.

:

h , V

CI.

s a t i s f y i n g ( 1 . 5 ) . Then w e d e f i n e

a h b = a

T h i s b i n a r y r e l a t i o n i s a n t i s y m m e t r i c as

T r a n s i t i v i t y f o l l o w s from a s s o c i a t i v i t y .

A

is

(1.5)

implies (1.7)

a A a = a

for a l l a E H .

(Idempotency)

This can be seen using t h e absorption i d e n t i t i e s :

10

Ordered Algebraic Structures

a = a h (av (aha)) = a h a

for a E H .

Idempotency i m p l i e s r e f l e x i v i t y

relation.

Hence

(H,L)

i s a p a r t i a l l y o r d e r e d s e t . We c l a i m

that aAB = inf{a,b).

Clearly

(ahb) h a =

5

shows a h b Then c = c,

h

a.

i.e.

(1.8)

c

5

( a h a )h b = a h b

Similarly,

a = c = c

h

a h b .

a h b = a

b.

5

a A b

Therefore c

b. h

. I I ,

5

Now l e t c

5

a and c

(a h b ) = ( c h a )

Hence a A b = i n f { a , b ) .

A

b = c

b. b

h

Now o b s e r v e t h a t

a v b = b

Assume a h b = a .

for all a,bEH.

of the binary

Then t h e r i g h t a b s o r p t i o n

identity yields a v b =

(ahb) v b = b v

b h a ) = b.

The o p p o s i t e i m p l i c a t i o n c a n b e shown i n t h e s a m e m a n n e r . T h e r e f o r e a V b = s u p { a , b ) c a n be p r o v e d s i m i l a r l y t o t h e p r o o f of a h b = inf{a,b). system

(H,h,V)

Thus

satisfying

(H,L)

is a l a t t i c e and w e c a l l a

(1.5)

a lattice,

too.

W e h a v e shown t h e f o l l o w i n g r e s u l t :

(1.9) T h e o r e m (1)

Let

(H,5)

be a l a t t i c e .

and a v b : = (2)

Let

(H,h,V)

sup{a,b)

Then

(H,h,v)

with a h b : = inf{a,b)

f o r a l l a , b E H is a l a t t i c e . '

be a l a t t i c e .

Then

(H,z)

with a z b :

CI,

a

h

is a lattice.

We r e m a r k t h a t , i f

the lattice

define a further l a t t i c e

(HI

(H,h,V)

)

in

(1.9.1)

by a 5 ' b : e - b

i s used t o

a h b = a ,

then

b = a

Ordered Sets, Lattices and Matroids

this yields the original lattice (H,() a

5 b if and only if a <'b.

11

in (1.9.1),

i.e.

Similarly, the lattice (H,Z) in

(1.9.2) leads to a lattice (H, k

, v' ) ,

which coincides with

the original lattice (H,h,V) i n (1-9.2).

In the following we consider some special classes of lattices. A

lattice H is called d i s t r i b u t i v e if

(1.10)

a

A

(b v c) = (a A b ) V (a A c)

for all a , b , c E H . We will show that (1.10) is equivalent to (1.10')

a

v (b A c)

=

(a V b)

A

(a V c)

for all a , b , c E H .

(1.11) P r o p o s i t i o n Let (H,L) be a lattice. Then a

(2)

a V (b A c)

(3)

(1.10) is equivalent to . ( 1. lo').

A

( b V c)

2

(1)

(a A b) V (a A c)

5 (a V b)

A

(a v c)

for all a , b , c E H ; for all a , b , c E H ;

Proof. ( 1 ) This inequality follows from the inequalities aAb

5

a, a A b

5 b v c and a h c 5 a , a A c 5 b V c .

(2)

Follows

similarly to (1). ( 3 ) We only show that (1.10) implies (1.10'). The opposite implication is proved in the same manner. Let x,y,zEH. As

X V ( X A Z ) .=

x we find

Using (1.10) and x = ( x v y) A x we transform the right hand side :

I:!

Ordered Algebraic Structures

Using again (1.10)

we find

(XVY)

=

(x VZ).

A

It should be noted that (1.101 and (1.10') are not equivalent for the same three elements. Equivalence holds only with respect to a l l elements. Let

S

denote a subset of the set P ( A ) of all subsets of a

given set A. If X n Y E S and X U Y E S for all X , Y E S then the system ( S , n , U )

is called a r i n g of s e t s .

A ring of sets is ob-

viously a distributive lattice with partial order C_ (set inclusion).

(1.12) Theorem A

lattice is distributive if and only if it is lattice-isomor-

phic to a ring of sets. Proof.

A

ring of sets is obviously a distributive lattice.

We show the opposite implication only in the finite case. For a proof in the infinite case we refer to G R f l T Z E R [1978].

Now let H be a distributive lattice with finite set H. An element a of H is called j o i n - i r r e d u c i b Z e if a a = b or a = c for all b,c E H .

we

A

denote tAe set of

of I with a E A implies { x f H [

claim that r: H

+

P*(I)

b v c implies

Let I denote the set of all

join-irreducible elements of H. Let P * ( I )

all subsets

=

x

1. a) 5

A. Then

defined by

is a lattice isomorphism. A s P * ( I )

is a ring of sets we can

prove the theorem in the finite case in this way. Now r is an injective mapping as

a

is the j o i n of all elements in r(a1.

13

Ordered Sets, Lottices and Matroids

The same argument yields r(a) fl r(b) = r ( a A b ) . In order to show surjectivity let J E P * ( I ) and let a be the join of all elements in J. Clearly J C_ r(a). For x E r(a) we have x = x A a. Distributivity shows x = sup{xhbl x = x h b for some b E J , i.e.

x

1. b.

b E J ) . A s x E I we find

Due to the definition of

P*(I) this implies x E J . Thus r(a) C_ J. Finally we show

c_ r(a

r(a) v r(b) = r(a v b). Clearly, r(a) U r(b) Let x E r(a vb). =

Distributivity leads to x = x A

( x h a ) v ( x h b ) . Then x E I implies x = x A a

i.e.

xEr(a)

or

x E r ( b ) . Thus r ( a V b )

(a V b

-

x = x A b,

or

r(a)

v b)

u

r(b

Theorem (1.12) characterizes all distributive lattices up to isomorphism. Every distributive lattice can be visualized as a ring o f sets. A simple finite example is given in figure 3 .

Figure 3. A

Ring o f sets

finite lattice is always bounded, In a bounded lattice H

an element b E H is called a compZement of a E H iff a A b = O and a v b = 1.

In the lattice in figure 3 the elements {1,2,3)

and

$4 are complements of each other; all other elements have no complements.

A

lattice is called c o m p l e m e n t e d if all elements

have complements.

Ordered Algebraic Structures

14

(1.13) P r o p o s i t i o n Let

(H,L)

be a bounded, distributive lattice. Then a E H has at

most one complement. Proof. Suppose that b,b' E H are complements o f a E H . Then ~ =l b A ( a v b ' ) = ( b h a ) v ( b h b ' ) = O v ( b h b ' ) = b h b ' .

b = b

Similarly b' = b h b ' .

Thus b = b'.

A bounded, complemented and distributive lattice is called a

B o o l e a n lattice. I n a Boolean lattice the unique complement of a is denoted by a*. Boolean lattices are characterized in a similar manner a s distributive lattices. Let P(A) denote the set of all subsets of a nonempty set A. Then for X E P ( A ) the set

-

X:=

A\X

is the complement o f X. A ring o f sets

S

C_ P(A) is

called a f i e l d of s e t s iff G E S for all X E S . A field of sets is obviously a Boolean lattice with partial order

5.

(1.14) Theorem A

lattice is Boolean if and only if it is lattice-isomorphic

to a field o f sets. Proof. Again we consider only the finite case and refer to GRfiTZER [I9781 for the infinite case. Let H be a finite Boolean lattice. We use the same denotations as in the proof o f theorem (1.12). Let a E I . Then a c o o e r s 0, i.e. 0 < a and 0 implies xE{O,a) a = a

A

5 x 5 a

for all x E H . Suppose 0 < x < a. Then

( x V x * ) = ( a h x ) V ( a h x * ) = X V ( a A x * ) shows that a is

not join-irreducible contrary to a E 1 . Thus a , b E I with a

+

b

are incomparable. Therefore P*(I) = P(1). This means that the lattice isomorphism r maps H onto the fields o f sets P(1).

Ordered Sets. Lcrttices and Matroids

15

From the proof of (1.14) we see that in the finite case a lattice is Boolean iff it is isomorphic to the Boo ean

attice

of all subsets of a finite set. Next we consider a class of lattices which are distributive but not Boolean, in general. Let H be a bounded lattice. Then H is called pseudo-Boolean

if for all a , b E H there exists an

element c E H such that (1.15)

a A x 5 b

x(c

y

for all x E H . If such an element exists then it is unique and will be denoted by b:a.

(1.16) Proposition Let H be a Boolean lattice. Then H is pseudo-Boolean and b:a = b v a * for all a , b E H . Proof. Let a h x a*

v (a A x )

=

5 b for x E H . Then a * v ( a h x ) 5 a * v b . Now

(a* v a)

A

(a* V x) = 1

inequality follows from x =

( a h b ) v (a*Aa) = a A b

A

(a* V x)

2

x. The opposite

5 b v a* as then a h X 5 a h ( b v a * ) 5 b. w

The relationship between distributive and pseudo-Boolean lattices is given in the following proposition.

(1.17) Proposition A

pseudo-Boolean lattice H i s distributive.

Proof. Proposition ( 1 . 1 1 ) a

h

(b v c)

shows that it suffices to prove

5 (a A b)

V (a A c)

Ordered Algebraic Strucrures

16

for all a,b,c E H . Let x:= ahc

5

x.

Therefore b

is equivalent to

a h

( a h b) v ( a h c ) . Then a h b

5 x:a,

(bVc)

c

5

x,

5 x:a. Then b v c 5 x:a which

5 x.

rn

In particular we consider the element O:a for a E H in pseudoBoolean lattices H. Then a h x = 0 is equivalent to x

5 0:a. In

a Boolean lattice a* = O:a and therefore (1.18)

a v (0:a) = 1

for all a E H .

(1.19) Let

H

Proposition be a pseudo-Boolean lattice. Then H is a Boolean lattice

if and only if it satisfies (1.18). Proof. Due to (1.17) we know that H is distributive. If (1.18) is satisfied then O:a is a complement of a. Thus H is Boolean. rn A

simple example o f a pseudo-Boolean lattice is a bounded,

linearly ordered set (B,5). Then a h b = min(a,b) and a

vb

= max(a,b)

.

b:a =

Further

1. a ,

I'

if b

'b

if b < a,

for all a , b E B. In general, B is not Boolean as for 0 < a < 1 we find a v (0:a) = a V O = a < 1 . The following example shows that a distributive lattice is not necessarily pseudo-Boolean. Let N,:=

INU

{-I

there exists c E N m with a - c = b. Then ( N , , i )

and let a 5 b iff

-

is a bounded

distributive lattice with minimum 1 and maximum

(as- = - 1 .

Ordered Sets, Lattices and Matroids

( g r e a t e s t common d i v i s o r of

We define a h x : = g.c.d.(a,x)

and x). Then a h x

=

17

a

1 for a 6 1 1 , ~ ) has the set o f solutions

1x1 g.c.d.(a,x) = 1) which does not have a maximum. Thus l : a does not exist and N m i s not pseudo-Boolean.

On the other side we show now that every finite distributive lattice H is pseudo-Boolean. Then H is bounded and 1 = sup H , 0 = inf H. Let a , b E H and X:= 1x1 a A x

5

Let c:= sup X. Then a h sup X = sup{ahxI is equivalent to x E X .

b}. O E X implies X

xEX)

*@.

5 b. Thus x 5 c

The same arguments show that a complete

lattice H is pseudo-Boolean provided that the g e n e r a l

law of

distributivity (1.20)

a h s u p X = sup{ahxI

XEX)

holds for a l l a E H and for all X E P ( H ) .

Obviously,(1.20) implies

distributivity.

Next we discuss a class of finite lattices. A finite lattice a.lways has a minimum 0 and a maximum 1. An element a c o v e r s an element b if a > b, and if a

2

x

2

b implies x E {a,b).

An ele-

ment a is called an atom if a covers 0. A finite lattice H is called semimoduZar (1.21)

if

a and b cover a h b

for all a , b E H with a

*

b.

A

-D

a v b covers a and b

finite lattice H is called geome-

t r i c if it is semimodular and every element is the join of a set of atoms. Let H be a finite lattice and let G

H

= (H,E) be a digraph

representing H with E:= {(a,b)I b covers a].

For x E H let Px

Ordered Algebraic Structures

18

denote the set of all elementary paths from 0 to x. The

length of a path p is denoted by Z(p). Then the height func-

t i o n h: H + N U { O } is defined by

for all x E H . H satisfies the Jordan-Dedekind chain condition if for all a , b E H all paths from a to b have the same length.

( 1 . 2 2 ) Theorem A

finite lattice is semimodular if and only if

(1)

it satisfies the Jordan-Dedekind chain condition,

(2)

h(x) + h(y)

h(x V Y) + h(x

A

y)

for all x , y E H .

Proof. At first we assume that H is semimodular. ( 1 ) Let a , b E H and assume w.1.o.g.

I

a < b. We claim that if

there exists a path from a to b of length k then all paths from a to b have length k. This is obvious for k = 1. We assume that the assertion holds for all k p =

( a , x , ...lb) and q =

5 m for some m 2 1. Let

(aly,...lb) be paths from a to b in

G H (both denoted by the sequence of their vertices). Let

I(p) = m+l. If x = y then the inductive hypothesis shows that every path from x to b has length m and therefore l(p) = l ( q ) = m+l.

Otherwise x

*

y and semimodularity shows that x V y

covers x and y. Using the inductive hypothesis we find that every path of the form (x,x v y l...,b) has length m ; every path of the form ( x v y ,

...,b)

...,b )

form ( y l x v y l (2)

has length m - 1 ; every path of the

has length m. Thus l ( q ) = m + l .

Due to ( 1 1 h is well-defined. Let p and q be paths from

x A y to x and from y to x V y. Then l(p) = h(x)

-

h(x

A

y) and

19

Ordered Sets, Luttices and Matroids

Let p = (alfa2,...,am). A s a i+l covers

l ( q ) = h ( x v y ) -h(y).

ai for i = l,ZI...,m-l

we find that either a i + l v y covers

a i v y or a i + l . ~ y = ai v y l(q)

for i = 1,2,. ..,m-1. Therefore

5 l(P).

Secondly, we assume that (1) and (2) are fulfilled in H. Let a , b E H with a =

h(a

A

b)

+ 1.

$

b and let a and b cover a A b . Then h(a) = h(b)

Then ( 2 ) implies h(a v b)

5 h(a) + 1. Hence a v b

covers a. Similarly we find that a v b covers b.

rn Let a,b be elements of a lattice H and let a [a,b]:=

IxEHI a

5

x

5 b} is called

an

5

b. Then

i n t e r u a t of H. [a,bI

is a sublattice of H. If H is a finite, semimodular lattice

then theorem ( 1 . 2 2 )

leads to the fact that [a,b] is semimodu-

lar. If H i s a finite, geometric lattice with set of atoms D ;hen {avdl d E D , d

a, d

5 b)

is the set of the atoms of an interval [a,b].

This implies

that [a,bl is geometric.

Next we review basic results of matroid theory. We assume that the reader is familiar with the concept of .a matroid and for proofs we refer to WELSH 119761.

Let F denote a nonempty subset of the set P ( N ) of all subsets of N : = {l,Z,...,n}.

Then F is called an i n d e p e n d e n c e S y s t e m if

(1.23)

and

IEF

JC_I

*

J E F .

20

Ordered Algebraic Structures

The elements of F are called i n d e p e n d e n t s e t s . Obviously F is partially ordered with respect to set inclusion. An element of F i s called b a s i s if it is maximal in F. An element of P ( N ) is

called d e p e n d e n t if it is not an element of F. A minimal dependent set is called a c i r c u i t . The function r: P ( N )

+

22;

(nonnegative integers) defined by r(1) = max{lJl f o r all I E P ( N 1 is called

u: P ( N )

-+

P(N)

I

J E F and J C_ I)

r a n k f u n c t i o n of F. The function

defined by

O(1) =

i E N l r(IU{i]

=

r(1))

f o r all I E P ( N ) is called c l o s u r e o p e r a t o r .

o(1)

is called the

c l o s u r e of I. An

independence system F is called a rnatroid if

(1.24)

for all 1 , J E F with 111 < IJI there exists j E J \ I such that I U { j ] E F

is satisfied. Matroids can be described in terms of bases, dependent sets, circuits, rank function or closure operator. We can give axiomatic definitions of a matroid in terms of each of these concepts.

(1.25) T h e o r e m ( B a s e a x i o m s ) A nonempty collection B C _ P ( N ) is the set of bases of a matroid on N if and only if it satisfies the following condition: (Bl)

If 1 , J E B and i E I \ J such that (I U {j))

then there exists j E J L I

{ i ]E B.

Ordered Sets. Luttices and Matroids

(1.26) T h e o r e m A

21

(Rank axioms)

f u n c t i o n 'r: P(N)

+

Z+ i s t h e r a n k f u n c t i o n o f a m a t r o i d o n

N i f a n d o n l y i f f o r a l l s u b s e t s I , J o f N:

1. 1 1 1 ,

(R1)

r(1)

(R21

I

(R31

r(1U J)

J

-o

(1.27) T h e o r e m A

function

+

5

r ( 1n J )

r(J),

5

r(1) + r ( J ) .

(Closure axioms) P(N)

U:

r(1)

+

P(N) i s t h e c l o s u r e o p e r a t o r o f a m a t r o i d

on N i f and o n l y if f o r a l l I , J E P ( N ) and f o r a l l i , j E N :

5

fJ(I),

(S1)

I

(52)

I C_ J

(S3)

O ( I )

(54)

i f j @ u ( I ) but j E u ( I U { i l ) then i E u ( I U { j ) ) .

*

U ( J ) ,

= O(O(I)),

(1.28) T h e o r e m A

U(1)

collection C

(Circuit axioms) 5 P(N)

i s t h e set of c i r c u i t s of a matroid

o n N if a n d o n l y i f f o r a l l d i s t i n c t 1 , J E C a n d f o r a l l i E N

$

J,

(C1)

1

(C2 1

i E I f l J implies t h e e x i s t e n c e o f K E C such t h a t

-

K c (I

A

u J ) \(il.

m a t r o i d w i l l o f t e n b e d e n o t e d by M i n d e p e n d e n t l y o f t h e

c h o i c e of t h e p a r t i c u l a r d e s c r i p t i o n c o n s i d e r e d . T h e b a s e s o f a m a t r o i d h a v e a l l t h e same c a r d i n a l i t y r ( N ) ; f u r t h e r m o r e f o r a s e t J E P ( N ) a l l m a x i m a l i n d e p e n d e n t s e t s I C_ J h a v e t h e same c a r d i n a l i t y r ( J ) a n d r ( u ( J ) ) = r ( J ) . I f J is i n d e pendent then

Ordered Algebraic Structures

22

and f o r a l l j E o ( J )

X J

t h e r e e x i s t s a u n i q u e c i r c u i t i n J U {jl

denoted by C ( j , J ) . I f { i , j l i s independent f o r a l l i , j E N then t h e matroid i s c a l l e d simple. L e t B b e t h e s e t o f a l l bases o f

{NLII I E Bl

a m a t r o i d M on N .

Then

a g a i n is t h e s e t o f t h e b a s e s o f a m a t r o i d ,

is c a l l e d t h e dual matroid M* o f

M.

Bases,

which

c i r c u i t s and rank

f u n c t i o n o f M* a r e c a l l e d c o b a s e s , c o c i r c u i t s a n d c o r a n k func-

tion of M.

L e t SC and S B d e n o t e t h e s e t o f

a l l c i r c u i t s and c o c i r c u i t s o f

T h e n M is c a l l e d r e g u l a r o r o r i e n t a b l e i f t h e r e

a m a t r o i d M.

+

e x i s t p a r t i t i o n s C ,C

-

+ -

of each CESC and B , B

of each B E S B

such t h a t IC

+

flB

+1 +

IC-flB-1

=

IC

+ f l B- I +

IC-nB'I.

N o w we c o n s i d e r a c e r t a i n p a r t i a l l y o r d e r e d s u b s e t o f P ( N ) i n d u c e d by a m a t r o i d M on N .

s e t of M if

O(I)

I E P ( N ) is c a l l e d a f l a t o r closed

= I.

( 1 . 2 9 ) Theorem T h e s e t H o f a l l f l a t s o f a m a t r o i d M is a f i n i t e g e o m e t r i c

lattice. Proof.

H is p a r t i a l l y o r d e r e d b y s e t i n c l u s i o n .

The i n t e r s e c -

t i o n of t w o f l a t s is a f l a t ; t h e r e f o r e j o i n a n d m e e t e x i s t a n d

a r e g i v e n by

Ordered Sets. Lattices and Matroids

23

I A J = I ~ J ,

I V J =

n {KEHI

I U J EK} = ~(IUJ),

for all 1 , J E H . H has minimum @ and maximum N. The rank function r of M is the height function of H .

This follows from

the fact that a flat I covers a flat J if and only if J

I

I I is and r(1) = r ( J ) + 1 . Thus the length of a path from @to

bounded by r(1). In factreach such path has length r ( 1 ) . Therefore A satisfies the Jordan-Dedekind chain condition. From

2

( R 3 ) we know r(1) + r ( J )

r(I

u J)

r ( 1 h J ) = r(IflJ) and r ( I U J ) =

+ r ( I fl J ) . If 1 , J E H then

r ( U ( 1 U J ) )

= r ( I U J ) . Then

shows that H is semimodular. Finally let I E H

theorem ( 1 . 2 2 )

with r ( 1 ) = k. Then there exists an independent set J:= ).I

{jl,j2r...,jk}

= l r 2 ,

...,k.

I. Clearly, o(j

u

)

fl J = {j

v1

for all

Thus o(jp), l~ = 1 , 2 , . ..,k are distinct atoms

and sup{o(j) I j E J } = I.

Now we prove a representation theorem for finite geometric lattices.

(1.30) T h e o r e m A

finite lattice H is geometric if and only if it is lattice-

isomorphic to the lattice o f flats of a matroid. Proof. Due to theorem ( 1 . 2 9 )

it suffices to find a suitable

representation of a geometric lattice H. Let of all atoms of H .

Let F

A

denote the set

P(A) consist in all subsets I of A

with h(sup I) = 111 with respect to the height function h of H . Theorem ( 1 . 2 2 ) lity h(sup I)

...,ik}

shows that for I = {il,i2,

5 1 h(i v

)

.

) I

=

C_ A the inequa

111 is valid. Let 1,J C_ A. Then

Ordered Algebraic Structures

24

-

J implies h(sup

I)

5 h(sup

I

C

J ) . Further h(sup

->

h((sup I) v (sup J ) ) + h((sup I)

A

I) + h(sup J )

(SUP J ) ) 2 h ( s u p ( I U J ) )

+ h(sup(1 n J)). The last inequality follows from (sup A

(sup J )

2 sup(1

n J j .

I)

+

A

Therefore the function r(I):= h(sup I)

satisfies the rank axiomatics of a matroid. Hence r is the rank function of a matroid M(H) on

A.

Now let u be the closure operator of M(H) and let H' be the geometric lattice of the flats of M(H). Let for any I E H by

( 3 ( I ) : =U{il,i2,

..., ik}

Q:

H

+

H' defined

where {il,i2,...,ik}

is the set of all atoms below I. Then Ip is a lattice-isomor-

phism.

Two matroids M,M' on N and on N' are called i s o m o r p h i c if there exists a bijective mapping Ip: N

+ N'

preserving independence.

Equivalently, we may assume that Ip preserves the rank functions, circuits and so on. Now two matroids with lattice-isomorphic geometric lattices of flats are not necessarily isomorphic as matroids. It should be noted that the rank function of the matroid M(H) for a geometric lattice H satisfies r{i) = I ,

r{i,j} = 2

for all distinct atoms i,j E H . Hence the lattice H' of the flats

of M(H) is a simple matroid. Conversely, two simple matroids are isomorphic if and only if their geometric lattices of flats are lattice-isomorphic. Thus the study of simple matroids is just the study of finite geometric lattices. A

lattice is called r e l a t i v e l y c o m p l e m e n t e d if any interval

[a,b] is complemented.

Orderrd Sets. Lattices and Matroids

25

(1.31) P r o p o s i t i o n A finite geometric lattice is relatively complemented. Proof. It suffices to consider the lattice H of the flats of a simple matroid M on N. Let A be a flat and let I be a maximal independent set with I

A. Let B be a base containing I.

Then we define A*:= o ( B \ I). We find o(A

u A*)

= N and A n A * =

@.

In the language of lattices this means that H is complemented (cf. definitions of h,V in the proof of (1.29)). Now every interval of H i s a finite, geometric lattice. Therefore, H is relatively complemented.

A special class of finite geometric lattices is derived from the partitions of a finite set N .

Let R = {Ril i E I) and

all partitions of N .

denote two partitions of N . a function q : I

+ J

Let n ( N ) denote the set of

We define R

such that R

S

i

rp(i)

5

S

S

= {S.i j

I

E J )

iff there exists

for all i E I. Meet

and join of this partial order are given in the following. At first R A S = IRins,l

~ E I , jEJI.

Now consider a graph with vertex set R U S . Two vertices Ri and S

j

are joined by an edge if Ri

nS

j

9

@.Let

(Ck

, k E K ) denote

the family of the vertices of the connected components of G and define Tk:= U{Ril

RiECk) = UIS.1 S . E C k ) 1 1

for k E K . Then

R V S = {Tkl k E K ) . n(N)

is called the partition lattice of N.

A recent result of PUDLAK and TfMA (119771, [19801) answers a question raised by BIRKHOFF 119671:

26

Ordered Algebraic Stnccturek

( 1 . 3 2 ) Theorem Every finite lattice can be lattice-embedded in a partition lattice over a finite set.

For a proof we refer to PUDLAK and T6MA [ 1 9 8 0 ] .

Finite parti-

tion lattices are special finite geometric lattices. We consider the c o m p l e t e g r a p h K N = {1,2,

with i A

*

...,nl

n'

i.e. the graph with vertex set

and set of edges E containing all pairs (i,j)

j. Here (i,j) and

( j , i ) denote the same edge for i S j .

subset I of E is called independent if it contains no cycle.

These sets are the independent sets of the c y c l e m a t r o i d of Kn.

(1.33)

Theorem

The partition lattice of the finite set N = {lf2,...fn} i s a geometric lattice which is lattice-isomorphic to the lattice of the flats of the cycle matroid of the complete graph Kn. Proof. Let ( V u ,

p = l , Z ,

...,k)

be a family of mutually dis-

joint subsets of N. Then the subgraphs complete subgraphs of K

n

.

Let E

G

u generated by V P are

u denote the corresponding set

of edges. Then E' = E l U E

2

u...

"

Ek

is a flat of the cycle matroid M. All flats of M have this form. N O W { E

u

I

p =

1,2,

..., k}

u

( N X E ' ) i s a partition of N.

This 1 - 1 correspondence of flats and partitions leads to a lattice isomorphism between a ( N ) and the finite geometric lattice of all flats of M.

Ordered Sets, Lattices and Matroids

21

Together with the theorem o f P U D L A K and T8MA theorem ( 1 . 3 3 ) shows that every finite lattice can be lattice-embedded in the finite, geometric lattice of the flats of the cycle matroid over a suitable complete graph Kn. As

an example we consider a finite, distributive lattice H of

n elements. ORE 1 1 9 4 2 1 shows that c p : H cp(a):= {{bEHI

a v b = c}l

+

n(H) defined by

c E H ) is a lattice-embedding of H in

n(H). For a proof let RZ:= { b E H I a v b = c}. For c

*

c' we find immediately Ra

n R:'

=

2 . Further for b E H

we know b E R a Therefore V(a) E a(H). Now let a avb'

5 a' and let

b , c E H with a v b = c. Then a' V b = a' V a v b = a' v c . Therefore

c_

a' Ratvc which implies cp(a)

5 cP(a'). Finally a finite distri-

butive lattice has always a minimum 0. Therefore OER: R:'*

@.Thus

Then Ra a

C

-

R

a = max Ra. Let now a,a' E H with V(a)

, i.e.

5 (O(a').

a' and thus a 5 a'. a'

Finally In this section we discuss the role of certain duality principles in ordered structures (H,z). Its dual is (I3,Z) with respect to the inverse or dual order of

5. Reflexivity, anti-

symmetry, transitivity and linearity hold in if the same prope'rty holds in (H,?).

( H , L ) if

and only

The d u a l of a l a t t i c e

(H,A,V) again 1 s a lattice denoted by (H,V,A), i.e. join and meet are exchanged. If a lattice (H,A,V) has a minimum (maximum) then its dual (H,V,A) has a maximum (minimum). The dual o f a distributive lattice is distributive. This follows from (1.11.3). The law of distributivity ( 1 . 1 0 ) implies ( 1 . 1 0 ' )

as ( 1 . 1 0 ' )

can

Ordered Algebraic Structures

28

be derived as law of distributivity in (H,V,A), i.e. by exchanging meet and join. Now the duality p r i n c i p l e of ordered sets is the following. It can be applied to ordered sets (H,I) of special type iff the dual (H,:)

is of the same type. Let A be a statement in-

volving z , ~ , v , O , 1 which holds in all ordered sets of the special type considered. Then the dual A' of this statement

5 by 2 ,

is found by replacing 1 by 0 in A .

Then

A'

A

by v, V by

A,

0 by 1 and

holds in all ordered sets of the special

type considered. This follows from the fact that A' is the same as A formulated for the dual

(H,Z).

We can apply the duality principle for example to preordered or quasiordered or partially ordered or linearly ordered sets, to lattices,and distributive lattices. In particular, (1.10') is the dual of (1.10). The dual of a Boolean lattice is distributive and bounded. Let a* be the complement of a. Then a A b = 0 and a V b = 1 . These equations show that with respect to the dual lattice a* is a complement of a , too. Thus the dual of a Boolean lattice is Boolean. On the other hand the dual of a pseudo-Boolean lattice is not necessarily a pseudo-Boolean lattice. This is due to an asymmetry in (1.15). (1.15')

The dual of (1.15) is

avb)b

c.

x)c

tl a , b E H , 3 c E H .

If such an element c exists then it is uniquely determined and

is denoted by b/a. Boolean and (1.15')

A

bounded, linearly set (B,Z) is pseudois satisfied with

Ordered Sets, lmttices and Matroids

29

i f a c b , b/a:= if a l b

for all a,bEB. tive lattice.

(1.15')

,

i s s a t i s f i e d f o r any f i n i t e d i s t r i b u -

T h e r e f o r e t h e d u a l o f a f i n i t e , pseudo-Boolean

l a t t i c e is pseudo-Boolean. (1.22)

shows t h a t t h e d u a l o f a f i n i t e ,

s a t i s f i e s t h e Jordan-Dedekind

semimodular l a t t i c e H

chain condition but the dual

h e i g h t f u n c t i o n h* g i v e n by h * ( x ) = h ( 1 ) - h ( x ) f o r x E H s a t i s f ies h*(x)

+

h*(y)

5

h*(xvy) + h*(xhy)

a n d t h u s t h e d u a l l a t t i c e is n o t n e c e s s a r i l y s e m i m o d u l a r .

2.

Ordered Commutative Semigroups

In this chapter we consider ordered sets H in which a b i n a r y

operation ( i n t e r n a l cornposition)

*:

H

x

H

+

H is defined.

Basic concepts and definitions are introduced and discussed with respect to possible specializations. Commutativity is assumed throughout this chapter although commutativity is not necessary for some results. This assumption is motivated by the optimization problems considered in the second part. Further commutativity often leads to simpler proofs and is necessary for some important results. For a more detailed discussion of ordered algebraic structures we refer to FUCHS [19661,

KOKORIN and KOPYTOV [ 1 9 7 4 ] ,

MURA and RHEMTULLA [ 1 9 7 7 ]

and SATYANARAYANA [ 1 9 7 9 ] .

A

nonempty set H with internal composition

*:

H

x

H

-*

H is

called a s e m i g r o u p , if (2.1)

a*(b*c) = (a*b) * c

(Associativity)

for all a , b , c E H . A semigroup is called a monoid if it contains an element e with e + a = a

*e

= a for all a E H .

Such an element

is always uniquely determined and is called the n e u t r a l e l e -

m s n t ( i d e n t i t y ) of H.

A

is called a m o n o i d .

semigroup is called c o m m u t a t i v e if

(2.2)

A

semigroup containing a neutral element

(Commutativity)

a * b = b *a

for all a , b E H. An ordered set H , which is a commutative semigroup, is called an o r d e r e d c o m m u t a t i v e s e m i g r o u p if

30

(2.3)

-

a c b

Ordered Commutative Semigroups

(Monoto n i c i t y )

a * c z b * c

f o r all a , b , c E H . Axiom

31

(2.3) describes the interaction of

internal composition and order relation. Thus we consider only binary operations which are i s o t o n e in both arguments. Let a i E H for i E N : = {lIZr...,n~. Then we define

* iEN F o r ai

5 bi

a

i

:=

a l * a 2 * . . - * an

*

i E N , monotonicity leads to

I

rk ai iEN

5

*

bi.

iEN

In particular, let an:=

* ai if ai

Then monotonicity implies

a

n

= a for all

EN (ao:= el.

1. bn if a 5 b.

In chapter 1 we have seen that in a lattice (H,L) meet

A

and

join V are binary operations. Due to ( 1 . 5 ) we know that (H,A) and (H,V) are commutative semigroups. Furthermore ( H , v , ~ )and (H,A,()

are lattice-ordered commutative semigroups, since a

implies a A c

5

b A c and a v c

5 b

5 b v c for all a , b , c E H .

Two further examples are derived from real numbers. Besides (lR ,+,()

with usual addition

+

and usual order relation

5

we

find the totally ordered commutative semigroups ( W ,max,l) and (lR ,min,c) with internal composition max(a,b) and min(a,b).

We call a commutative lnonoid G a c o m m u t a t i v e ( a b e l i a n l g r o u p if for all a E G there exists an element b E G with a * b = e. Such an element is always uniquely determined and denoted by a

-1

.

An ordered commutative monoid G is called an o r d e r e d commu-

t a t i v e group if G is a group.

Ordered Algebraic Structures

32

Ordered commutative groups have many properties which do not hold in ordered commutative semigroups. In particular the

c a n c e 2 t a t i o n law (2.4)

a * c = b * c

*

a = b

for all a , b , c E G is valid in a group G and implies a stronger version of (2.3),

i.e.

for a l l a , b , c E G . Therefore a < b implies a * c < b * c in ordered

commutative groups.

A

semigroup H satisfying ( 2 . 4 ) is called

cance l l a t i v e . Let H be an ordered commutative semigroup. a E H is called

p o s f t i v e if

for a l l b E H . If all elements in H are positive then H is called

p o s i t i v e l y o r d e r e d . We remark that "positively ordered" often includes "partially ordered" which is not assumed here.

S t r i c t l y p o s i t i v e , n e g a t i v e and s t r i c t l y n e g a t i v e elements are defined similarly by inequality (2.6) with and >. The set of all positive noted by H+

(€I-). aEH

5 replaced by <, >,

(negative) elements of H is de-

+ ( a E H - 1 and a

5 bEH

(a

1. b E H )

b E H+ (b E H-). Therefore H+ (H-) is called the p o s i t i v e

c o n e of H.

A

imply

(negative

positively ordered cornmutative semigroup H is

called n a t u r a l l y o r d e r e d if

for all a , b E H. We remark that "naturally ordered" often includes "partially ordered" which is not assumed here.

Ordered Commutative Semigroups

In an ordered group

an element a is positive if and only if

G

> e. The ordering on a positive cone

G+

is completely determined by the

G

5 b if and only if b * a - 1

as a

a , b E G . Reflexivity is equivalent to e E = e

G+nG;'

(Gi*:=

33

{a-ll aEG))

G+

EG+

for all

, antisymmetry to

and linearity to

G+UG;'=G.

For subsets P I Q of an ordered algebraic structure we define P * Q : = { a * b l a E P , b E Q , and P

5

Q by a

b E Q . For P = {a} we use a * Q and a

5

{a)

Q meaning (a) * Q and

Q. Then transitivity in G is equivalent to G + * G +

Vice versa a subset P of group

5

5 b for all a E P ,

G

C_ G+.

defines a suitable ordering on a

if at least P * P C_ P. It should be remarked that commu-

G

tativity implies monotonlcity if P defines an ordering by a z b If

G

c . b*a

-1

EP.

is an ordered commutative group then

ordered

G+

is naturally

.

A bijective mapping

H

@:

+

H' for two semigroups ( A , * )

,

(H,*')

is called a s e m i g r o u p i s o m o r p h i s m if

for all a , b € H . If

@:

H

+

@(HI

not necessarily surjective then

is a semigroup isomorphism but is called a s e m i g r o u p em-

b e d d i n g of H into HI. Similar denotations are used for special classes of semigroups (for example, g r o u p i s o m o r p h i s m ) . ~f H and H' are ordered semigroups and rp: H

+

H' is an order iso-

morphism then Q is called an i s o m o r p h i s m between (H't*',Z').

( A , * , Z ) and

An embedding is defined accordingly. In general,

an isomorphism between ordered algebraic structures is an isomorphism with respect to all relevant compositions and orderings.

34

Ordered Algebraic Structures

( 2 . 8 ) Theorem A

naturally ordered commutative monoid H fulfilling the is isomorphic to the positive cone o f

cancellation rule ( 2 . 4 )

an ordered commutative group. Proof. Define a binary relation iff a * d = b * c . Then

-

-

on H

x H

by (a,b)

-

(c,d)

is an equivalence relation. Reflexivi-

ty and symmetry are obvious. Let (a,b)

-

(c,d) and (c,d)

-

(e,f).

Then a * d = b * c and c * f = d *e. Composition with f resp. b

*

shows a * d Thus

-

B

* e

* b . Cancellation yields (a,b)

-

(e,f).

is transitive. Let G denote the induced partition of

B l o c k s of G are represented by elements as usual.

H U H

Now

f = d

(c,d) = (a*c,b*d) defines an internal composition

a,b)

: G

X

G

+

G which is commutative and associative. Further

(e,e) is the neutral element and (b,a) the inverse element o f (a,b). Thus ( G , B

)

is a group and cp:

W(a1 = (a,e) fulfills cp(a * b ) = @(a)

H

Let P:= f p ( H ) . Then P

P

@

5

*c

= b

5 (b,e). Then (b,a) E

with (b,a)

-

ordered a a fb

The group

-

5

cp(b).

5

b then there exists

(if a , = b then c:=

Therefore Q(b) B c p ( a ) - i = (b,a) (a,e)

G defined by

P. Therefore we can define an order

relation with positive cone P. If a c E H such that a

@

-B

-

e

for example).

(c,e) E P . Finally let

P implies the existence of c E H

(c,e). This means a * c = b.

As

H

is positively

a * c holds. Then a 5 b. Thus we have shown

cp(a) f (3(b).

(G,

@ , 5 )in

the proof of theorem ( 2 . 8 )

is obviously

35

Ordered Commutative Semigroups

reflexively, antisymmetrically or linearly ordered iff the same holds for (H,+,Z). In the l i n e a r t y o r d e r e d c a s e

u

G = (G-'{e))

w

{el

(G+L{el) and elements of the negative

cone of G are just the inverse elements of elements of the positive cone, i.e.

a E ~ + L { e l } . In view of

we may identify H and G+. Therefore

theorem ( 2 . 8 ) G =

G - \ { ~ I = {a-'l

{a-ll a E H , a 9 el

u

{el

u

(Hk{el).

The assumption of the existence of a neutral element in theorem ( 2 . 8 ) can often be fulfilled by adjoining an extra element.

(2.9) Proposition Let (H,*,s) be an ordered commutative semigroup without a neutral element. Let e b H and HI:= HUCe), by a

*'

b = a

*b

define

*':

H' xH'-rH'

f o r a r b E H and

a*'e = e * ' a = a for all a E H . Let by a

2' denote the binary relation on

H' defined

5' b iff a 5 b for all a , b E H , a <' e <' b

for all a E H -

, b E H , and, if 5 i s reflexive or linear, e 5' e.

HI is an ordered commutative monoid, if

5

is reflexive, antisymmetric or linear then the same

holds for

(I,

if H is positively or naturally ordered then the same

holds for HI

,

if H is cancellative then the same holds for H'.

Ordered Algebraic Structures

36

Proof. We only remark that H- < H,.

Then all properties can m

easily be verified.

Xn particular

and ( 2 . 9 . 4 )

(2.9.3)

show that it is sufficient

to assume that H is a naturally ordered commutative semigroup in ( 2 . 8 ) . If A has no neutral element then i t is isomorphic to the strict positive cone G+\{e]

of an ordered commutative

group (G, e . 5 ) . Some examples are considered in the following. We begin with

(IN,

. , z ) with

a

C

b iff b/a EN ( d i v i s i b i l i t y ) . This is a

naturally and lattice-ordered, commutative monoid. Clearly it

is isomorphic to the positive cone of the lattice ordered group of positive rational numbers I a E Q I a > 0 ) endowed with multiplication and ordered by divisibility. Secondly let (G,+) denote the additive group of all real functions f: [ 0 , 1 1 + I R f(x)

with (f +g)(x):=

5 g(x) for all x E

f(x) + g ( x ) and f

[0,1]. Then

(G,+,()

5

g

iff

is a lattice ordered

commutative group. The third example does not look so familiar. For k E N let V denote the set of all subsets A of lNU { O ? with IAI trary B

zk. For

arbi-

{O} let k - m i n ( B ) denote the set of the k smallest

elements of B (in particular, for IBi < k let k - m i n ( B ) =

€3).

Now for A , B E V let A i B

k-min(AUB) =

:-b

A

.

This binary relation defines a lattice on V. Meet i s given by A A B =

k

-

min(AUB)

,

and join is (under consideration of its commutativity) given by

Ordered Commutative Semigroups

A v B = (AnB) U

(2.10)

31

i'

if IAI, IBl C k ,

{bEBl b>a)

if IAl =k, IS1 Ck,

{bEBl b>a}U{B+l, . . . , B +

r)

if IAI, IS1 = k ; a 5 8

with a = max A, 8 = max B and r = k - I ( A n B ) U { b E B I

b>a}l.

The

meet formula is obvious and the join formula is proved in the following proposition. Beforehand we define

* B:=

A

k

-

min(A

+ B)

with A + B : = { a + b l a E A , b E B ) for all A , B E V .

(2.11) (V,*,L)

Lemma is a lattice-ordered commutative monoid with neutral

element {O), ment

least element

2 =

{O,l,

...,k-1)

and greatest ele-

a.

Proof. The properties o f the neutral element, the least element and the greatest element are easily verified. The same holds for the properties of the partial order, the meet and for commutativity, associativity and monotonicity of the internal composition. The remaining part is the verification of the join-formula in ( 2 . 1 0 ) . At first we remark that A B E A Further let

C:=

or

C

d E (AnB)\

C

to A n B

5

C

B is equivalent with

(IAI = k and b E B \ A

implies b > m a x A).

A V B , a = max A, B = max B, y = max

Secondly we prove A n B C_ B we find

5

C.

2

(max

@=-).

As A n B is an upper bound of A and

5 A n B . Assume A n B we find a , B

C

$

C.

Then I C I = k and for each

d > y . The assumption

and thus to the contradiction A n B =

C C.

c_

A n B leads

Therefore

Ordered Akebraic Structures

38

there exists c E C \ ( A n B ) . W.1.o.g. A

let c 6 A . Together with

5 C this implies c > a contrary to a >

y.

Thirdly consider the different cases in formula (2.10). IAI, IBl < k then C

A and C

B shows C

If

A n B . As we know

A n B C_ C this proves C = A n B. In the second case IAl = k and IBl < k we know that c E C \ A

implies c > a and that

C

E B.

Therefore C C ( A n B ) U { b E B I c > a)=:D. Thus D

1. C.

As D is an upper bound of A and B we find D = C.

In the third case IAl = IBl = k we know that c E C \ A c > a and that c E C \ B

implies c > 3 . Therefore ( a

N

N

Then D = k -min(D) fulfills D

implies

5 B)

N

1. C. Further D is an upper bound

N

of A and B. Thus D = C.

The order relation in this example is neither positively ordered nor is ( 2 . 7 )

fulfilled. Therefore ( V , * , l )

may not be

embedded in the positive cone of an ordered commutative group.

An ordered commutative semigroup H is called r e s i d u a t e d if for all a , b E H there exists c E H such that (2.12)

a * x l b

a-~

x l c .

If such an element c exists for a , b E H then it is called a

r e s i d u a l o f b with respect to a. If the order relation in H is antisymmetric then a residual is uniquely determined and denoted by b:a.

If for all a , b E H there exists c E H such that

39

Ordered Commutative Semigroups

(2.12')

a * x z b

Q

x z c

then H is called dually residuated. Such an element c is called a d u a l residual and is denoted by b/a. (2.12)

(2.12')

in the lattice-theoretic sense.

A

is the dual o f

(dual) residual b:a

(b/a) is the greatest (smallest) solution o f the inequality a

*x

5 b

(a * x

2

b). Therefore the (dual) residual exists if

and only if the set { x E H I a * x

5 b)

({xEHI a * x

1. b))

has

a greatest (smallest) element. An important implication of the existence of residuals is the distributivity of the least upper bound.

(2.13) Proposition Let H be a residuated, lattice-ordered commutative semigroup. Then a

*

( b V c ) = (a * b ) V (a * c ) for all a , b , c E H .

* b,a * c 5 a * (b v c). x:= (a * b) V (a * c).

Prdof. Let a,b,c E H. Monotonicity implies a Therefore (a Then a * b

* b)

v (a * c) 1. a

*

(b v c). Let

5 x and a * c 5 x. Then b 5 x:a and c 5 x:a. Thus

< x:a which implies a * ( b v c ) 5 x. bvc -

On the other hand the greatest lower bound is not necessarily distributive.

( 2 . 1 4 ) Proposition Let G be a lattice-ordered group. Then G is residuated and dually residuated are distributive.

and greatest lower and least upper bound

rn

Ordered Algebraic Structures

40

Proof. In a group b:a = b/a = b * a

v follows from

of

(2.13)

-1

.

Therefore distributivity

and distributivity of

follows from

A

(2.13) applied to ( G , * ) ordered with respect to the dual of

the lattice ( G , Z ) .

(2.14)

shows that lattice-ordered commutative groups are

examples for residuated lattice-ordered commutative semigroups. A

more detailed discussion is given in chapter 3 .

Let (H,h,z) denote the lattice-ordered commutative semigroup derived from a bounded lattice (H.5). If (H,h,5) is residuated then from ( 2 . 1 3 )

we know that for all a , b , c E H

a h ( b v c ) = ( a h b ) v (ahc). We have found a stronger result in chapter 1 .

this case that (H,()

(H,L)

(2.12)

means in

is a pseudo-Boolean lattice. Therefore

fulfills a V (b A c) = (a v b)

A

(a v c)

for all a , b , c E H , too. If the lattice ( H , Z ) is Boolean then the residuals of ( H , h , L ) are given.by b:a (see 1.15).

=

bva*

Clearly, for pseudo-Boolean lattices the ordered

commutative semigroup (H,V,()

fulfills ( 2 . 1 2 ' ) ,

a , b E H there exists c E H such that a v x z b

Y

x l c .

i.e.

for all

3.

Lattice-Ordered Commutative Groups

In this chapter we consider lattice-ordered commutative groups (G,*,L). A characterization of such groups can be found in CONRAD, HARVEY and HOLLAND 119631. For a detailed discussion we refer to CONRAD [1970]. We introduce

basic concepts and

properties which allow a precise formulation of this characterization. Due to proposition ( 2 . 1 4 )

we know that a lattice-ordered commu-

tative group is d i s t r i b u t i v e , i.e. least upper and greatest lower bound are distributive over the internal composition.

(3.1) P r o p o s i t i o n (1)

partially ordered commutative group G is lattice-

A

ordered if and only if a v e exists for all a E G . Let G be a lattice-ordered commutative group. Let a,b E G and n E N . Then n

2

(2)

a

bn

(3)

(ave)”

y

a

2

n a ve

=

(G i s

b

isotated)

,

.

Proof. ( 1 ) Necessity is obvious. Otherwise let a : = [(a*b-’) ve] *b. Then a = a v b . Further let n (2)

a

> -

B:=

-1 (a

b-l)-l

.

Then 8 = a

b.

A

bn is equivalent to (a * b - l ) n > e. Thus it suffices

to show a Let a

n

2

2

e

o

a 2 e .

e. Then monotonicity leads to a

n

2

e. Let now a

n

2

e.

n = 1 i s trivial. Let n > 1. Then distributivity leads to (a A e)” = an

A

an-’

A

... A

e = an-1

41

A

a n-2

A

...

A

e = (a A e) n- 1

.

42

Ordered Algebraic Structures

Cancellation implies a h e = e; i.e.

2

e.

For 0 < k < n we find

(3)

(an-k va -k ) n = a (n-k)nv Therefore

implies a

(2)

(aveIn = a

A

a

n

va

n-k

n-1

v a

. . .va (n-k)k-k (n-k)v . . . v ~- kn

>

e.

2 e , i.e. a n v e 2 a k . Then

-k

v...v~

= a

n

ve.

semigroup (group) (H,*) is called d i v i s i b Z e or r a d i c a b l e if

for all a E H and all nclN there exists b E H such that a = bn.

(3.2) P r o p o s i t i o n A

lattice-ordered commutative group G can be embedded into a

divisible, lattice-ordered commutative group Proof. Let set G

X

denote the set of all equivalence classes of the

{ -n1I

nclN) with respect to the equivalence relation

defined by 1

(a,;) Then

(G, 0

c.

-

1

(b,i)

:

-

am = bn

N

.

is a commutative group with internal composition

)

1 (a,;)

0

(b,;):= 1

(ambn,

L, mn

where elements in

are represented in the usual form. This

definition is independent of the choice of the representing elements as G is cancellative. Let W: G

-

+

G defined by Q(a):=

(a,l). Then

The lattice ordering o n G is given by the positive cone G+ of G. Now

G+:=

1

{(a,;)

1

a E G + j defines a partial order on

e.

This

Lattice-Ordered Commutative Groups

43

definition is independent of the choice of the representing elements as G is isolated. Clearly, (P(G+) = (P(G) fore a

5 b

-

V(a)

cellation arguments. Thus tative group. From (3.1.1)

(E,@,z)is

1 upper bound of (e,l) and (a,;).

2

and can-

a partially ordered commu-

we know that it suffices now to

1 show the existence of (e,l) V (a,;)

xn

There-

1. V(b)

for all a , b E G . Monotonicity follows from (3.1.2)

fore using (3.1.3)

n G+.

e ~a

m

in

Then x

G. 2

Let (x,:) e and xn

m = ( e v a ) , i.e.

denote an

.

2

m a

1

1 z(eva,-). n

(x,;)

There-

Similarly we find that ( e V a , L ) is such an upper bound. Thus n 1 1 (e,l) v (a,-) = ( e v a,-) n n

A

.

special divisible, lattice-ordered commutative group can be

constructed from the root lattice V = V(A) in chapter 1.

The

components x A for A E A of an element x E V are real numbers. Therefore an internal composition

+

on V can be defined by com-

ponentwise addition of real numbers, i.e. (a + bIA:= a A for all A € A

+

bA

and a , b E V . Then (V,+) is a divisible, commutative

group. V(A) is a subset of the Cartesian product X X E A H A H A = IR for all

A E A.

with

If ( H A , + ) is a subgroup of (IR, + ) then the

same construction leads to the r o o t l a t t i c e

= ? ( H ~ , A ) with

r e e p e c t t o t h e f a m i l y ( H A , A ) and to a commutative group which is not necessarily divisible.

(v,+)

Ordered Algebraic Structures

44

( 3 . 3 ) Proposition

-

V ( V ) is a (divisible) lattice ordered commutative group.

Proof. It suffices to show monotonicity. Let a proof of (1.3) we know that aM ordered subset of A .

5

b. From the

5 b M for each maximal linearly

This means that either L:= { L I E M I a

P

9 b 1 P

is empty or a A 5 b A for A = max L. Now let c E V . Then L = (uEMI

a

l

+c J

U

*blJ+cU) M

and therefore a M + c M < bM+c

.

As this is satisfied for each

maximal linearly ordered subset of A we find a + c

1. b + c .

V is called a H a k n - t y p e group and is considered in CONRAD, HARVEY and HOLLAND [1963]. They prove the much deeper result that each lattice-ordered commutative group can be embedded into a Hahn-type group. A proof of this important theorem is beyond the scope of our book; but in order to understand a more detailed formulation of this result we introduce some basic concepts. In the following G is a divisible, lattice-ordered commutatfve group. Divisibility can be assumed w.1.o.g.

due to proposition

(3.2). A

subset G' of G is called convex if a , b E G ' imB.ligs that

[a,b] C_ G'.

A

convex subgroup G' which is a sublattice of G is

called a l a t t i c e i d e a l of G. The set I of all lattice ideals is partially ordered by set inclusion.

A

lattice ideal G' is

called r e g u l a r if there exists an element g E G such that G' is maximal in I with respect to the property g EG'. Let (GA,h) denote the family of all regular lattice ideals of G.

Luttice-Ordered Commutative Groups

CONRAD,

45

HARVEY a n d HOLLAND [ 1 9 6 3 ] show t h e e x i s t e n c e o f

unique l a t t i c e i d e a l Gh covering G h f o r h E A.

Gx

a

is called

t h e cover of Gh f o r A E A .

A s a s i m p l e example w e c o n s i d e r t h e Hahn-type

with T = { 1 , 2 , 3 } .

T h e root s y s t e m

r

group V ( r )

is o r d e r e d as d e s c r i b e d

i n figure 4.

Figure 4.

T h e n V = lR

and G I = V,

'.

The r o o t s y s t e m

r

We f i n d t h r e e r e g u l a r l a t t i c e i d e a l s

G 2 = G 3 = G1.

Due t o c o m m u t a t i v i t y a * G

Therefore w e d e f i n e A:=

* a-l

r.

= G h f o r a l l a E G h and h E A ~

h

IGx i s c a l l e d a norrnaz subgroup of G 1.

group

x G /GX

T h e r e f o r e t h e factor

d e f i n e d by h G / G ~ : =C a * G A ] h E A )

is w e l l - d e f i n e d

with r e s p e c t t o t h e i n t e r n a l composition

Ordered Algebraic Structures

46

A

Due to convexity

G /GX

is totally ordered by the well-defined

induced ordering given by (a * G x )

5 (b * G x )

5 b

a

91.

.

Using the fact that G h covers G x it can be seen that the factor group

x

is isomorphic to a subgroup o f the additive

G /Gx

x

group of real numbers. In our above example every G / G x is isomorphic to IR. For A , p E h let A

u if and only if A

be seen that

is a root system, which has the following

(A,f)

= p or G

x -3 G ' .

It can

two properties: (1)

V g E G ,

(2)

g@G'

An index

gPe

*

3 )tEA

3 AEA,

:

h > p :

g E G

h

\G

in ( 1 ) is called a v a l u e of g.

A

subset 'E; of h N

satisfying ( 1 ) and ( 2 ) is called p l e n a r y .

x

A '

x g E G \ G x .

It can be seen that A

N

is a root system. Therefore V ( G / G x , A )

is a lattice-ordered

commutative group (cf. proposition 3 . 3 ) .

In the above example A is order-isomorphic to

r.

Properties ( 1 )

and ( 2 ) are easily checked. Clearly, A = 1 is a value of (l,O,O).

x

Further A is the only plenary set and V ( G / G x , A ) to

is isomorphic

v(r).

In general, a group-embedding CP:

V a l u e - p r e s e r v i n g if

G

x

+

N

V(G / G x , A )

E'E; is a value of g

E

G

is called

if and only if q ( g ) x

is a maximal nonvanishing component of q ( g ) and, in this case, rP(g)x = g * G x . in particular

Such an embedding Q(G)

is always a lattice embedding;

i s a sublattice of

v.

47

Lattice-Ordered Commutative Croups

Now we can give a detailed version of the embedding result of CONRAD, HARVEY and HOLLAND 119631. For a proof we refer to CONRAD [1970].

( 3 . 4 ) Theorem Let G be a divisible, lattice-ordered commutative group and let ( G A , A ) be the family of all regular lattice ideals of G. Let

?i

be a plenary subset of A .

Then G can be group-embedded

into V = V(?i). This embedding is value-preserving and therefore the image of G is a sublattice of V.

In general, there may exist many different plenary sets and for each of them we get an embedding. The discussion in CONRAD, HARVEY and HOLLAND 119631 shows on the other hand that if A is finite then A is the only plenary set and G is isomor-

-

A

phic to V(G /GA,A). Thus, for finite A lattice-ordered

commu-

-

tative groups are represented by V(HA,h), where H A is a subgroup o f the additive group of real numbers and A a root system. Other examples with interesting properties are discussed in CONRAD, HARVEY and HOLLAND 119631.

If the root system A is linearly ordered then V(A) is called a H a h n - g r o u p . Then V ( A ) is a linearly ordered commutative group. In linearly ordered groups a lattice ideal is called

i d e a l . Let G be a divisible, linearly ordered commutative group and let ( G A , li € A )

be the family of all regular ideals

of G. Then A is linearly ordered. Theorem ( 3 . 4 ) the theorem of HAHN 119071.

reduces to

Ordered Algebraic Structures.

48

(3.5) T h e o r e m Let

be a divisible, linearly ordered commutative group and

G

let ( G A , A € A ) Then

G

be the family of all regular ideals of G.

can be embedded in the divisible, linearly ordered

commutative group

V(A).

For a particular proof of this theorem we refer to FUCHS 119661. Theorem (3.4) and (3.5) are valuable characterizations of lattice- (linearly) ordered commutative groups and provide a general view of such ordered groups.

Our simple example illustrating the various definitions and denotations above can easily be extended to finite root systems A.

In particular, if A is a linearly ordered set with n ele-

ments then V ( h ) is isomorphic to the additive group Wn of real vectors which is linearly ordered by the l e x i c o g r a p h i c

o r d e r r e l a t i o n of vectors defined by (3.6)

x < y

:-

x =

y

or

xi
for i=rnin{jl x

We remark that for A = { A 11X21...,Xn} with X 1 > A 2 > . . . the component x

1

*y.I. 1

>An

for x E V ( 6 ) corresponds to xi for xEIRn. Ai

An infinite example is the additive group of all real functions f: [0,11 + IR with (f + f(x)

5 g(x) for

g)

(x): = f (x) + g ( x ) and f

1. g defined by

a l l x € [0,1].This is a divisible, lattice-

ordered commutative group isomorphic to V ( A ) for the root system A = [O,l1 trivially ordered by x

5 y iff x

=

y.

Finally w e consider a subgroup of the additive group of all

49

Lattice-Ordered Commutative Groups

real functions f: A

[ O , l ] +lR w i t h f ( 0 ) = 0 . L e t x E [ 0 , 1 ] .

f i n i t e p a r t i t i o n Px o f t h e i n t e r v a l [ O , x l c a n b e r e p r e s e n -

1 of i t s interval-endpoints; i n

t e d by t h e s e t { a o , a l , . . . , a p a r t i c u l a r 0 = ao C a l < . . . < a

= x.

r

r

I

v(x,pX):=

-

If(ai)

Now d e f i n e

f(ai-l)I

i= 1 If V ( b , P ) i s b o u n d e d by a r e a l b

f o r such a r e a l function f .

c o n s t a n t B i n d e p e n d e n t o f Pb t h e n

e x i s t s a n d i s c a l l e d t h e total variation o f f i n [ O , x ] . t h i s c a s e we s a y t h a t f L e t G denote

the

i s o f bounded variation.

set of a l l r e a l functions f :

bounded v a r i a t i o n w i t h f ( 0 ) = 0. C l e a r l y ,

A

function f :

if x

5

[ 0 , 1 ] +IR

of

t h i s is a subgroup

of t h e a d d i t i v e group of a l l r e a l functions f : I t s n e u t r a l element

In

[ 0 , 1 ] + IR.

( f n 0 ) w i l l b e d e n o t e d by 0.

[ 0 , 1 1 +IR

y implies f ( x )

5

is c a l l e d monotonicaZZy non-decreasing f ( y ) . Such a f u n c t i o n w i t h f ( 0 ) = 0

is o f bounded v a r i a t i o n , I n p a r t i c u l a r V(x) = f ( x ) f o r a l l x E [0,1]. order

1. o n

The s e t G+ o f a l l s u c h f u n c t i o n s i n d u c e s a p a r t i a l

,

G a s G + + G + C_ G+

OEG+

a n d G + fl -G+

o r d e r t o show t h a t G i s a l a t t i c e - o r d e r e d t o show t h a t f v 0 e x i s E Z - f o r a l l f

51 ( V ( X )

(3.7)

f * ( x ) :=

f o r f E G.

Clearly f = f+

f o r a l l x,y with x

5

y.

*f -

= (0). I n

g r o u p i t now s u f f i c e s

rc< L e t

~~

_ ~ _

(XI)

f-

.

F u r t h e r f + , f - E G+ as

Therefore f +

2

0,

f+

2

f.

Now assume

50

Ordered Algebraic Structures

that g is an arbitrary upper bound of f and 0. We will show g

f+

.

As

g

-

f+ = ! ( g 2

ces to prove g - V

2

-

f)

0. Let x

+

1 ~

( -gV ) and g

1. y.

Then g

-

f

-

f

2

1. 0 0, g

it suffi-

2

0 im-

plies

-

If(y)

Now for a partition P Px:= (Py

n

f(x) I f S(Y )

-

.

g(x)

we define a corresponding partition

Y

[O,xl) U { X Iof [O,x]. Then the above inequality

used for points in (P f l [x,y])U{x) Y V(Y,PY) and therefore g - V

2

implies

5 V(X.PX) + g(y) -

g(x)

0. Summarizing we have proved the follow-

ing proposition.

(3.8) P r o p o s i t i o n Let G denote the additive group of all real functions f: [O,ll -+lR of bounded variation with f(0) = 0. Let G,

denote the set

of all monotonically non-decreasing functions of G. Then G is

a divisible, lattice-ordered commutative group with respect to the positive cone G + .

Meet and join are given by

(f+g+V(f- 9 ) ) / 2

,

f v g = (f+g-V(f-g)) / 2

.

f A 9 =

For further examples of lattice-ordered commutative groups we refer to CONRAD 1 1 9 7 0 1 .

4.

Linearly Ordered Commutative Divisor Semigroups

In this chapter we develop a characterization of linearly ordered commutative semigroups (H,*,z) which satisfy the additional axiom (4.1)

a C b

-D

3 cEH:

a * c = b

(Divisor rule)

for all a , b E H . An element a E H is called a d i , v i s o r of b E H if there exists c E H with a * c = b. Thus ( 4 . 1 )

means that an

element a is a divisor of all strictly greater elements. Therefore we call such semigroups d i v i s o r s e m i g r o u p s . For convenience we denote a linearly ordered commutative divisor semigroup shortly as d - s e m i g r o u p . Positively ordered d-semigroups have been characterized by CLIFFORD ( 1 1 9 5 4 1 ,

[1958]

and 1 1 9 5 9 1 ) .

D-semigroups which are

not positively ordered were characterized by LUGOWSKI 1 1 9 6 4 1 in a similar manner extending the results of CLIFFORD. The positively ordered case is also covered by the discussion of positively totally ordered semigroups i n the monograph of SATYANARAYANA 1 1 9 7 9 1 . From proposition ( 2 . 9 ) we know that we can assume w.1.o.g.

the

existence of a neutral element in a positively ordered d-semigroup. Later'on we will see that a d-semigroup containing elements which are not positive is always a d-monoid. Therefore the discussion of d-monoids and d-semigroups leads to the same results. A positive element a E H satisfies b 5 a * b for all b E H (cf. 2.6). Since d-semigroups are linearly ordered there exists b E H with b > a * b if an element a is not positive. 51

Ordered Algebraic Structures

s2

(4.2)

Proposition

Let H be a d-semigroup and a E H . Then the following two properties of a are equivalent: i s not positive,

(1)

a

(2)

a > a*a.

Further both properties imply

2

13)

b

Proof.

(1)

a*b I ,

for all b E H .

(2). Then b > a * b for some b E H and a * b * c = b

for some c E H . Therefore a * b * c > a (2)

by cancellation of b * c . ( 2 )

(2)

*

*

*

(a * b * c ) which implies

( 1 ) is obvious.

( 3 ) . The case a = b is obvious. L e t b < a. Then a = b * c

for some c E H . Thus b * c > a * b * c . Cancellation of c implies b > a * b . Let a

4

b. Then a * c = b for some c E H . Suppose

*c

b < a * b . Then a

Proposition ( 4 . 2 )

< a

*a*c

implies a < a * a contrary t o ( 2 ) .

shows that an element a is negative i f it

is not positive. In general, if the d-semigroup is not a group, such an element is not strictly negative. Motivated by ( 4 . 2 . 2 ) we call it self-negative.

Let

(A,L) be a nonempty, linearly ordered set and let (HA,*;X,LA)

be a linearly ordered commutative semigroup for each A € A .

The

sets H A are assumed to be mutually disjoint. Let H be the union of all HA. Then we continue internal compositions and order relations of the H A on (4.3)

a < b

for all a E H A

and

,

b E H

€3

by

a*b = b*a = b U

with A <

JJ.(H,*,z)

is called the

Linearly Ordered Commutative Divisor Semigroups

o r d i n a l 8um of the family

53

(HA; A E A ) .

It can easily be seen that (H,*) is a commutative semigroup and (H,L) is linearly ordered.

(4.4) P r o p o s i t i o n Let H be the ordinal sum of the family ( H A ; (1)

'

A).

Then

H is a linearly ordered commutative semigroup i f and only if all H

li

are positively ordered f o r

!J

t: min A .

If H is a linearly ordered commutative semigroup, then (2)

H is positively ordered Iff all H A are positively ordered,

(3)

H is a d-semigroup iff all H A are d-semigroups.

Proof. (1) Let H be a linearly ordered commutative semigroup and let a € H A

-< b * c .

,

b,c E H

u

with

)i

< p . Then a < b implies c = a

*c

For the reverse direction It suffices to show monotoni-

city. This follows easily by considering the possible cases a , b E H A and c E H ; a E H find that a (2)

11,

b E H A with p <

)i

and c E H . We always

5 b implies a * c 5 b * c .

H positively ordered immediately implies that

set i s positively ordered.

As

aEHA

, b E H p with

HA

as a sub-

A < p implies

a < b = a * b , the reverse direction immediately follows. (3)

Let H be a d-semigroup and let a , b E H A with a < b. Then

a * c = b f o r some c E H . Now c b H A implies a * c = a

or

a * c = c,

i.e. a contradiction. Thus H A is a d-semigroup. The reverse direction follows from a * b = b for a E H A , b E H

IJ

with A < p

. 8

A

d-semigroup is called ordinaZty i r r e d u c i b l e or shortly i r r e -

d u c i b l e if it is not the ordinal sum of two or more subsemi-

Ordered Algebraic Structures

54

groups. The following theorem shows that each d-semigroup can uniquely be decomposed into irreducible subsemigroups; in other words each d-semigroup has a unique representation as the ordinal sum of a family of subsemigroups. Such a theorem was given for positively ordered commutative semigroups by KLEIN-BARMEN

(119421,

[1943])

and CLIFFORD 1 1 9 5 4 1 ,

ordered commutative semigroups by CLIFFORD 1 1 9 5 4 1 ,

for naturally and for d-

semigroups by LUGOWSKI 1 1 9 6 4 1 .

(4.5) Theorem (1)

Each d-semigroup H has a unique representation as ordinal sum of a family of irreducible d-subsemigroups.

( 2 )

If H contains a self-negative element then there exists a containing all self-nega-

first irreducible semigroup H A 0

tive elements of H. Proof. We consider certain partitions

(L,U) of H , i.e. L n U = #

and L U U = H. We call (L,U) a cut if L < U , if L is a subsemigrou and if a * b = b for all a E L , b E U . The set of all cuts is total1 ordered by (L,U) < (V,W) iff L such that (L,U)

5

(R,S)

5

V. A pair of cuts (L,U) < (V,W)

5 (V,W) implies (L,U)

=

(R,S)

or

(R,S) = (V,W) is called essential. Then (V,W) is called the

immediate 6 u c c e s 6 o r of (L,U). Let A denote the set of all essential pairs. For h = [(L,U),(V,W)] € A define the subset H := V

n

U.

Clearly V i s a convex subsemigroup of H. If positive elements then

U

U

contains only

i s also a convex subsemigroup o f H.

If c E U is self-negative then for b E L we find b < c = b * c . But this contradicts ( 4 . 2 . 3 ) .

Therefore L = fl and H = U.

Linearly Ordered Commutative Divisor Semigroups

55

Thus U is also a convex subsemigroup. This shows that H A is a convex subsemigroup. We claim that H is the ordinal sum of (HA ;

A E A)

If

i s not irreducible but the ordinal sum of its subsemi-

HA

and that each H A is irreducible.

groups A A < B A then ( L U A A I B A

u W)

is a cut between (L,U) and

(V,w) contrary to the definition of A .

Clearly the ordinal sum

of the family (HA ; A E A ) is well-defined. For a E H let A denote the union of all t o w e r

L of cuts (L,U) with a 6 L

CZQSSe8

and let B denote the intersection of all lower classes L o f cuts with a € L. Then A : =

[(A,H\A),(B,H\B)]

is an essential

pair and a E H A . Therefore H is the ordinal sum of the family (HA

;

A E A).

If H contains self-negative elements then due to ( 4 . 4 . 2 ) know that A has a minimum

)i

0

we

contains all self-negative

and H hO

elements. To prove that each H A is a d-subsemigroup of H let now a , b E H A

with a < b. Then there exists c E H wich a * c = b. If c a H A then either a * c = a

or

a * c = c. In both cases a * c 4 b. Thus

CEHA. Finally we show that this representation is unique. Assume that H is the ordinal sum of irreducible T

as well as (Lr U T r l U r \ T r )

u 1

pEM.

Then (L ,Ur)

are cuts. As Tr is irreducible we

find [ (Lr,Ur), (Lr U T r l U r \Tr)] E A .

This construction yields a

bijection between the family of the H A and the family of the T IJ since both families are partitions of H.

Ordered Algebmic Structures

56

The family ( H A ;

will be called the o r d i n a l d e c o m p o s i -

-h E A )

t i o n of H. From the proof it is clear that any positively, linearly ordered commutative semigroup has a unique ordinal decomposition, too. The only point in the proof using ( 4 . 1 ) was an argument treating the existence of self-negative elements. This case does not appear if H is positively ordered. Then all subsemigroups

--h

)

are positively ordered,

too.

If H is a d-semigroup then all irreducible subsemigroups are naturally ordered with the only possible exception of the

.

first subsemigroup H

An element of a semigroup is called

xO

i d e m p o t e n t if a * a =

a.

For example, the neutral element is always idempotent.

(4.6) Proposition An irreducible, positively ordered d-semigroup H contains at most one idempotent element a. Then a = max H and a * b = a for all elements b E H . Proof.

Assume the existence of an idempotent element a. Irre-

ducibility implies that the only cuts are (H,$) and ($,HI. Let U:=

{ x > a?. Monotonicity implies that L = H L U and U are sub-

semigroups of H. Now let x E L , y E U . Then a < y and a * z = y for some z E H . x

< -

5 a and H positively ordered imply a 5 a * x

a * a = a. i.e. a = a * x . Hence x * y = x * a * z = a * z = y.

Therefore (L,U) is a cut. As a E L we find U = $. Then a = max H and a

5 a

* b

5 a *a

=

a for all b E H complete the proof.

Linearly Ordered Commutative Divisor Semigroups

57

In the positively ordered case we know that we can assume w.1.o.g.

that (HI*,()

contains a neutral element. Then e

for all a E H and thus (Ie),H\{e))

5 a

is the immediate successor

of (@,HI. This shows that A has a minimum X

0

= {el.

with H A 0

If (H,*,Z) is a d-semigroup and contains self-negative elements then from ( 4 . 5 . 2 )

again we know that A has a minimum

A.

-

Then let P- denote the set of all self-negative elements and P+:= I C E H I 3 a , b E P - : a * c = b)\PFurther

M:= P - u P +

and

M

.'

= H X M . Clearly, P -

C

P+.

The following result i s due to L U G O W S K I 1 1 9 6 4 1 .

(4.7) Proposition Let H be a d-semigroup with P-

+

@.

Then M C

and M is a linear-

ly ordered commutative group with positive cone P + . Proof. At first we prove M

+

let (Cf.

$.

C

i.

This is trivial if

Let c E M and d E i . Then c

4.2.3).

+

i

= $.

d. If c E P - then c * d

Further d is positive, and in particular c

=

Assume d < c. Then d * x = c for some x E H . Further a * x

5

-C

a * x * d = a * c = b. But a * x

5 d

5 c *d.

5 d. If c E P + then there exist a , b E P - with a * c

Thus c

Thus

b.

5 b E P - implies a * x E P - . Thus

(a * x ) * d = b implies d E P + contrary to the choice of d. Secondly, we prove that M i s a d-subsemigroup of H with positive cone P + . for c , d E P - .

For example consider a E P - ,

Then c

*

b E P + with c * b = d

(a * b ) = d * a € P- implies a * b e P + .

Other cases can be treated with similar arguments. Thus M is a subsemigroup. Now let a , b E M with a < b and a * x = b for some x E H. For example we consider a , b E P+. Then let c * b = d with

58

Ordered Algebraic Structures

Now c * a 5 c * a * x = c * b = d E P - implies c * a E P -

c,dEP-.

and therefore x E P +

.

Thirdly, we prove the existence of a unique idempotent element, i.e. of aEt4 with a

2

=

a. For c E P - we find c

fore for some x E P + : c * c * x = c. Then (c *XI'

*c

.

=

< c and therec *x

and

c * x E P+. On the other hand P + is a naturally ordered commutative semigroup. Due to ( 4 . 5 )

it has a unique representation as

ordinal sum o f its irreducible subsemigroups. From ( 4 . 6 )

we

know that each o f these subsemigroups contains at most one idempotent which then is its maximum. Therefore for any idempotent a E P + and all b E P + we find b < a

*

b * a = a

a < b

*

a * b = b .

(4.8)

NOW let a,b denote two idempotent elements in P + .

W.1.0.g.

assume b < a. Let c , d E P - with c * a = d. Then d = c * a = c * a =

2

d * a . P - < P + implies d < b. Therefore d * x = b for some xEP,.

Then b*a = d*x*a = d*x = b < a contrary to ( 4 . 0 ) . The unique idempotent of P

+

e is a neutral element of M.

is now denoted by e. We show that If c E P - then similarly as in the

existence proof of an idempotent element we find c * c * x = c and c * x = e. Thus c * e = c. If c E P + and e < c then ( 4 . 8 ) plies c * e = c. Now suppose the existence of c E P + ,

im-

c < e.

For a E P - we find a(a*cla*e=a, i.e.

a

=

a * c . Since a < c let a * x = c for some x E P + . Then

59

Linearly Ordered Commutative Divisor Semigroups

c*c = a*x*c = a*x = c implies c = e contrary to the assumption c < e. Hence e = m i n P + Finally we show the existence of inverse elements. Let a E P Then a * x = e for some x and x = a Then a = d

* C-l

-1

.

'

.

For e < a E P + let c * a = d .

and a - 1 = d - 1 * c as

a*(d

-1

*c) = d * d

-1

= e. H

Proposition ( 4 . 7 )

shows in particular that a d-semigroup with

P - 9 @ always contains a neutral element. If e is the neutral

element of M then for a E H \ M

there exists x C H with e * x = a.

Then a = e * x = e * e * x = e * a implies that the neutral element of M is a neutral element of H. The existence of a neutral element in a d-semigroup always implies the existence of a minimum A

If P - = @ then we may adjoin a neutral element.

of A .

Therefore we can assume w.1.o.g. and A has a minimum A Further M C_ R A

0

, 1.e.

that a d-semigroup i s a d-monoid

. M is always contained in the first irre-

0

ducible semigroup H A

of the ordinal decomposition of H. Suppose

0

. Then a * b = b for all e * b. Multiplication with

the opposite, i.e. let b E M \ H A For a

+

e this implies a

*b

aEM.

0

=

b-l

yields a = e contrary to the choice of a.

(4.9) P r o p o s i t i o n Let H be a d-semigroup with P -

* @.Then

H is a d-monoid and

-

(1)

M * M = M,

(2)

a E M , b E M and a * c = b imply c E M ,

(3)

H is the ordinal sum of M and naturally ordered,

M if and only

if

is

Ordered Algebraic Structures

60

if

(4)

i

has a minimum then

Proof. ( 1 ) Let a E M, b E

i.

is naturally ordered.

Assume a

contrary to b E i . Thus M * M

* b E M.

c_ i. {el * i

=

Then a

i

-1

*

(a * b) E M

shows the reverse

inclusion. Let a E M , b E i and a * c = b. Then c = a

( 2 )

= a

-1

(3)

-1

*

(a * c )

*bEM. If H is the ordinal sum of M and

then let a , b E M with a < b .

For some c E H we find a * c = b. If c E M then a * c = a contrary to a < b. Thus c E i . As e <

i

we know that

i

is positively

ordered. For the reverse direction we assume that ordered. From ( 4 . 7 )

we know M <

i, M

is naturally

is a group and, as

i

is

is a subsemigroup of H. It suffices to

positively ordered,

show that a * b = b for all a E M , b E M . Let b E M . For a = e this is trivial. Let e < a. Then b b

*x

= a

*b

for some x E

cancellation a (2).

a

-1

(4)

-1

*x

Now a - 1

*b*x

< a * b . Then

= b < a

< a E M; in particular a-l

Now consider the case a < e. Then e < a * b = b.Composition with a yields b = b

*b

*xEM

-1

implies by contrary to

and therefore

* a.

It suffices to show a * b = b f o r all a E M , bE'M. Then H

is the ordinal sum of bE

i.

5 a * b . Assume b

E. Let

M

and

i

and we can apply ( 3 ) . Let a E M ,

d denote the minimum of

i.

for some x E M . Assume e < a. Then x case a < e we know e < a

-1

From ( 2 ) we know a * x = d

5

d implies x = d. In the

and therefore a

-1

* d = d. Again this

implies d = d * a . Now let d * y = b for y E H . Then

for all a E M .

61

Linearly Ordered Commutative Divisor Semigroups

We can now describe the ordinal decomposition of d-semigroups we assume that a d-semigroup is a

in more detail. W.l.0.g.

d-monoid (possibly after adjoining a neutral element in the case p - = $1. The following result is due to LUGOWSKI [ 1 9 6 4 ] .

(4.10)

(Decomposition o f d-monoids)

Theorem

Let H be a d-monoid. Then: H has a unique ordinal decomposition (H

.

A '

a minimum A

0

and I\ has

. ,

and all H

H\HX

AEA)

A

*

A.

are naturally ordered.

0

If P- =

0 then

€lh

= {el.

0

If P-

*

@ and H A

is a group then H

0

If P -

$

@ and H A

= M. AO

is not a group then 0

a)

M

5

HA

and M -c R : =

HA

0

b)

R

\M,

0

is a positively ordered commutative semigroup, is

not naturally ordered, and has no minimum, c)

R is irreducible and contains at most one idempotent

element a, which then fulfills a-=

and a

*b=a

XO

.

for all a E H

max H

AO

Proof. (4.5).

(4.7)

and ( 4 . 9 )

imply (1)

-

fli

(4) and (5a). R = = . H A 0

shows that R is a positively ordered commutative semigroup. If R contains a minimum

or is naturally ordered then H

is the A.

ordinal sum of M and R .

As

H

i s irreducible we find (5b). AO

Now we suppose that R is the ordinal sum of S I T with

S

C

T.

62

Ordered Algebmic Structures

Then consider M U S , T . Clearly M U S < T. M U S is a subsemigroup as a * b

5

b * b E S for a E M , b E

S is a subsemigroup due to

S.

our assumption. Let a E M U S , b E T . For a € S we know a * b = b due to our assumption. Now consider a E M . If a E P + then for c E S

x

-

we find a * x = c for some x E M (use 4 . 9 . 2 ) .

5 c and therefore x E

S. Thus x * b

=

b which implies

b = c * b = a * x * b = a * b . For a E P - we know a

a

-1

a E P + implies

-1

E P + . Thus

* b = b. Multiplication with a yields b = b * a .

is the ordinal sum of M U S and T contrary to H

Hence H

x0

irreducible. hO

Thus R is irreducible.

Now assume that R contains an idempotent element a and consider the partition S:= I b E R I b

5 a), T

S and T are subsemigroups of R .

.

some x E H A

= R L S .

Clearly

S

C

T and

Let b E T . Then a * x = b for

Therefore a * b = a * a * x = a * x = b. Let c E S.

0

Then a 5 a * c < a * a = a shows a = a * c . Thus we find c * b =

c * a * b = a * b = b. As R is irreducible and a E S we know T = @ .

.

Therefore a = max R = max H

For c E R we have already shown

hO

a * c = a. Let a E P + . Then a For c E P - we know c

-1

5 a *c 5 a *a

E P + and thus c

-1

= a. Therefore a

* c=a.

* a = a. Again multipli-

cation with c yields a = a * c . m

The following example illustrates the situation occurring in (4.10.5). Let H = { O ,1 , . (H,*,*)

. . ,n)

X

Z

u

{ (n + 1,O)

for n

2 1. Then

is a d-monoid with internal composition defined by

(a,b) * (c,d) : =

(a + c,b + d)

if a + c < n + l , otherwise

Linearly Ordered Commutative Divisor Semigroups

and with lexicographical order relation P- =

{ol

x

Izl z < o ) ,

.

Now we will show H = H For (0,z)E M with z (0,z)

+

*

AO

.

We find

P+ =

We know that always M = P U P +

0 and

(x,y) 4

4

63

(x,y) E M \

(n+1,0

-

C

HA

.

0

we find

(X,Y).

Therefore an upper class U of a cut of H can contain at most one element, i.e.

(n+l,O). But then (1,O) is an element of the

n+l

lower class L and therefore (1,O)

=

(n+l,o) implies U = @.

Hence H is irreducible. Obviously H is not a group. Further R = (4.10.5)

i.

All statements in

are easily verified. In particular (n+1,0) is the

only idempotent element in R. Theorem (4.10) shows how we may construct examples from a set

of naturally ordered d-semigroups and a first semigroup as described in ( 4 . 1 0 . 5 ) .

In fact, if we can characterize such

semigroups in the irreducible case then we have characterized all d-monoids. Unfortunately such a result is not known in general. Nevertheless, if additional properties are assumed, then such characterizations have been developed.

A

d-semigroup H is called C a n c e Z Z a t i v e if it fulfills the can-

cellation law ( 2 . 4 ) , a * c = b * c

i.e.

*

a = b

for all a , b ,cE H. From theorem ( 2 . 8 ) we know that if H is positively ordered then H+ = { a \ e

1. a)

cone of a linearly ordered group.

is isomorphic to the positive

Ordered Algebmic Structures

64

(4.11) P r o p o s i t i o n Let H be a cancellative d-monoid. Then H can be embedded in a linearly ordered commutative group G such that H + is isomorphic to the positive cone G + of G. Proof. If H is positively ordered then we refer to proposition (2.8).

Otherwise P-

@.

$

Now H+ is naturally ordered. Therefore H+ is isomorphic to the positive cone

G+

of a linearly ordered group G . Let U): H+ N

denote the isomorphism and extend rp to Ip: H for all a

-1

+ G

by q(a

-1

+

) =

G+

(cp(a))-'

a -1 E P - iff a E P + this is well-defined. CP is d. ,

EP-.

As

an embedding of H into G. w

An ordinal sum H of cancellative d-semigroups fulfills the

weak c a n c e 1 l a t i o n law a * c = b * c for all a,b,c E H .

A

*

a = b

a * c = c .

V

d-semigroup fulfilling this law i s called

u e a k l y c a n c e l t a t i v e . Vice versa we may ask whether all weakly cancellative d-semigoups have an ordinal decomposition in cancellative d-semigroups.

(4.12) P r o p o s i t i o n Let H be an irreducible weakly cancellative d-semigroup. Then H is cancellative. Proof. If H = {a) then A is the trivial group. Now we assume IHI > 1 . At first let A be naturally ordered. Then H can contain

at most one idempotent a and then a * b = a for all b E H (cf. 4 . 6 ) . Let L = H\{a}

and U = {a).

Then L

C

U

(cf. 4 . 6 )

and if L is a

65

Linearly Ordered Commutative Divisor Semigmups

subsemigroup then H is the ordinal sum of L and U.

Therefore

for some b,c < L we find a * b = b * c = a. But then weak cancellation implies a = c contrary to the choice of c. Therefore H has no idempotent element.

Now define a binary relation by a - b Clearly

'c

w.1.o.g.

:-

max(a,b) < a

*b.

is reflexive and symmetric. Let a l b l c E H and assume

a < c. If max(a,c) = a

*c

Weak cancellation implies a * b = a

then or

a

*b

=

a *b

* c.

-

b = b * c . Therefore

is transitive. Let L denote an equivalence class of

-.

Then

for a , b E L we find a * b € L since weak cancellation leads to a

-

a *b. If a < c < b then b < a * b 5 a * c shows a

-

c I 1.e.

c E L . Thus equivalence classes are convex subsemigroups. Let U be another equivalence class and let a < d for a € L, d E U

w.1.o.g. Therefore

Then L

-

C

U leads to an ordinal decomposition of H.

has only one equivalence class H. Then a , b E H satis-

fy a < a * b which implies that H is cancellative. Secondly, we consider the case that H contains self-negative elements. Then H = P - U P + U R as given in ( 4 . 1 0 . 4 )

or ( 4 . 1 0 . 5 ) .

If R = @ then H is a group and thus cancellative. We will show that the naturally ordered d-subsemigroup

fi:=

(P+\{e))

irreducible. Suppose 'i is the ordinal sum of A < B. As and a < a * b for all a , b E P+ we find (P+\{e)) sider A':= P -

u

{el U A . If

U R is A

*

@

A. Now con-

is the ordinal sum of A and B then

H is the ordinal sum of A' and B. Therefore

a

has to be irre-

ducible. Then % is cancellative which implies that H+ is cancellative. Now let a,b,c € H and a * b = a * c . If a b R then composition with a-l shows b = c. Assume a E R .

If b,c E H,

then

Ordered Algebraic Structures

66

we know b = c . Assume w.1.o.g.

b E P - . Then a = a

If c E R then cancellation in K + leads to e = c 4.9.1) contrary to eff R. Assume c f f R . leads to c

-1

b

=

-1

, i.e. b

*b

Then a * c

-1

* C

-1 =

*b

-1

.

E R (see a *b

-1

= c.

C O N R A D [I9601 shows that a weakly cancellative, positively

ordered d-monoid is the ordinal sum of cancellative, positively ordered d-monoids. Theorem (4.10) and propositions ( 4 . 1 1 ) and (4.12)

imply ( 1 ) and ( 2 ) in the following more general theorem.

( 4 . 1 3 ) Theorem Let H . b e a weakly cancellative d-monoid. Then H has a unique ordinal decomposition ( H ;~X € A )

11)

has a minimum X (21

and A

0 '

all H A are cancellative and can be embedded in groups G X , and

for X > X o

(31

all H X are isomorphic either to the strict

positive cone of G X or to the trivial group G X .

Proof. It remains to show (3). From (4.10) we know that H A is positively ordered. If H A contains a neutral element e X then e

< a for all a E H A

h -

.

due to (4.6) we find a

On the other hand e X is idempotent and

5

e X for all a € HA. Then H A is isomor-

phic to the trivial group, i.e. H A = {eXj. Otherwise we adjoin a neutral element to H

A'

Then from ( 4 . 1 1 )

it follows that H A

is isomorphic to the strict positive cone ( G A ) + L e X where e denotes the neutral element of G We may assume w.1.o.g.

X W

A'

that all G X , X E A are mutually disjoint.

and let for X > X

0

either F A : = G X (iff H A is

isomorphic to the trivial group) or F

(GA)+

{ e X l . Then the

Linearly Ordered Commutative Divisor Semigroups

ordinal sum F of F A

,

)r

67

E A is also a weakly cancellative

d-monoid. We will assume later on that a weakly cancellative d-monoid is always given in this form. Clearly H can be embedded in F. In the same manner as we previously adjoined a neutral element, here we adjoin the possibly missing inverse element of the first irreducible subsemigroup. We remark that for the irreducible case a characterization of weakly cancellative d-monoids follows from proposition ( 4 . 2 ) and theorem ( 4 . 5 1 ,

i.e. all cancellative d-monoids can be

embedded into a Hahn-group. CLIFFORD ([19581, 119591) characterizes another class of irreducible d-monoids. A d-monoid H is called c o n d i t i o n a l Z y c o m p l e t e if each subset A which is bounded from above has a least upper bound. Then each A which is bounded from below has a greatest lower bound. CLIFFORD considers conditionally complete d-monoids which are positively ordered. Due to the positive order relation the first subsemigroup in the ordinal decomposition is always the trivial group {el. Therefore we will consider only d-semigroups without neutral element. A positively ordered d-semigroup H is called A r c h i m e d e a n if

3 nEH

:

for all a , b E H , a

*

(4.14)

a

n

L b

e. The following results are due to

CLIFFORD ([1958], [1959I).

(4.15)

Proposition

Let H be an irreducible, conditionally complete positively ordered d-semigroup without neutral element. Then H is Archimedean.

68

Ordered Algebraic Structures

Proof. Let a , b E H . If an = a

n+ 1

for some nElN then an is

idempotent and due to ( 3 . 6 ) the maximum of H. Therefore an

2 b.

n n+ 1 Otherwise an < a for all n E N . Suppose a < b for all n E N . Then c:= sup{anl nElN] exists and an < c for all nElN (an = c n implies a n+l - a 1.

Now a * d = c for some d E H . If d < c then

d < an for some n and therefore a * d i.e.

5

an+l < c. Thus c = d ,

a * c = c.

Let L:= (91 g < cl. For x , y E L we find max(x,y) < a nElN and therefore x * y

5

a

n

for some

2n . implying x * y E L . Further

x * c f an * c = c shows x * c = c. Let U : =

(91 g

2

c).

Clearly

positivity implies that U is a subsemigroup. Let y E U , c < y. Then c

*

z

=

y for some z E H . Suppose z E L . Then c * z = c

contrary to c e y. Therefore z E L . Let x E U . Then x * y = x * c * z = c * z = y. Thus we have found that H is an ordinal sum of L

and U , contrary to our assumption. Therefore H is Archimedean. w

Now Archimedean, positively ordered d-semigroups have been characterized by HOLDER [19011 and CLIFFORD 119541. It should be noted that an Archimedean, linearly ordered divisor semigroup always is commutative (HOLDER [19011, LUGOWSKI 119641). Thus our assumption of commutativity is redundant. The following result is due to CLIFFORD [1954]:

(4.16) P r o p o s i t i o n Let H be an Archimedean, positively ordered d-semigroup. If H is not cancellative then

Linearly Ordered Commutative Divisor Semigroups

(1)

H contains a maximum u,

(2)

for all a

(3)

if a * b = a * c

*

69

e there exists nEaJ such that a

*

n

u,

=

u for a , b , c E H then b = c.

Proof. Due to the assumption there exist a , b , c E H with a * b = a * c =:u

and b

*

c. Now b * x = c for some x

u + x = u. Suppose u < y E H then y

5 xn for some n

we find the contradiction u = u * x

n

1. u

*y

1. y >

(1) and ( 3 ) . Now for a E H (a 9 e) we know a

i.e.

a

n

n

*

e. Thus

E N . Therefore

u . This shows

5 u for some nElN,

= u.

Now the following theorem characterizes Archimedean, positively ordered d-semigroups. The result is due to HOLDER [1901] and CLIFFORD [1954]. Its proof is drawn from FUCHS 119661.

(4,. 1 7 ) T h e o r e m H be an Archimedean, positively ordered d-semigroup. If H is cancellative, then it is isomorphic to a subsemigroup of the positive cone

(JR+,+,z) of

the real numbers.

In particular if H has a minimum and contains no neutral element then H is isomorphic to the additive semigroup of natural numbers (IN

,+,z).

If H has a maximum u and u has no immediate predecessor then H is isomorphic to a subsemigroup of ([O,ll,*,5) with respect to the usual ordering and a *b:- min(a+b,l). In particular, if H has a minimum and contains no neutral element, then H is isomorphic to (k/nl O * k scme n E m .

5 n) for

70

Ordered Algebraic Strucrures

(3)

If H has a maximum u and u has an immediate predecessor then H is isomorphic to a subsemigroup of ( [ O , l ] U { = ) , @ , ~ ) with respect to the usual ordering and a @ b : = a+b

1.

1 and a @ b : =

a + b if

otherwise.

m

Proof. We always discard the neutral element e if it exists in H. At first we assume that H contains a minimum a. Let b E H , b

*

a. Then ak < b 5 a k + l for some k Em and ak

c E H . Now a

k + l = a k * a z ak * c = b L a

fore H is generated by a. If a isomorphic to (IN

n

< a

, + , z ) .Otherwise

*c

= b for some

k + l shows a k + l = b. There-

n+l

for all n E B then H is

...

a < a2 <

some n E N . Then H is isomorphic to ({k/nl

1

< an = a

n+ 1

for

5 k 5 n),*,i).

Secondly, we assume that H does not contain a minimum. For a E H let b < a. Then b * c = a for some c E H . Let z = min(b,c). z2

5 a. Therefore for all t E B there exists

Now choose a fixed element v E H

(v

*

Then

z E H such that Z t z a .

u if u exists).

following we define a function f: H + W + . Let a E H

I n the

(a

*

u)

and define two subsets of IN X B by

As

La:= f(m,n)

I

v

Ua:= { ( k , l )

I

y1

H is Archimedean L a,Ua

(1)

La(U

a

1. x n

and

x

5 v

and

a 5 yk

*

m

5 a

I

(s,r+l) E La

,

For t EN let zt 5 min(b,c). r+l z r ~ b < z

and

for some yEH).

0. We claim that

and that for all tEIN there exist s,r (11)

for some x E HI,

(s

+1,r)

Eva

t such that

.

For b,c E H we define r,s E B by

zs 5 c < z

s+ 1

.

Then r,s I t . In particular,

for b:= v and c:= a we find ( 1 1 ) . Further for b:= x and c:= y

71

Linearly Ordered Commutative Divisor Semigroups

in La resp. U

corresponding to (m,n) resp.

rm z

zsl <

(s+lIk,

<

-

t a < z

(k,l) we find

(r+l)n

Therefore rm < (s+l)k and sl < (r+l)n. This implies m/n < (1 + l/r) ( 1

+

l/s) k/l

.

As t can be chosen arbitrarily large and r r s

5 t we find

(I).

Further the distance between the elements in ( 1 1 ) is arbitrarily small for sufficiently large t. Therefore there exists a unique real number a separating L

.

and U

Now we define f(a):= a .

In particular f(v) = 1. Further f(a) > 0 as Ua a

*c

=

<

Now z

@

(e * u +a). For a < b

($

u ) let

b. Then for t € 2 4 let zt 5 min{a,brc,v3 and define '

r,s,V,A by zr zA

*

b
5 v < z r + l , zs 5 a < z Then t

'+' < a*c

=

5

r,s,p,A.

S+l

,

' z

5 c
'+'and

Again (s+l,r) ELIa and (A,r+l) E L b .

b < zA+l implies

v +

s

< A + 1. Therefore

For sufficiently large t we see (t+s)/(r+l) > (s+l)/r. Therefore (A,r+l) E U a . This shows f(a) < f(b). Now (11) shows the existence o f sequences in L a

1

a'

with limit f(a). Therefore,

if a * b C u , then we find sequences in La*b ' 'a*b f(a) +f(b).

with limit

Thus f(a * b ) = f(a) +f(b). Hence f is an embedding

of H (without u ) into W +

,.

If H is cancellative then from (4.16) we know that H contains

no neutral element. Therefore f embeds H into W + . Otherwise H contains a maximum u. Then f(H) i s bounded; in particular, i f an = u , then f(H)

aEH\{u)

5 n. Let sup f (H)

= y.

If f(a) < y for all

then we define f ( u ) : = y . Otherwise we define f(u):=

m.

I?

Ordered Algebraic Structures

Then g , defined by g(a) = f(a)/y, embeds H into [0,11 or [O,l] U

{-I

with respect to ( 2 ) or ( 3 ) .

In this way all irreducible conditionally complete, positively ordered d-semigroups are characterized. CLIFFORD 119591 gives also an example f o r an irreducible positively ordered d-semigroup which is not Archimedean and thus not conditionally complete. Further he considers certain completions of such semigroups. Let H be a positively ordered d-semigroup. If H is a subsemigroup of a conditionally complete, positively ordered d-semigroup fi such that for all a € % there exist subsets A , B

of H with sup t i O n

A =

a = inf B then

fi is called a n o r m a l c o m p t e -

of H. If H has a normal completion

R which is isomorphic

to a subsemigroup of one of the semigroups in (4.17) then we say that H can be n o r m a l l y embedded into one of these semigroups. Clifford shows that this is possible ff and only if H is also Archimedean. We remark that the five semigroups in theorem (4.17) are con-

ditionally complete and have no conditionally complete subsemigroups with the exception.of the trivial group. Each is the ordinal sum of its neutral element and the set of all other elements. We remark that then the obtained discrete semigroup in (4.17.2) for k = n = 1 is isomorphic to the trivial group. Therefore each irreducible conditionally complete, positively ordered d-semigroup is isomorphic to one of these five semigroups (after discarding the neutral element), called the

f u n d a m e n t a l semigroups. Thus every conditionally complete, positively ordered d-semigroup is an ordinal sum of fundamental

13

Linearly Ordered Commutative Divisor Semigruups

semigroups. CLIFFORD [I9581 shows that then the index set A of the family is conditionally complete, too. The possible choice of the fundamental semigroups for h E A depends on the local structure of A ;

in particular, it is important whether

A has an immediate predecessor or successor (cf. 4 . 2 3 ) .

Although we have now quite well characterizations of d-monoids, in particular of cancellative and conditionally complete ones, for practical applications as in part I1 we need a list of properties which are always fulfilled in d-monoids. Such properties will be used in the discussion of methods proposed for the solution of the optimization problems considered. For a E H we consider the following sets

,

~ ( a ) :=

{bEHI

a > a*b)

neg(a):=

CbEH(

a

a*b)

,

dom(a):=

IbEHI

a = a*b)

,

pos(a):=

IbEHI

a f a*b)

,

:=

{bEHI

a < a*b)

.

P(a)

2

The elements of these sets are called s t r i c t l y n e g a t i v e , n e g a -

t i v e , d o m i n a t e d , p o s i t i v e and s t r i c t l y p o s i t i v e

( w i t h respect

t o a).

( 4 . 1 8 ) Proposition Let A be a d-monoid. Let a , c E H and b:= a * c . Then

Ordered Algebmic Structures

7.1

N(c) E neg(a) ;

(3)

P(c)

(4)

dom(a) and pos(a) are d-monoids;

(5)

neg(a) is a linearly ordered commutative monoid.

Proof.

pos(a),

( 1 ) It suffices to prove the inequalities. Let x E d o m ( a )

and y E P(a). Then a * x = a < a * y implies by cancellation x < y. N(a) < dom(a) is similarly proved.

(2) is an immediate

conclusion from the definitions and ( 1 ) . ( 3 ) Let x E P ( c ) .

Then c * x > c = c * e . Therefore x >

for a E H we get a * x

a, i.e.

xEpos(a).

e.

Then

N(c) C_ neg(a) is

shown in the same way. (4) =

At first consider dom(a).

Let x , y E d o m ( a ) . Then a

*

(x * y )

a * x = a. Now assume x < y and x * c = y for some c E H . Then

a S c = a * x * c = a * y = a shows c E d o m ( a ) . Now let x , y E p o s ( a ) . Then a

5 a

* y

5 a

* x * y .

Assume x < y and x * c = y for some

c E H . Then x < x * c shows c E P ( x ) C_ pos(a) (5)

(cf. ( 3 ) ) .

Similarly a s in ( 4 ) the discussion of pos(a).

We remark that in general neg(a) is not a d-semigroup. This is due to the asymmetry in the divisor rule. For example assume

*

(a

and let a E P - . Then a < e and for a solution c E H with

a *c

=

e we know c = a

P-

-1

commutative semigroups

(4.1) '

a < b

=o

> e. We may consider linearly ordered

(H,*,c)which 3 c E H :

for all a , b E H . Then ( H , * , z ) If (H,*,()

fulfill

a = b * c

is called a d u a l d - s e m i g r o u p .

is a d-semigroup and a dual d-semigroup then (H,*,L)

is a linearly ordered commutative group. All results for d-

semigroups can be translated into a result for the corresponding

Linearlv Ordered Commutative Divisor Semigroups

15

dual d-semigroup which we get by reversing the order relation of H. Now we define for a d-monoid H with ordinal decomposition (HA i h E A )

u if a

the i n d e x h(a) o f an element a E H by h(a) :=

is an element of H

u

.

(4.19) P r o p o s i t i o n Let H be a weakly cancellative d-monoid with ordinal decomposition ( H A I X E A ) .

5 min(X(b),X(c))

If h(a)

then a * b = a * c implies b = c.

< min(A(b),A(c)) If h(a) -

5 d*c

and d f a then a * b

implies b f c. For a < b there exists a unique c E H with a * c = b. Then h(c) = h(b). If h(a) = h(b) and a * b = a then b = e or H If an = b If b

n

(a)

=

{a].

then a = b.

< a for j = 1,2,...,n

j -

and a

5 b:= b l * b 2 * . . .

* bn

then X(a) = X(b). Let a:= a l * a 2

* ... * a n ,

b:= b

1

* b 2 * ... *

bn. If

a * c < b * c then there exists p with a * c < b

’

*(*

a )*c.

j*u

j

Proof. (1) Assume b

and h(c)

*

c. Then a

*b

5 h(a). Then h(a)

=

a = a

* c.

Therefore h(b) 5 h(a)

= A(b) = h(c) yields a contradiction

as the irreducible H A are cancellative. (2) From a * b

5 d

*c

and d

5 a we find a * b 5 a

* c.

Now ( 1 )

implies b = c if equality holds. Otherwise b < c. ( 3 ) We know a * c = b for some c E H . Then h(a),h(c)

f X(b).

Ordered Algebmic Structures

76

Suppose X ( c ) < X(b). Then X (a) = A(b) and therefore a

*c

= a< b

contrary to the choice of c. Thus h ( c ) = h(b). If h(a) < h(b) then c = b. Otherwise h ( a ) = X(b) = X(c) and c is uniquely determined as H (4)

A (a)

is cancellative. If X(a) = X o then b = e. Other-

This follows from (4.13.3).

wise suppose that H

X (a)

is not the trivial group. Then H

A (a)

is isomorphic to the strict positive cone of a group. Hence

> a contrary to the assumption.

a * b

(5)

an = bn implies h(a) = X(b). Now H

X (a)

is cancellative and

thus a = b.

...,

= maxib.1 j = 1 , 2 , n}. Then A(b) = X(b n 7 < X'(a). On the other hand a 5 b implies X(a) 5 h(b).

( 6 ) W.1.o.g.

assume b

(7) Suppose that a * c

Then an

* cn z

b

2 bu

* a n-1 * c n .

* ( *

a.)

*c

for all

' " > " holds then

If

*c

cellation in H

< b

*c

X (a*c)

= 1 , 2 , . ..,n.

cancellation leads

to the contradiction a > b. Otherwise A ( b * c ) other hand a

u

j*!J

implies X(a

* c)

5 X(b

5 X(a *c). On the

* c).

Then can-

implies a * c = b * c contrary to the assump-

tion.

In the remaining part of this chapter we give examples for different classes of d-monoids. The relationship between these classes is visualized in the following diagram (Figure 5 ) .

)

Linearly Ordered Cammutative Divisor Semigroups

77

d-monoids

weakly cancellative d-monoids

conditionally complete d-monoids

d-monoids in Hahn groups

d-monoids in the real additive group

cancellative fundamental monoids

F i g u r e 5.

fundamental monoids with maximum

Relationship of classes of d-monoids

A line joining two classes means that the upper class contains the lower one. More details can be found in the following remarks and examples. The f undar nent at monoids are the monoids explicitly mentioned in (4.17).

( 4 . 2 0 ) 0-monoids i n t h e r e a l a d d i t i v e . group The only conditionally complete subgroups are (IR

, + r z ) ,(iZ,+,z);

positive cones of these are the cancellative fundamental monoids in (4.17). If a d-monoid H contains an element aEIR+\iZ+ H is d e n s e in IR+ y E

then

(i.e. for a , B E I R + with a < B there exists

H such that a < y < B ) .

A typical example i s ( p + f + , z ) , the

positive cone of the additive group of rational numbers. Another example can be derived from a transcendental number a € (0,l). Let R denote all real numbers of the form

-

p, a’

with

Ordered Algebraic Structures

7b

r € l N U 10) and p . € Z for each j. Due to the transcendency of 1 a this representation is unique and R is a well-defined dense subgroup of IR. Two linearly ordered commutative groups isomorphic to ( I R , + , z )are considered in the following. The multiplicative group (IR

+

\ { O ) , = , z ) of all positive real numbers

is isomorphically mapped ontoIR by x

-B

In x. Now let

( - 1 ) 1 /P

p:=

for p €IN. On R : = IR+p U IR+ we define an extension of the usual

order relation of IR+ by W + P zlR+ and a , 6 Em+.

ap

1. 6 p

a

>

6 for

With respect to the internal composition a@b:=

the system

(R,@,c)

D-monoids

(aP+bP l/P

is a linearly ordered commutative group which

is isomorphically mapped onto ( I R , + , L ) by x

(4.21)

iff

+

xp.

i n Hahn-groups

A d-monoid in a group is always cancellative and therefore irreducible. In particular, d-monoids in the real additive group are examples. Further from (4.11) w e know that a cancellative d-monoid can always be embedded in a linearly ordered commutative group. Such groups can be embedded into Hahn-groups. Therefore all cancellative d-monoids are isomorphic to a dmonoid in a Hahn-group.

A

the additive group (IRn , + ,

typical example of a Hahn-group is * )

of real n-vectors ordered with

respect to the lexicographical order relation. If n > 1 then the strict positive cone P of this group is a cancellat ve d-semigroup which is positively ordered and irreducible but not Archimedean. For example, l e t n = 2 .

Then (0,l) 4

but there exists no n €IN such that n(0,l) = (0,111

+

(

I

1

(1,O). Thus

P is not Archimedean and hence not conditionally complete

(cf. 4.15). A more general example is the following.

0

Linearly Ordered Cummutah'veDivisor Semigroups

Let ( A , i )

19

be a nonempty, linearly ordered set and let for

each A E A

H A be a cancellative d-monoid. We consider the

subset H of the Cartesian product X A E A H A w h i c h consists in all elements

(...

is w e l l - o r d e r e d .

xA

... )

such that Sx:= { A E A l x A 9

(Each subset of

*

On H an internal composition

S

e,)

contains a minimum.)

X

is defined componentwise.

Further H is linearly ordered with respect to the lexicoy

if

x = y

or

graphical order relation, i.e.

x.6

x A < y A for A = min{u E A 1 x

yu}. Then ( H , * , d 1 is called

u

9

the l e x i c o g r a p h i c a l p r o d u c t of the family tion:

H =

(HA, A € A)

[Denota-

n A E A H ) , ] .H is a cancellative d-monoid if and only

if for all A > min A

H A is a linearly ordered commutative

group. Further H is a linearly ordered commutative group if and only if for all A

H A is a linearly ordered commutative

group. For example, let H denote the set of all functions f: [O,l] +IR

*

such that Sf:= 1x1 f(x)

01 is well-ordered. Together with

componentwise addition and lexicographic order relation H is a linearly ordered commutative group. We remark that H is the lexicographic product ll

AE[O,

H A = IR for all A E [ 0 , 1 1 .

As

11

H A of real additive groups, i.e

all H A coincide with lR we call H

a r e a l h o m o g e n e o u s l e x i c o g r a p h i c a l product. H is isomorphic to the Hahn-group V([O,ll)

(cf. chapter 3 ) with respect to

the inverse ordered set [0,1]. In this way each Hahn-group V(A)

is "inverse" isomorphic to the real homogeneous lexico-

graphical product

nAEAH A

'

Ordered Algebraic Structures

80

(4.22) Weakly cancellative d - m o n o i d s A l l weakly cancellative d-monoids are ordinal sums of cancella-

tive d-semigroups. Therefore all Irreducible weakly cancellative d-semigroups can be embedded into Hahn groups. As

first example we consider ([O,l],*,z)

with internal compo-

sition defined by a*b:= a + b

-

ab

and by

Both are weakly cancellative, positively ordered d-monoids. The irreducible subsemigroups are { O ) , (0,l) and {l}: in particular the nontrivial subsemigroup can isomorphically be mapped onto ( I R , + , z )by x A

+

-ln(l-x) and by x+-ln((l-x)/(l+x)).

further example is the r e a l bottleneck s e m i g r o u p

with internal composition defined by (a,b)

+

(lR,max,z)

max(a,b) and with

respect to the usual order relation of real numbers. Its ordinal decomposition is ((a);

aEIR), i.e.

all irreducible subsemi-

groups are isomorphic to the trivial group. We may adjoin

-a

as a neutral element. All elements of the bottleneck semigroup are idempotent. This semigroup is not cancellative but conditionally complete. Now we consider (IR x I R + , * , + )

with internal

composition defined by

for all (a,b), (c,d) EIR of the form

{(a,O))

xlR+.

and { a )

Its irreducible subsemigroups are x

( W + \{O))

for aEIR. We define a

81

Linearly Ordered Commutative Divisor Semigroups

suitable index set A : =

{1,2}

x

IR.

Clearly, (1,a) is the index of

{(a,O)}, and (2,a) is the index of {a}

x (IR+x{O}).

A is linear-

ly ordered by - $ . We may adjoin a neutral element (-0.0) or a first irreducible semigroup

Then the new ordinal

( { - a )xIR,+,.$).

sum is called time-cost semigroup. This semigroup is not conditionally complete. The most general example is the following. Let G h be a real homogeneous lexicographic product for each h € A . Let A have a minimum h o . for h > h

. Le.t H

Then H A is either (GA)+ or (Gh)+x{eA}

be the ordinal sum of G h

and all H A . Then H A

0

is called a homogeneous weakly c a n c e l l a t i v e d-monoid. Clearly,

all weakly cancellative d-monoids can be embedded into a homogeneous weakly cancellative d-monoid.

(4.23)

Conditionally complete d-monoids

All fundamental monoids are conditionally complete. In the positively ordered case all irreducible, conditionally complete d-semigroups are isomorphic to the strict positive cone of a fundamental semigroup. A weakly cancellative example, the Our next example is

bottleneck semigroup, IS given in ( 4 . 2 2 ) . not weakly cancellative. We consider (IR

x

[O,l],*,+)

with

internal composition defined by if a > c,

(a,b)

*

_.

if c > a,

(c,d):= (a,min (b+d,1)

i f a = c,

for all (a,b), (c,d) E l R x [O,lI. Similar to the time-cost-semi-

group in ( 4 . 2 2 ) {a} x (0,1]

the irreducible subsemigroups are {(a,O)]

for aEIR. Again A = { 1 , 2 )

xlR

and

is a suitable index

set. We may adjoin a neutral element ( - 0 , O ) . The reader will

82

Ordered Algebraic Structures

easily construct further examples using fundamental semigroups (without neutral element) from ( 4 . 1 7 )

in the following way.

Let A be a conditionally complete, linearly ordered set and let H x be a fundamental semigroup for each A E A such that the following rules are satisfied (cf. CLIFFORD [ 1 9 5 8 ] ) . If A E A has no immediate successor, but is not the greatest element of A ,

then H A must have a greatest

element. If A € A has no immediate predecessor, but is not the least element of A , If A . u E A ,

C

then H

u and

A

x

must have a least element.

is the immediate predecessor of

u then either H A has a greatest o r H

u has a least ele-

ment. It is easy to see that these rules are necessary for conditionally complete d-monoids. If we do not assume that the considered semigroup is positively ordered then we may consider the ordinal sum of an arbitrarily with negative

irreducible conditionally complete d-semigroup H A 0

elements and an arbitrary conditionally complete positively ordered d-semigroup H without a neutral element. Again we have to fulfill rules ( 1 ) complete d-monoids.

As

(3).

In this way we get a t 2 conditionally

an example consider

which is defined as the ordinal sum o f the group and the subsemigroup (IR

+

x [ O , l ] , * , 4 )

({--}

xlR,+,4)

of the above example.

Linearly Ordered Cbmmutative Divisor Semigroups

(4.24)

83

0-monoids

All examples considered before are d-monoids. An example which is neither weakly cancellative nor conditionally complete can be found from the first example in ( 4 . 2 3 ) subsemigroups of the form {(a,O)) Then

(1R

X

by eliminating all

in the ordinal decomposition.

(0,13,*,4)is no longer conditionally complete as

the rules in (4.23) are not satisfied, but it can normally be embedded into the original one. Here the question arises whether a d-monoid exists which is neither weakly cancellative nor conditionally complete and, furthermore

cannot be normally

embedded into a conditionally complete one. CLIFFORD 119591 gives the following example. Let

be lexicographically ordered, i.e. (0,l)4 (0,2) 4

.. . 4 (l,-l)4 ( 1 , O ) -4 ( 1 , l ) 4 .. . . . . 4 (2,O) 4 (3,O)4 . . . .

We define an internal composition by

(n,x)

*

(m,y):=

t

(n+m,x+y)

if n+m < 2 ,

(n+m,O)

otherwise.

Then (H,*,4 ) is an irreducible, positively ordered d-semigroup without a neutral element, A s H is not Archimedean it cannot be conditionally complete. Further H we try to embed

H

into

a

i s

not cancellative. Now

normal completion. The only undefined

supremum (infimum) is a:= sup{ (0,n) I n E N } . We consider H U { a ) with the obvious extension of the linear order relation on H. If the extended internal composition

*

satisfies the monotonicity

Ordered Algebraic Structures

a4

condition then it is uniquely defined by

a

*

(O,n):= a

,

f o r all nElN and f o r all z E Z .

a *a:=

(2,O)

But now a < (1,O) and there

exists no x E H U { a ) such that a * x = ( 1 , O ) .

is not a d-semigroup. CLIFFORD [ 1 9 5 9 ] a positively ordered d-semigroup

Therefore H U { a }

shows that, in general,

can normally be embedded in-

to a conditionally complete d-semigroup if and only if all its irreducible subsemigroups are Archimedean. Then this normal completion is uniquely determined.

5.

Ordered Semimodules

In this chapter we introduce ordered algebraic structures which generalize rings, fields, modules and vectorspaces as known from the theory of algebra. Additionally these structures will be ordered and will satisfy monotonicity conditions similar to the case of ordered semigroups. Let (R,@) be a commutative monoid with neutral element 0 and let (R,(D) be a monoid with neutral element 1

(0

*

1 ) . If

(5.1)

for all a , B , y E R then (R,@,@) is called a s e m i r i n g w i t h u n i t y 1

and z e r o 0. We will shortly speak of a semiring (5.1.2)

R.

and

(5.1.1)

are the laws of d i s t r i b u t i v i t y . We remark that

@

is not

necessarily commutative. Otherwise we call R a c o m m u t a t i v e

s e m i r i n g . If all elements of R are idempotent with respect to then R is called an i d e m p o t e n t s e m i r i n g . NOW, if ( R , @ , Z ) is an ordered commutative monoid and

for all a,B,y,6 E R with y

2

0, 8

ordered semiring. 85

5

0 , then R is called an

@

86

Ordered Algebraic Structures

(5.3) P r o p o s i t i o n (I)

A

(2)

An idempotent semiring is partially ordered with respect

semiring is idempotent iff 1 B 1 = 1.

to the binary relation defined by a

5 B iff

a B

B

B

=

for all a , B E R . Proof. ( 1 ) The if-part is obvious. Now let 1 B 1 = 1. Then a

6a

(1

1 ) = a B 1 for a E R .

@

Distributivity shows that a is

idempotent. (2)

If a is an idempotent element then a 5 a . Commutativity im-

plies antisymmetry. Associativity leads to transitivity.

B

B implies ( a B y )

is partially ordered a s a

@

= 6

Clearly 0

B y for all a , B , y E R .

=

5

y

@

for all y E R .

(B

8)

(R,B,f) y)

Then

distributivity implies ( 5 . 2 . 1 ) .

In particular, if ( R , @ ) is a commutative group then R is called a ring. (5.4)

A

semiring R with

a @ B = o

I ,

a = O

or

B = O

for all a , E E R is called a semiring w i t h o u t ( n o n t r i v i a l ) d i v i s o r s Of

z e r o . A commutative ring without divisors of zero is called

an i n t e g r a l domain. called a f i e l d .

A

A ring R with commutative group ( R \ { O } , B )

well-known theorem from the theory of algebra

is the following.

(5.5) P r o p o s i t i o n A

is

(linearly ordered) integral domain R can be embedded in a

(linearly ordered) field.

Ordered Sem'rnodules

87

Proof. We give only a sketch of the proof which is similar to the proof of theorem ( 2 . 8 ) . An equivalence relation is defined by ( a , B )

R x (R\{O))

-

(y,6)

-

on

iff a O 6 = 8 O y .

We denote the equivalence class containing ( a , B ) by

a/B

and

we identify a E R with a / l . Then the internal composition may be extended by

for all a / B , y / 6 .

It can be seen that then the set F of all

equivalence classes is a well-defined field. With respect to the linear order relation defined by

for all a / B , y / 6

this field is linearly ordered.

The following result for the Archimedean case is similar to (4.17.1).

(5.8) Proposition An Archimedean linearly ordered field ( F , @ , O , Z ) i s isomorphic to a subfield of the linearly ordered field ( I R , + , * , zof ) real

numbers. proof. We only indicate how this result follows from the proof of ( 4 . 1 7 ) . Let

6

The details are left to the reader.

resp.

that F':=

1

denote zero resp. unity of F. It suffices to prove

{aEFI a >

of a subfield of IR

6 , is

isomorphic to the strict positive cone

(IR' : = {aEIR[ a > 0)). F' does not contain a

minimum. Therefore using the proof of ( 4 . 1 7 )

we can choose an

Ordered Algebmic Structures

88

isomorphism rp with a ( ? ) = 1 such that

l ! l

maps (F',B,Z) isomor-

phically onto a dense subsemigroup of ( W ' , + , ( I . cp maps =

$:=

frp

-1

We find that

(9)I q E Q , q > 0 ) onto Q such that rp(a Q 5 )

(P(a)rp(B) for all a , B E Q ' . Then

for all a E F ' leads to cp(a

Q

5 ) = cp(a)cp(5)

for all a , B E F ' .

Now let (H,*) be a commutative monoid with neutral element e and let (R,tB,@) be a semiring with unity 1 and zero 0. If n:

R X H + H

is an e x t e r n a l c o m p o s i t i o n such that

(a 8 6) o a = a n (6

(5.9)

(a B 5) n a =

(a

=

(a

an(a*b)

a) 0

,

a)

*

(5

a) * ( a

0

a)

,

b)

,

O O a = e , l O a = a ,

for all a , B E R and for all a , b E H then ( R , @ , Q ; U ; H , * )

is called

a s e m i m o d u l e o v e r R. We remark that the adjective 'external' does not exclude the case R ' = H .

In our definition we only

consider a left external composition

0:

R XH

one may consider a r i g h t external composition

+

H.

0:

Similarly, H x R + H.

If

we want to distinguish between such semimodules then we will use the adjective "left" or "right". Usually we will only treat left semimodules and omit the adjective "left". A

direct consequence of (5.9) is the relation a o e = e for all

a E R . This follows from e = O n e = (a

(D

0) n e = a n ( O n e ) = a n e .

Ordered Semimodules

89

In particular, if R is a ring then H is a commutative group with a

-1

=

( - 1 ) O a for all a E H (-1 is the additive inverse

of 1 in R ) .

Then H is called a m0dUZe over R .

then H is called a v e c t o r s p a c e over R .

If R is a field

We remark that if R is

a ring and H a commutative group then O o a = e is a redundant assumption in (5.9). N o w let

(H,*,L)

be an ordered commutative monoid with neutral

element e and let ( R , @ , @ , Z ) be an ordered semiring with unity 1 and zero 0. Let

0:

R x H

+

H be an external composition satis-

fying (5.9),

for all a , B E R , (5.10.2)

c , d E H with d 5 e 5 c and

a z b

*

yoa(yob

for all a , b E H , y E R + : = { E E R I Then (R,@,@,Z;O;H,*,5)

E

and

2

6 0 a 2 6 0 b

0 ) and 6 E R - : =

{EERI

E

5

0).

is called an o r d e r e d semimodute over R .

We remark that we will use the same denotation different order relations in R and H .

5 for the

From the context it will

always be clear which one is meant specifically. If, in particular, ( R , @ , @ , Z ) is the positive cone of a field then H is called an o r d e r e d s e m i v e c t o r s p a c e over R .

Some direct consequences should be mentioned:

(5.11) P r o p o s i t i o n Let H be an ordered semimodule over R . (1 1

If H is a group then a ( 6

*

aoc(B0c

Ordered AIgebmic Structures

90

for all a , B E R and c E H + implies (5.10.1). If R is a ring then

(2)

*

a z b

yoa(y0b

for all a , b E H and y E R + implies (5.10.2). N

(3)

If R is the positive cone of a linearly ordered ring R N

and H is the positive cone of a linearly ordered group H then the external composition can be continued in a N

W

N

unique way on R x H such that H is a linearly ordered N

modul over R.

5 d Is equivalent to c

Proof. ( 1 ) In an ordered group c

This follows from composition of c

5

d with c

-1

*d

-1

.

2 d -1

-1

NOW let

a , B E R with a

5 8 and let d E H-. Then d-' E H+ a n d , due to our

assumption, a

0

d

-1

(B

= e shows that a o d

-1

Od-l. Now = (a od)

(2) In particular, ( H , * ) and let 6 E R - .

(-6) O a

1.

(a

-1

.

od

-1

)

*

(-6) o b . Now [ (-6)

(Rfl'ii- = ( 0 ) ) and

;=

Ex;

by

on

0

-1

*

a]

.

( 6 Oa) = "-6)

Therefore 6 O a

HUZ-

a ob:=

(5.12)

Bob:= for all aER\{O),

61 o a

=

e

2 6 Ob. = RUE-

I

- 1 -1 (aob )

8 oa:= ( ( - B )

@

( H n z - = {el). Therefore we may

OOb:= e

N

Bod.

Then -6 E R + a n d , due to our assumption,

( 3 ) A s R and H are linearly ordered we know

0

2

*d)

is a group. Now let a , b E H with a f b

shows that (-6) o a = (6 n a )

continue

-1

( a o d ) = a 0 (d

Therefore a o d

.

oa)

(-8) O b

-1

, ,

-1 I

B E z i R , a E H , bE'ii\H.

Then

is a module

over R. In a module the equations used in (5.12) are necessarily

Ordered Semimodules

fulfilled. Therefore the extension of

0

91

is uniquely determined.

As B is an ordered semimodule over R we know that a

5 5 im5 b im-

c 5 5 O c for all a , B E R and c E H and that a

plies a

0

plies y

0a

1. y

0b

for all a r b€ H and y E R .

It suffices to show

that the first implication holds for all a , 5 E'ii and that the N

second implication holds for all a , b E H . Then (1) and ( 2 ) N

imply that

is a linearly ordered module over R .

At first N

we remark that for a 5 B only the additional cases a E R and a , 5 (-a)

0

c

€s

occur. Then a

(-B)

0

c 5 00 c = e 5 5

O c lead to a O c

0

c

, 5ER

resp.

5 5 o c . The second implication

follows in a similar way.

Proposition ( 5 . 1 1 )

shows that for special semimodules some of

the monotonicity conditions are redundant. Next we consider some examples.

(5.13) P a r t i a l l y o r d e r e d s e m i r i n g s Let ( R , @ , @ , ( )

be a partially ordered semiring. Then (R,@,z)

is a partially ordered semimodule over ( R , C e , @ , L ) to the "external" composition defined by a a,BER.

In particular, ( 5 . 9 )

follows from ( 5 . 2 ) .

0

5:=

follows from ( 5 . 1 )

with respect a @

5 for all

and ( 5 . 1 0 )

As an example, (Z+,+, 5 ) is a linearly

ordered commutative semimodule over the linearly ordered commutative semiring ( Z Z + , + , ~ , 5A) s. in ( 5 . 1 1 . 3 )

this semi-

module can be extended to the linearly ordered commutative

,+,z) over

module (Z

the linearly ordered commutative ring

(22 , + , * , 5 ) .Further the additive group of a subfield of the

real numbers is a vectorspace over this subfield. The positive

Ordered Algebmic Structures

92

cone of such an additive group is a semivectorspace over the positive cone of such a field. Finally we remark that ( R " , @ , Z ) with respect to componentwise composition and componentwise order relation is again a partially ordered monoid. Therefore (R

n

,@,i)is

a (right as well as left) partially ordered semi-

module over ( R , @ , O , z ) .

In particular 23

n

n (Q

and

ordered commutative modules (vectorspaces) over

n IR ) are latticeZ(Q

and IR)

.

( 5 . 1 4 ) Semirings Let ( R , @ , 8 ) denote a semiring. Then a i b

:-

a @ b = b

defines a pseudo-ordering on R with minimum 0. In general ( R , @ , Z ) is not a pseudo-ordered monoid but satisfies

(5.14.2)

a z B

for all a , B , y , 6 E R .

and

y i 8

*

a @ y z B

Further (5.2) is satisfied. If R is an

idempotent semiring then R is a partially ordered semiring. From (5.13) we conclude that ( R , @ , L ) is a partially ordered semimodule ( R , @ , @ , z ) .

In this partial order the least upper

bound is well-defined, i.e. sup(a,b) = a @ b but the greatest lower bound may not exist, in general.

( 5 . 1 5 ) R e s i d u a t e d l a t t i c e - o r d e r e d commutative monoids Let ( R , O , < )

be a residuated lattice-ordered commutative monoid

with neutral element 1 . At the end of chapter 2 we have

Ordered Semimodules

93

mentioned lattice-ordered commutative groups, pseudo-Boolean and, in particular, Boolean lattices as examples for this ordered structure. If not present we adjoin a minimum 0 and a maximum

m.

The internal composition is extended by

0 = 0 8 a = a 8 0 for all a E R ' : =

= a 8

03

for all a E R U { m } .

R

U

and

{O,m)

Let a @ B : =

=

-

8 a

a V 8 for all a , B E R ' .

Then (5.1) is satisfied by all a , B , y E R ' .

In particular

distributivity follows from (2.13). Therefore ( R ' , @ , @ , Z ) is a lattice-ordered, idempotent, commutative semiring with unity I and zero 0. From ( 5 . 1 4 )

we conclude that ( R ' , @ , L ) is a lattice-

ordered commutative semimodule over R'.

(5.16) L i n e a r l y ordered s e m i m o d u l e s o v e r real n u m w Let R + be the positive cone of a subring of the linearly ordered field of real numbers with {O,l)

5

Let ( H , * , z )

R+.

be a linearly ordered commutative monoid and let

0 :

R+xH + H

be an external composition such that ( R + , + , = , L ; O ; H , * , ( ) linearly ordered commutative semimodhle over R +

is a

.

In particular we consider the d i s c r e t e semimodule H over i Z + with external composition defined by z Oa:= az for all z EZ+ and for all a E H .

As

we discuss only semimodules over semi-

rings containing distinct

zero and unity such a discrete

semimodule is contained in each semimodule. Linearly ordered semimodules over real numbers are discussed in chapter 6 .

Now we introduce matrices for semirings R and for semimodules H over R . Matrices with entries in R and H will be denoted by A =

(aij), B,

C

and

X =

(xij), Y ,

2.

For matrices of suitable

Ordered Algebmic Structures

94

size we define certain compositions. Let A , B and X , Y be two m x n matrices over R and H. Then

C = A

B and Z = X * Y

(5.17)

yij:= a

ij

for all i = I r 2 , . . . , m

are defined by

,

e Bij

and j = I , 2 ,

...,n.

Let A be an m x n matrix

over R, let B be an n x q matrix over R and let X be an n x q matrix over H. Then C = A 0 B and

Z

= A O X

are defined by

n (5.19)

yik:=

(5.20)

n z. - = ik’ j=l

j=1

*

for all i = lr2,...,m

(a 0 Bjk) ij

(a

xjk)

ij

I

’

and k = 1 , 2 , ...,q.

Properties of these

compositions correspond to properties of the underlying semiring and semimodule. If the semiring or semimodule considered is ordered then, for matrices of the same size, an order rela-

tion is defined componentwise. A matrix containing only zeroentries is called z e r o - m a t r i x and is denoted by [ O ] . Similarly the n e u t r a l m a t r i x [el contains only neutral elements. A matrix containing unities on the main diagonal and zeros otherwise is called u n i t y m a t r i x and is denoted by [I]. considered for any size m

(5.21)

x

Such matrices are

n.

Proposition

Let H be an ordered semimodule over R. Let A , B , C , D , E over R with D 2 [ O ] and E H with U

[el and V

1.

[O].

Let X , Y , Z , U , V

be matrices

be matrices over

5 [el. The following properties hold for

95

Ordered Semimodules

m a t r i c e s o f s u i t a b l e size. (1)

( A @ B ) @ C = A @ ( B $ C )

I

A @ B = B @ A ( A 8 B )

8 C = A 8

(2)

8

I

I

( A 8 C )

@

( B 8 C )

,

( A @ B ) =

(CtBA)

@

( C 8 B )

,

101

[O]

A ( B A

( B O C )

=

( A @ B ) 8 C

I

B

A L B

C

8 A =

= A 8 101

;

-D

A @ C L B @ C

*

A 8 D C B 8 D

and

D @ A ( D Q B ,

I .

A O E ) B @ E

and

E 8 A Z E t B B ;

I

the set of all m x m matrices over R i s an ordered semiring with unity

(4)

(X*Y)

x

111

* z

= X * ( Y * Z )

*Y

= Y

and zero [ O ] ;

I

*x

I

( A 8 B ) 0 2 = A n (BOZ) ( A a B ) 0 2 =

101 (5)

=

*

(BOZ)

[el

I

;

X

i

Y

-b

X * Z L Y * Z

A

I

B

-D

A O U Z B U U

-D

D O X ~ D O Y and

X Z Y

(6)

O X

( A n Z )

I

I

and

A O V L B O V , E n X L E O Y

;

the set of all m x m matrices over H is an ordered semim o d u l e o v e r t h e s e t of a l l m x m m a t r i c e s o v e r R.

Proof. These properties are immediate consequences o f the definitions of the composition considered. A s an example we prove (A 0 B) 0 2 = A 0 ( B O Z )

Ordered Algebraic Structures

96

for m x n and n Z

X

q matrices

over H. An entry x

is

A

and B over R and an q x r matrix

of the left-hand side of the equation

is given by

Using (5.1) and (5.9) we find

which is an entry of the right-hand side of the equation considered. 8

For special semimodules a theory similar to the classical theory of linear algebra has been developed. We refer to m o dule theory (cf. RLYTH [1977]) and to min

- max

algebra (cf.

CUNINGHAME-GREEN 119791). A linear algebra theory in the general case is not known. Further properties will be discussed in the chapters on optimization problems. Here we have described the necessary basic concept for the formulation of equations, inequations and linear mappings over ordered semimodules. Duality principles will be discussed in chapter 10 and chapter 11.

6.

L i n e a r l y O r d e r e d S e m i m o d u l e s O v e r Real N u m b e r s

Let (R,+,=,Z) denote an arbitrary subring of the linearly ordered field of real numbers with zero O E R and unity 1 E R . Let (H,*,z) be a linearly ordered commutative monoid. In this chapter (R+,+,.,l;O;H,*,() dered) semimodule.

As

will always be a (linearly or-

multiplication is commutative the semi-

module is commutative, too. Further (R+,+,z) is a d-monoid as for u,B E R + with (4.17)

a

5 B we know B

- a €

R+. From the proof of

we conclude that R+ is either equal to Z+ or R + is a

dense subset of IR+. In any case, z+c_R+ as { 0 , 1 1

c_

R+.

Therefore the properties of the d i s c r e t e semimodule (Z+ ,+,

,f;O;H,

*,z) play

an Important role.

We do not assume in the following that the semimodule considered I s ordered. Therefore the monotonicity conditions (cf. 5.10) may be invalid. Nevertheless, for discrete semimodules we find: (6.1)

e C a O a

and

a o b c e

f o r all a E Z Z + x f 0 ) and all a , b E H with e

(6.2)

a l e

*

a o a f B o a ,

and

C

a, b

C

e,

uob,Bob

--1

for all a , B E Z+ and all a,b E H with e 5 a , b 5 e l and

for all a E Z +

and all a , b E H . We remark that (6.2) and (6.3)

for all a , B E R + and all a,b E H are equivalent to (5.10). We will show how these properties are related and that they hold In all r a t i o n a l semimodules (R+

91

g+).

Ordered Algebraic Stntctures

96

(6.4) Proposition L e t H be a l i n e a r l y o r d e r e d commutative monoid which i s a

semimodule o v e r R+. (11

*

a ( b

If

a o a z a o b

V a,bEH

VaER+

then t h e following p r o p e r t i e s are s a t i s f i e d :

v

< a o a

(2)

e

(2')

a o a < e

(3)

e

(3')

a o a

(4)

a

5 B

9

a o a

5

(4.')

a

5 8

*

aOa

2

5

aER+L{O)

V a E H

with e C a,

V aER+\{O}

V aEH

with a < e ,

v

aER+

V aEH

with e

5

a,

V aER+

V a € H

with a

5

e,

Boa

V a,BER+

V aEH

with e

5

a,

Boa

V a,BER+

V aEH

with a

5 e.

a o a

5

e

Furthermore

(2),

( 3 ) and

as w e l l as

(4)

(Z'),

(3')

and

(4')

are

equivalent. Proof.

(1)

*

( 3 ) . L e t a = e . Then a

same way b = e s h o w s of

(21,

( 3 ) and

(4).

(1)

*

(3'1.

0

a = e leads t o

(3). In the

We p r o v e o n l y t h e e q u i v a l e n c e

The e q u i v a l e n c e o f

(Z'),

( 3 ' ) and

(4')

follows similarly. (2)

*

( 3 ) i s obvious.

(3)

*

( 2 ) . L e t a E R + L { O j and l e t e < a E H .

Now 1 5 n a

Suppose e = a O a .

f o r some n E N . T h e n

contrary t o e < a. (4)

*

( 3 ) . Let a = 0 i n

(3)

*

(4).

e

5 (B

- a)

This leads t o

(4).

Let a,BER+ with a 0

a. T h e r e f o r e

(a

0

5 B a)

*e

and l e t e

5

(a

0

a)

(3).

5

*

aEH. [ (8

-

a)

Then 0

a]. m

Linearly Ordered Semimodules over Real Numbers

99

(6.5) T h e o r e m Let H be a linearly ordered commutative monoid which is a semimodule over R + .

If R + C_ Q + then H is a linearly ordered semi-

module. Proof. Due tc proposition ( 6 . 4 )

it suffices to prove ( 6 . 4 . 1 ) .

Let a E R + and let a , b E H with a

1. b.

a

Ua

a =

5

a

If a = b or a = 0 then

O b . Now assume a > 0 and a < b. A s R + C _ , Q +

we find

n/m f o r some m , n E W with greatest common divisor 1. There-

fore there exist p , q E Z with p n + q m = 1. This implies that

Let a' = (l/m) O a and b' = (l/m) O a . Now a < b implies a' Cb'.. The latter yields (at)mn 5 (b')mn, i.e.

a O a f aob. rn

In the general case R + may contain irrational or, in particular, transcendental numbers. Then even in the case (H,*,z) rn

( m , +-, < )

the monotonicity conditions may be invalid. This is shown by the following example.

(6.6) S e m i m o d u l e c o n t a i n i n g t r a n s c e n d e n t a l n u m b e r Let a € (0,l) be a transcendental number and consider the subring R generated by

ZL

U{a).

Then the elements of R have the

form (cf. 4 . 2 0 ) r j 1 P, a j=o with p tion

j

0:

€ 2 3 , j = O,l,...,r.

R x l R +lR

by

Now we define an external composi-

100

Ordered Algebraic Structures

for all a E I R . Then lR is a module over R. B u t now

shows that ( 6 . 4 . 3 )

is invalid. Therefore the monotonicity con-

ditions (5.10) are invalid. On the other hand we will show that for an important class of monoids all monotonicity properties hold.

( 6 . 7 ) Proposition Let H be a d-monoid which is a semimodule over R+. Then ( 6 . 4 . 1 1 , (6.4.21,

(6.4.31,

and ( 6 . 4 . 4 )

and ( 6 . 4 . 4 ' ) .

(6.4.3'1,

Proof. Due to ( 6 . 4 ) (6.4.1).

are equivalent and imply ( 6 . 4 . 2 ' 1 ,

it suffices to show that (6.4.2) implies

Let a , b E H , u E R + and assume ( 6 4 . 2 ) .

If a = b or a = O

then a O a = a a b . Otherwise a * c = b for some c E H with c > e. Then e <

a

a c which implies a m a 5 (a a a

*

(a 0 c ) = a o b . m

Proposition ( 6 . 7 )

shows that a d-monoid is a linearly ordered

commutative semimodule over R + if it i s a semimodule over R + and the external composition restricted to R + x H + has only values in H+. Unfortunately, example ( 6 . 6 )

shows that even in

the case of the additive group of real numbers we can construct a

semimodule in which this condition is invalid.

101

Linearly Ordered Semimodules over Real Numbers

(6.8) T h e o r e m Let H be a positively and linearly ordered commutative monoid which is a semimodule over R+. Then and ( 6 . 4 . 4 )

hold,

(1)

(6.4.21,

(2)

if H is a d-monoid then H is a linearly ordered semi-

(6.4.3),

module over R + . Proof. A s H is positively ordered we know H = H+. Therefore (6.4.3)

is satisfied. Now ( 1 ) follows from ( 6 . 4 )

and ( 2 ) follows

from ( 6 . 7 ) .

In chapter 4 we discussed monoids which have an ordinal decomposition. If a monoid H is the ordinal sum of a family (HX'

hEh)

of linearly ordered commutative semigroups and if H is a semimodule over R + then this semimodule is called ordinal. The

i n d e x X(a) for a E H is defined with respect to the ordinal decomposition by A(a):=

p if a € Hu

.

(6.9) P r o p o s i t i o n Let H be an ordinal semimodule over R+. Then

for all a E R +

{ O } and all a - E H.

Proof. Let aER+\{O}

and let a E H . Then h ( ( n + a ) m a ) =max(X(a),

A(a0a)) for all nEIN. Therefore it is sufficient to consider

< max{A(aoa),X((I-a) a € ( o , ~ ) . NOW X ( a o a ) -

oa)} = X(a). On

the other hand let n €IN with 1 5 n a . Then X(a = max{X(a),X((na-l)

0

a) = A ( (na) 0 a)

oa)l 2 A(a). w

Ordered Algebraic Structures

102

Proposition (6.9) shows that the external composition decomfor h E A .

poses into external compositions on R+'{O}xHh-rHh

From (4.9) we know that H h is positively ordered with the possible exception of h = min A .

Therefore we find the

following result.

(6.10) P r o p o s i t i o n Let H be an ordinal semimodule over R+. Then H h U { e ) semimodule over R+ with (6.4.2),

(6.4.3).

is a

and (6.4.4) for all

X E A ~ m i nA .

We remark that if A = min A then H A is a semimodule over R +

but the monotonicity properties (6.4.2),

(6.4.3), and (6.4.4)

may be invalid.

A

d-monold H has an ordinal decomposition ( H A ; h E A ) .

positively ordered then H

=

{el for .A

= min A .

If H is

Otherwise

hO

is described in theorem (4.10).

the structure o f H i 0

( 6 . 1 1 ) Theorem Let H be a d-monoid which I s a semimodule over R+. If the over R+ satisfies the monotonicity rule (6.4.3)

semimodule H xO

then H is a linearly ordered semimodule over R+. Proof. Due to (6.7), (6.8), and (6.10) H ordered semimodule over R + for all h over R

. The

HA

is a semimodule

monotonicity rule (6.4.3) together with propo-

0

-

ho.

U {el Is a linearly

0

sition (6.7) shows that H A over R +

i

is a linearly ordered semimodule

103

Linearly Ordered Semimodules over Real Numbers

In ( 4 . 1 8 ) we proved that dom(a) and pos(a) for a E H are sub-d-monoids of the d-monoid H. A similar result holds with respect to the corresponding semimodules. For convenience { a ma1

we define R + o A : =

aER+

,

a E A )

for

A

C_ H.

(6.12) Proposition Let H be a d-monoid which is a semimodule over R + . (1)

R,

12 f

H is a linearly ordered semimodule over R+ iff R

Proof. -

odom(a) = dom(a)

Then

+ Opos(a)

=

for all a E H ,

pos(a)

( 1 ) Let b E dom(a).

for all a € H.

Then bn E dom(a) for all n El".

fore it suffices to consider a E R + =

(aob)* (anal

*

n

(0,l). Now ( a

( ( 1 - a ) o a ) = ( a 0 (a*b))

*

0

b)

There-

*a

=

( ( 1 - a ) m a ) = a.

( 2 ) Assume that H is a linearly ordered semimodule over R+.

Again it suffices to consider a E R +

b E dom(a) then a

0

n

(0,1)

and bEpos(a1. If

b E dom(a) C_ pos(a) follows from ( 1 ) . Other-

wise e < b and therefore e < ( a o b ) as ( 6 . 4 . 2 ) Then a f a

*

is valid.

( a o b ) . For the reverse implication it suffices

to show that ( 6 . 4 . 3 )

holds in H over R+. For the special choice

of a = e we find R+UH+ = H+

In proposition ( 4 . 1 4 )

,

i.e.

(6.4.3).

cancellation rules and related proper-

ties in weakly cancellative d-monoids are stated. ( 4 . 1 9 . 1 ) implies the following cancellation rule.

Ordered Algebraic Structures

I04

(6.13) P r o p o s i t i o n Let H be a weakly cancellative d-monoid which is a linearly ordered semimodule over R+. If a O a

*

6 O a or X(a)

5 A(b)

then

*

(aoa) * b = (Boa)*c f o r all a , b , c E H and all a , 6 E R +

b = [ ( B - a ) ma1 * c

with a

2 6.

Proof. The case a = e is trivial. If a < e then the inverse element a

-1

> e exists. Composition with u o a

b = (Boa)* ( a o a plies b =

[ ( f ?-

-1

)

* c . Then 6 O a = “ 6

-1

- a ) Oa]

leads to

*

( a O a ) im-

a ) o a ] * c . N o w let e < a. At first we assume

a o a 8 8 m a . Then a O a <

tla and thus a < 6 .

> A(b). Then B o a > a o a = $ O a

*c

Suppose A(a)

leads to a contradic-

tion. Therefore A(a) 5 X(b). In the second case X(a)

1. A ( b )

is valid, too. Therefore in both cases A(u o a ) 5 X(b) and

< A ( ( ( B - a ) o a ) *c). A(aoa) -

(4.19.1)

implies b = [ ( B - a ) m a ] * c . rn

Weakly cancellative d-monoids which are semimodules over R + play an important role in the consideration of optimization problems in part 11. The ordi.na1 decomposition of such a monoid H may contain H A with IH

xI

=

1. Such t r i v i a l subsemigroups with

an idempotent element (a + e for X * A o )

lead to difficulties in

the formulation and validation of algorithms. We avoid these difficulties by a certain extension of the underlying semimodule. Let H be a weakly cancellative d-monoid with ordinal decomposition ( U x :

A E A ) and assume that H is a linearly ordered semi-

module over R

+.

Let H

P

=

{a] for some v E A . Then it is con-

venient to extend the trivial semigroup H

P

to

105

Linearly Ordered Semiwwdules over Real Numbers

in the case p > X

= min h and to

~ [ a , r I)

N

(6.14.2)

H":=

in the case p = Xo.

~ E R )

Then let

(6.15)

(a,r)

(a,r'):=

for all (a,r), (a,r') E';i

(a, r + r'),

u and all bEH\{a).

The external com-

position is defined by if a = 0, (6.16)

a

0

( a , r ): =

i f a > 0.

(a, a r )

We identify a

t)

(a,l) if l~ > X

and a

c--)

(a,O) if p = X

0

.

N

*he order relation on H

v

is the usual one with respect to the N

second component. In this way H

p

replaces H

p

in the ordinal

decomposition. If this is done for all trivial semigroups the new semimodule

is called e x t e n d e d . This is again a linearly

ordered semimodule over R+. We remark that therefore an extended semimodule is always linearly ordered by definition.

linear-

A

ly ordered semimodule without trivial semigroups in its ordinal N

decomposition is clearly extended. In particular, H is a weakly cancellative d-monoid. The ordinal decomposition of an extended semimodule H has w.1.o.g.

(cf. remarks after 4 . 1 3 )

the following form. H A

=: G I

0

0

is a nontrivial linearly ordered commutative group with neutral element e which is also the neutral element of H. For X > X o we find that H A is the strict positive cone of a nontrivial

Ordered Algebraic Structures

106

linearly ordered commutative group G A . We may identify the neutral element of G A with e. Then G The order relation on G tive cone H A u {el

A = {a-ll a E H A l U {el U H A .

is completely determined by the posi-

(cf. remarks after 2.7). Using (5.12) to

continue the external composition on R x G A

+

G A for A > A.

find that G A is a module over the ring R for all A € A .

we

We

remark that it is not useful to consider the ordinal sum of the groups

GA.

> 1 then such an ordinal sum is not an or-

If I h l

dered semigroup. Several cancellation properties in extended semimodules will be helpful in part 11.

(6.17) Proposition Let H be a weakly cancellative d-monoid the ordinal decomposition of which contains no trivial subsemigroups. Then

-

for all a , b E H . If H is an extended semimodule over R+ then (4 1

a z B

(5)

a ( B

c,

a a a ~ B o a , a o b z B o b ,

for all a , 6 E R + and all a , b E H with b < e < a and (6)

a z b

for all a E R + , then

c.

u o a z a o b

a > 0 and all a , b E H .

If a < 6 or A(a)

5 A(b)

107

Linearly Ordered Semimodules over Real Numbers

for all a , b , c E H and all a , B E R + with a Clearly, a < a

Proof. ( 1 )

*b

5 8.

implies e < b and X (a)

the reverse implication assume e < b and A ( a ) A(a)

5

For

A(b).

5 A(b). If

< X(b) then a < a * b follows from the definition of ordi-

nal sums. If X(a) = X ( b ) then a < a * b follows from ( 4 . 1 9 . 4 ) applied to an extended semimodule. (2)

and ( 3 ) are proved similarly.

( 4 ) As

(5.10)

implies a a

0

a =

B

0

is satisfied it suffices to prove that a n a

5 B if e

C

a. Suppose

5 B Ua

a. If a o a < B O a then a < 8 . Now assume

B <

Then [ ( a

a.

together with ( 2 ) imply ( a

-

- B)

0

a1

*

(B Da)

=

( B ma)

8 ) O a = e contrary to ( 6 . 4 . 2 )

which

holds in a linearly ordered semimodule. ( 5 ) and

( 6 ) are proved similarly.

(7) If X ( a ) 5 X(b) then (7) follows from ( 6 . 1 3 ) . Now assume h ( a ) > X(b) and a < 6 .

Then h(a) > X o and therefore a > e . In

an extended semimodule this implies a o a

C

B O a . Again

(6.13)

shows (7). 8

From ( 1 )

-

(3)

in proposition ( 6 . 1 7 ) we conclude the following

corollary which describes the sets corresponding to the different definitions of positivity and negativity considered in the discussion of d-monoids in chapter 4 .

(6.18)

Corollary

Let H be a weakly cancellative d-monoid the ordinal decomposition of which contains no trivial subsemigroups. (1)

The set H+ of all positive elements is equal to {a1 e

5 a);

strictly positive elements exist only if A has a maximum

Ordered Algebraic Structures

and then the set of all strictly positive elements is (a1 e < a , i(a) = max A } .

The set H- of all negative

elements is nonempty only if l A l = l ; then H-

=

{a1 a z e )

and the set of all strictly negative elements is equal to H - \ {el. The set P -

of all self-negative elements is equal to H'H+;

the only idempotent is e and the set { a / a < a * a ]

self-positive

of all

elements is equal to H + \{el.

Let a E H . The partition N(a) < dom(a) < P(a) of H in strictly negative, dominated and strictly positive elements with respect to a is given by {el <

~+xCel

and is given by N(a) = @ and < X(a)}

< {bl X ( b )

5

X(a))

we remark that any weakly cancellative d-monoid can be embedded into a weakly cancellative d-monoid the ordinal decomposition of which does not contain trivial subsemigroups (via the corresponding discrete semimodule). Finally we discuss the possibility that a given linearly ordered semimodule H over R + is contained in a linearly ordered semimodule H' of R I with H $ H'

sub-semimodule

of H' over R ' .

or

$ R',

R

i.e.

H over R + is a

At first we consider two examples

and show that the semimodules considered are not sub-semimodules of a semimodule with H = H' and R Let H = ZZ

*

C

R'.

be the additive monoid of the nonnegative integers.

109

Linearly Ordered Semimodules over Real Numbers

Then H is a linearly ordered semimodule over Z+ with usual multiplication as external composition. Suppose that H is a sub-semimodule of a linearly ordered semimodule H over R+ with Z+

5

R+. Then R+ is dense in IR+. F o r a E (0,l) we try to

define a o l E Z + . Monotonicity shows 0 < a n 1 fore a 0 1

5 1 and there-

1. For the special choice a € (0,1/21 we find the

=

contradiction 1 = (2a)ol = Z o ( a o 1 ) = 2 0 1 = 1 + 1 = 2 .

Therefore no such semimodule exists. Now let H = Q + be the additive monoid of nonnegative rationals. Then H is a linearly ordered semimodule over.Q+ with usual multiplication as external composition. Suppose that H is a subsemimodule of a linearly ordered semimodule H over IR+ Let B € I R + \ Q + .

Then a

5 B 01 5

y for

.

all a , y E Q + with a < B

C

y.

There exists no rational number B O 1 with this property. Therefore no such semimodule exists. In both examples the semimodules considered are subsemimodules over IR o f the linearly ordered semimodule (IR+,+,z)

with

+

respect to usual multiplication as external composition. Secondl y , the following example shows that a given linearly ordered

*

semimodule H over R+ with R C I R exists which is not a subsemimodule of any linearly ordered semimodule H' over R; and R +

5

with H

C

A'

R;.

Let H be the fundamental monoid ([O,l)],*,c-)

with respect to

the usual order relation and a *b:= min(a+b,l). ordered semimodule over Z

+

= min(za,l).

E is a linearly

with external composition z n a = 'a

Now we try to define a n 1 for a € (0,l).

Odered Algebraic Structures

I10

(a 0 1)

*

( a n 1 ) = a 0 (1

*

1 ) = a 0 1 shows that a 0 1 is an idem-

potent element of H'. Let 1

-<

(nu)

1 = (a

1)

n

=

a

1.

5 n a for some nElN. Then 1 In particular, let a E

(0,

=

1 0 1

1/21

and a E [ 1 / 2 , 11. Then a o l = a O ( a * a ) = a 0 ( 2 o a ) = ( 2 a ) o a

< -

loa = a < 1/2. This contradiction to 1

1. a 0 1

shows that no

such semimodule exists. Thirdly, let H be a submonoid of the linearly ordered commutative monoid H'. Then H and H' are linearly ordered semimodules over 22 +

and H is a sub-semimodule of H'. In general, if H i s

a linearly ordered semimodule over R + then it is not known

whether the external composition on R

0:

R+ x H

-D

H can be continued

x H ' such that H' is a linearly ordered semimodule over

R+.

Nevertheless, we make the following conjecture. Let H be a weakly cancellative d-monoid which is a linearly ordered semimodule over R+. Then this semimodule can be embedded in a linearly ordered semimodule H' over lR+ with weakly cancellative d-monoid H' 2 H. We remark that using the ordinal decomposition of such a semimodule and using the embedding result

of HAHN (cf. theorem 3.5) the cases R + C_ Q, and Q,

it i s possible to provide a proof for

5

R+

.

PART I 1

7.

L I N E A R ALGEBRAIC O P T I M I Z A T I O N

L i n e a r A l g e b r a i c Problems

In part I1 we discuss optimization problems which are generalizations o f linear and combinatorial optimization problems. Differently from part I we assume that the reader is familiar with the basic concepts and results which are necessary for the solution of such problems in the classical case. For t i n e a r

programming we refer to a chapter in BLUM and OETTLI

[1975],

for c o m b i n a t o r i a t o p t i m i z a t i o n to LAWLER 1 1 9 7 6 1 . Further we will use without explicit explanation the concept of c o m p u t a t i o n a z COmpteXity as described in GAREY and J O H N S O N [ 1 9 7 9 1 or in AHO, HOPCROFT and ULLMAN

[19741.

The greater part

of the problems considered can be solved by a pOZynomiaZ t i m e

a l g o r i t h m in the classical case. In ordered algebraic structures the computational complexity of performing an internal composition or of checking an equation or inequation is usually not known, in general. Therefore we can only give bounds for the number of such algebraic operations. The generalization of the optimization problems considered consists in replacing classical linear functions by functions which are linear with respect to a semimodule. Let ( H , + ) semimodule over a semiring ( R , $ , @ ) 0:

R X H + H

.. ,n}. (7.1)

with external composition

(cf. chapters 5 and 6 ) . Let c i E H for i E N : = { 1 , 2 , , .

Then the function z: R Y

z

be a

XI:=

xToc =

(XI

+

H defined by

0 c 1 ) * (x2 o c z 111

* . .. *

(Xn 0 Cn)

Linear Algebraic Optimization

112

is called a l i n e a r a l g e b r a i c f u n c t i o n with c o e f f i c i e n t s c i E H for i E N . Such a function is not linear in the classical sense. For example, we consider the semimodule (IRU{-},min) ring (1R+ ,+, +

IR U

{a]

*)

with external composition

0:

lR+

x

over the semi(IR U { - ) )

defined by i f a > O ,

a

0

a:= otherwise.

Then we find the linear algebraic function z(x)

= minIc. I x

I

j

> 01

f o r x E I R y with coefficients c i E l R U

{-I,

i E N.

I n particular, we may consider the semimodule (R,@) over the semiring ( R , @ , @ ) .

Then the linear algebraic function has the

form z(x) = x

T

Q

c =

cl)

(XI @

@

(x,

Q

c,)

@

...

@

(Xn

0 Cn)

for x E R n with coefficients c . E R , i E N . Further, a commutative semigroup ( H , * ) is a discrete semimodule over the semiring ( Z +, +

Then the linear algebraic function has the form

, a ) .

X

z(x) = x T n c = c for x E 22;

1

X

X

* c ,*...*c 2

with coefficients c

i

EH,

n

iEN.

NOW we consider a subset S o f Rn. The elements of S are called

f e a s i b t e o r f e a s i b Z e solutions. In ordered semimodules H over R we define the l i n e a r a l g e b r a i c o p t i m i z a t i o n p r o b l e m b y N

(7.2)

z:=

inf XES

xToc

Linear Algebraic Problems

113

with c o s t c o e f f i c i e n t s c i E H, i E N . In general, existence of an optimal value Y ' E H is not known; even if is possible that

exists then it

is not attained for any feasible solution.

Such questions are discussed for the respective problems in the following chapters, For most of these problems we can give a positive answer. As for classical linear programs and combinatorial optimization problems a characterization of the set of all feasible solutions by linear equations and inequations is very useful. The necessary basic definitions for matrix compositions over ordered semimodules are introduced at the end of chapter 5 .

L i n e a r a l g e b r a i c e q u a t i o n s l i n e q u a t i o n s ) appear in various forms similarly to linear algebraic functions; for example A B x L b for an m

x

n matrix

A

over R and b E Rm. Then {x E RnI

A @ x

5

b)

is the corresponding set of its solutions. In the discussion of duality principles we find linear algebraic equations of the form A for an m

x

T

n u l c

n matrix A over R and c E Hn.

In view of the discussion of ordered algebraic structures in part I it is highly improbable that one can develop a reasonable solution method for the general problem (7.2). Nevertheless, it is possible to solve (7.2) for several classes of problemei the simpler the underlying combinatorial structure of the problem the more general the algebraic structure in which a solution method can be developed. By analyzing a

Linear Algebraic Optimization

114

g i v e n c l a s s i c a l o p t i m i z a t i o n problem and a g i v e n method f o r i t s s o l u t i o n w e may f i n d a n 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

i n which t h e g e n e r a l i z e d method r e m a i n s v a l i d f o r t h e s o l u t i o n o f t h e g e n e r a l i z e d o p t i m i z a t i o n problem.

Using t h i s a l g e -

b r a i c a p p r o a c h we c a n g i v e a u n i f y i n g t r e a t m e n t o f d i f f e r e n t o p t i m i z a t i o n problems and d i f f e r e n t s o l u t i o n methods;

in this

way t h e r e l a t i o n s h i p b e t w e e n v a r i o u s p r o b l e m s a n d m e t h o d s becomes more t r a n s p a r e n t .

F u r t h e r by s p e c i a l i z i n g t h e g e n e r a l

method f o r t h e subsumed c l a s s i c a l p r o b l e m s i t i s o f t e n p o s s i b l e t o f i n d new s o l u t i o n m e t h o d s ;

i n t h i s way we may d e v e l o p m o r e

e f f i c i e n t algorithms.

We d i d n o t t r y t o c o v e r a l l o p t i m i z a t i o n problems which have been d i s c u s s e d o v e r 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 .

We selected

two c l a s s e s o f 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 which h a v e been d i s c u s s e d by many a u t h o r s d u r i n g t h e l a s t t h r e e d e c a d e s .

As

t h e s e i n v e s t i g a t i o n s w e r e made s e p a r a t e l y many r e s u l t s h a v e been r e i n v e n t e d from t i m e t o t i m e .

We d o n o t c l a i m t h a t w e

succeeded i n f i n d i n g a l l t h e r e l e v a n t

l i t e r a t u r e b u t w e hope

that the resulting bibliography is helpful to the reader.

I n t h e f i r s t c l a s s we c o n s i d e r o p t i m i z a t i o n p r o b l e m s o v e r s e m i r i n g s (R,@,@).

W e d i s c u s s a Z g e b r a i c p a t h prob2ems

(chap-

t e r 8 ) which c a n b e s o l v e d v i a t h e e v a l u a t i o n o f c e r t a i n matrix s e r i e s I @ A @ A

A*:=

2

@

...

o r equivalently v i a t h e determination of a c e r t a i n s o l u t i o n 2

of the matrix equation 2 =

(2 @ A )

@ B .

115

Linear Algebraic Problems

Then we investigate a l g e b r a i c e i g e n v a t u e p r o b l e m s

(chapter 9)

which are closely related to path problems. Finally we consider (7.2) over semirings, i.e. the e x t r e m a t l i n e a r program (chapter 10) M

z = inf{c

T

Q

x( A

Q

x

2

b, xERn}.

In the second class we discuss a l g e b r a i c l i n e a r programs in extended semimodules over real numbers, i.e. N

z = fx

for R

clR.

T

n a ] cx

2

c, XER:)

We develop a duality theory similarly to the classi-

cal case (chapter 1 1 ) which can be applied for a solution of combinatorial optimization problems with linear algebraic objective function. we discuss a l g e b r a i c n e t w o r k flow p r o b l e m s , including solution methods f o r a l g e b r a i c t r a n s p o r t a t i o n and

a s s i g n m e n t p r o b l e m s (chapter 12). Many combinatorial structures can be described by i n d e p e n d e n c e s y s t e m s . The corresponding algebraic optimization problems are considered in chapter 13; in particular, we investigate a l g e b r a i c m a t r o i d and 2 - m a t r o i d

i n t e r s e c t i o n p r o b l e m s . A l g e b r a i c matching p r o b l e m s can be solved by similar techniques, the difference lies only in the combinatorial structure. All discussed algebraic optimization problems involve only linear algebraic constraints and linear algebraic objective functions. There is no difficulty in formulating nonlinear algebraic problems in the same way. Only very few results for nonlinear algebraic problems are known. At the time being a theory for such problems comparable to the linear case has not been developed to our knowledge.

8.

A l g e b r a i c P a t h Problems

I n t h i s c h a p t e r we c o n s i d e r p a t h s i n a d i r e c t e d g r a p h G = ( N , E )

N x N. W e a s s i g n w e i g h t s a , .

w i t h v e r t e x s e t N and arc s e t E to the arcs (R,@,(D)

17

( i , J ) € E . These w e i g h t s a r e e l e m e n t s o f a s e m i r i n g

w i t h u n i t y 1 a n d z e r o 0. Such a g r a p h i s c a l l e d a

n e t w o r k . The l e n g t h I f p ) o f a p a t h ( e l , e 2 , of

...,e m ) i s

t h e number

l ( p ) = IE; t h e w e i g h t w ( p ) i s d e f i n e d b y

its arcs, i.e. w(p):= a

8 a

el

B

...

.

B ae

m

2

L e t P d e n o t e t h e s e t of a l l p a t h s i n G and l e t P . denote the lj

s e t o f a l l p a t h s i n G f r o m v e r t e x i t o v e r t e x j. The d e t e r m i n a t i o n of t h e v a l u e

for Some i , j E N a path P E P

i j

( i f it e x i s t s )

( o r f o r a l l i , j E N ) and t h e d e t e r m i n a t i o n of

with w(p) = a*

i j

( i f i t e x i s t s ) f o r some i , j E N

( o r f o r a l l i , j E N ) i s c a l l e d t h e a l g e b r a i c path problem.

For c o n v e n i e n c e w e i n t r o d u c e t h e w e i g h t f u n c t i o n

0:

2’

+

R

d e f i n e d by i f S = @ ,

0

(8.2)

a ( s ) := i f S + @ .

This function i s n o t always well-defined

a s P is often infinite.

L a t e r on w e d i s c u s s a s s u m p t i o n s which imply t h a t a t l e a s t o ( P

i s well-defined.

Then a *

i j

= 0(Pij).

116

i j

)

117

Algebraic Path Problems

At first we consider the classical shortest path problem. In this case the weights are elements of (IR U {-),min,+)

with

zero = and unity 0. We may as well consider the linearly ordered commutative monoid

(W

u

{ - ) , + , z )with

absorbing maximum

= and neutral element 0. In any case, the special resulting path problem a* : = min ij PEPij

(8.3)

z

, Ci,j E N )

a

VV

(V,V)EP

is called the s h o r t e s t p a t h p r o b l e m . Then p E P

ij

with a* = w ( p ) ij

is called a s h o r t e s t p a t h from i to j. The first discussion

of this problem seems to be due to BORfVKA 119261. He admits only nonnegative weights. In this case shortest paths from a given vertex i to all remaining vertices j

$

i in G can simul-

taneously be determined using an elegant method proposed by DIJKSTRA [ 1 9 5 9 ] and DANTZIG [19601.

Let N = 11.2 ,...,n).

The weight ui of a shortest path from

vertex 1 to vertex 1 = 2,3,...,n procedure. W.1.o.g.

we assume a

is computed in the following

ii

= 0 for i E N and a

ij

:=

0

for (i,j) 6 E.

(8.4)

S h o r t e s t paths i n networks w i t h r e a l nonnegative weights a

for j = 1 , 2 , . . .,n; T:= N \ { 1 }

Step 1.

u

Step 2 .

Determine k E T with

j

:=

= min{u I j ET); k j T:= T\{k).

u let

11

Linear Algebraic Optimizarion

118

If T =

Step 3 .

then stop. min{uj, u k + a 1 for all j E T ; kj

Otherwise u := j go to step 2.

In particular u . =

at the end of the procedure indicates

3

that there exists no path from vertex 1 to vertex j. We will give an inductive proof of the validity of this method.

we claim that at each stage of the algorithm two properties

hold

( Q : = N'T

for the current T ) :

for all jE Q

(8.5)

uj is the weight of a shortest path

connecting 1 and j ,

for all jE T

(8.6)

a, is the weight of a shortest path

connecting 1 and j subject to the condition that each node in the path belongs to Q U {j).

Clearly

( 8 . 5 ) and

(8.6) are satisfied after step 1. Now assume

that the current Q,T,u fulfill ( 8 . 5 ) ,

( 8 . 6 ) before step 2 .

Let k E T be determined in step 2 and suppose the weight of a shortest path p connecting 1 and k is v -c uk. Then p contains a first node j b Q u Ik).

Therefore v

2

u k contrary to our

assumption. Thus (8.5) is satisfied by Q U {k),

too. In parti-

cular, if T = {k} then the weights of all shortest paths are determined. Otherwise let v denote the weight of a shortest path connecting 1 and j ET\{k) for Q u [k,j).

If j

*

subject to the condition in (8.6)

Let i denote the predecessor of j in such a path.

k then v = u

j'

Otherwise v = u

is satisfied for Q U {k}, T\{k] in step 3 .

+ akj. Therefore (8.6)

and the revised weights u . 1

AZgebraic Path h b l e m s

119

I

The performance of this method needs O(n ) comparisons and O(n

2

additions.

Next we discuss the generalization of this method for weights from o r d e r e d c o m m u t a t i v e m o n o i d s

(H,*,Z)

with neutral element

,+,-,~) e. W e r e m a r k t h a t H is an o r d e r e d s e m i m o d u l e o v e r (Z+

with respect to the trivial external composition

0:

Z+ X H + H

d e f i n e d by (8.7)

z

a:= az

for all a E H , z E Z +

0

( i n p a r t i c u l a r a : = e). A p a t h p in t h e

d i r e c t e d g r a p h G m a y b e r e p r e e e n t e d b y its i n c i d e n c e v e c t o r x E {Oil IE d e f i n e d by

T h e s e t o f a l l i n c i d e n c e v e c t o r s o f p a t h s in P by S i j .

ij

ie d e n o t e d

Then we may rewrite the shortest path problem for the

c a s e of s u c h a s e m i m o d u l e a s (8.9)

a * := min ij XES ij

with coefficients of a

ij

weight

xToa

(weights) a

VV

I

(i,j E N )

E H . If we assume the existence

and the existence o f a path connecting i and j with

* aij

(an a l g e b r a i c s h o r t e s t p a t h ) then A has to be

t o t a l l y o r d e r e d , in general. A n a l g e b r a i c v e r s i o n o f (8.4) can b e g i v e n by s i m p l y r e p l a c i n g t h e e q u a t i o n in step 3 by (8.10)

A g l a n c e t h r o u g h o u r p r e v i o u s p r o o f o f t h e v a l i d i t y o f (8.4) s h o w s t h a t t h e a l g e b r a i c v e r s i o n is v a l i d if B i s a p o s i t i v e l y

Linear Algebraic Optimization

120

linearly ordered commutative monoid. Commutativity is not necessary for a proof of the validity; but without commutativity H is not a semimodule over 22 +

.

If we try to avoid the

assumption that H is linearly ordered then we cannot be sure that we can perform the steps in the algorithm: in step 2 we

I j E T ) does not exist, in general. All j weights have to be nonnegative (i.e. e 5 a 1 ; otherwise ij even in the classical case counterexamples can easily be may find that min{u

given. In terms of a semiring the algebraic version has been discussed by GONDRAN [1975c], MINOUX 119761 and CARRE 119791. GONDRAN mentions the following examples of positively linearly ordered commutative monoids. F o r an application of the algebraic version of ( 8 . 4 )

let w.1.o.g.

a

ij

:=

(maximum of H)

00

for (i,j) a E and aii = e (neutral element of H) for all i E N .

(8.11)

Examples (lR+U ( - ) , + , L ) with neutral element 0 and maximum Here we get the c l a s s i c a l s h o r t e s t p a t h ([0,11,*,2)

00.

problem.

with neutral element 1 and maximum 0.

Here w(p) is the product of arc-weights; this is the socalled r e l i a b i l i t y problem. (fO,l),-,z) with neutral element 1 and maximum 0.

*

Here a , . = 1 iff a path connecting i and j exists 11

( c o n n e c t i v i t y problem). (JR+U{-),min,))

with neutral element

00

and maximum 0.

* the Here w(p) is the capacity of a path p and a ij maximum c a p a c i t y of a path connecting i and j.

Algebraic Path Problems

121

F o r a d d i t i o n a l examples o f s u c h an 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 we r e f e r t o chapter 4.

In p a r t i c u l a r , t h e solution of

(8.11.4)

i s r e l a t e d t o maximum w e i g h t e d t r e e s i n u n d i r e c t e d g r a p h s ( c f . KALABA [19601).

For t h e remaining p a r t of t h i s c h a p t e r w e drop t h e assumption t h a t t h e u n d e r l y i n g a l g e b r a i c s t r u c t u r e is p o s i t i v e l y l i n e a r l y o r d e r e d . The s e m i r i n g (R,$,@) and,

is only pseudoordered

a path with weight

i n general,

* a

ij

(cf.

w i l l not exist.

5.14)

* a

i j

is

t h e supremum o v e r t h e w e i g h t s o f a l l p a t h s f r o m i t o j ; c l e a r l y , t h i s supremum w i l l n o t a l w a y s e x i s t ,

as P

i j

may b e i n f i n i t e .

F o r c o n v e n i e n c e w e i n t r o d u c e some d e n o t a t i o n s f o r s e t s o f p a t h s which w i l l a p p e a r i n t h e f o l l o w i n g d i s c u s s i o n . k E N U {O}. Then

b

i j

P[kl i j

'

a n d Pk

i j

0

L e t i , j E N and

denotes t h e set of a l l elementary paths i n

i j

d e n o t e s t h e s e t o f all p a t h s p E P

denotes t h e set of a l l paths p

E

P

i j

i j

with l ( p )

1. k

with l ( p ) = k . For

t e c h n i c a l r e a s o n s we a s s u m e t h e e x i s t e n c e o f a c i r c u i t p o f l e n g t h 0 and u n i t y w e i g h t c o n t a i n i n g o n l y v e r t e x i b u t no a r c . S u c h a c i r c u i t i s c a l l e d t h e n u l l c i r c u i t o f i. Then u ( P L y l ) = 1 f o r e a c h i E N a n d U ( P [ ~ ] ) = 0 f o r e a c h ' ( i ,j ) w i t h i i j

As

+

j.

an i n t r o d u c t o r y example we, c o n s i d e r t h e , c l a s s i c a l s h o r t e s t

p a t h problem with p o s s i b l e n e g a t i v e weights i n t h e semiring

(IR U { - ) , m i n , + )

with zero

of n e g a t i v e w e i g h t be u n b o u n d e d .

If P

0

and u n i t y 0; I f G c o n t a i n s c i r c u i t s

( d e n o t a t i o n : n e g a t i v e circuits) t h e n i j

+

( 8 . 3 ) may

@ and i f G d o e s n o t c o n t a i n a n e g a t i v e

c i r c u i t t h e n t h e r e e x i s t s a s h o r t e s t p a t h from i t o j which is elementary.

122

Linear Algebraic Optimization

We a s s u m e a

= 0 for all i E N and a

ii

f o r u ( P [ ~ ’ ) , k = 0,1,2,... ij

-

f o r k = 1,2,. =

UCP;;])

.. .

for i

if (i,j) d E . Then

=

we find t h e following recursion:

F o r k = 0 w e k n o w u ( P [ O ’ ) = 0 f o r i E N and ii

*

j. A n i n d u c t i v e p r o o f o f t h i s r e c u r s i o n

is g i v e n in t h e f o l l o w i n g .

F o r k = 1 v a l i d i t y of (8.12) i s o b v i o u s . W.1.o.g. t h a t G is a c o m p l e t e d i g r a p h . T h e n P [ k l i, and let p E P I k 1 be a path with weight i j

+ 0

we assume

for k

[kl) U(Pi,

.

2

1.

Let k

2 :

If 1 ( p ) c k

u ( P i[, k - l l ) + a .

O t h e r w i s e 1 ( p ) = k. T h e n l e t n 1,In d e n o t e t h e p r e d e c e s s o r of j i n p. N o w w ( p ) = u(P:;’) +a hj then

w(p)

=

.

b o t h c a s e s (8.12) is s a t i s f i e d .

A s G contains no negative circuits we can derive the weights

o f t h e s h o r t e s t p a t h s f r o m (8.12). (8.13)

j a i j = 0 ( P iIn-1

We find

I) L e t U [kl

s i n c e a n e l e m e n t a r y p a t h h a s at m o s t l e n g t h n - 1 . d e n o t e t h e m a t r i x w i t h entrie.s u(P [kl 1 . ij

Then

(8.12) c a n b e

i n t e r p r e t e d a s m a t r i x - c o m p o s i t i o n o v e r t h e s e m i r i n g (IR U min,+) (8.14)

A

*

= U In-’]

=

(ai,) w e f i n d

( ( ( V [ O 1 + A ) +A)...

U I O 1 i s t h e i d e n t i t y o f t h e s e m i r i n g of n

As

IR U

-

An 1 s

(cf. c h a p t e r 5). F o r A:=

{-I

t h i s l e a d s t o A * = An-’

.

) +A.

x

2 n-1 w e m a y d e t e r m i n e A

n matrices over

Therefore a computation o f

solves the shortest path problem. As A S = n- 1

{-I,

by t h e s e q u e n c e

-

An 1

for all

123

Algebraic Path Robkms

A,A

2

,A

4

,...,A Ik

;

therefore O ( l o g 2 n) matrix multiplications are necessary, each 3

3

of which needs O(n ) comparisons and O(n ) additions. In particular, we can determine the weights of all shortest paths from node 1 to node j for all j

*

I from the first row

A:

of A f

1.e.

this recursion is the classical B E L L M A N - F O R D method. It needs

3 3 O(n ) additions and O(n ) comparisons. Improvements of this method can be found in Y E N [19751. In the general case, i.e. for weights in arbitrary semirings, a solution of (8.11, if it exists, is determined by computation of Ak and A[k1:=

(8.16)

I

Q A Q A’

Q

...

Q Ak

where I is the unity matrix in the semiring of all n x n matrices over the underlying semiring ( R , Q , b ) .

Problems which can

be reduced to (8.1) and methods for the computation of these matrices, called matrix-methods, have been considered by many authors during the last thirty years. We mention only those which explicitly uae some algebraic structures. The following list will hardly be complete. LUNTS

[I9501 considers applications to relay contact networks,

SHIMBEL KLEENE

([1951], [19531 and [1954]) to communication nets. [1956] bases a theorem in the algebra of regular events

on a Gaussian elimination scheme which is separately developed

for the transitive closure of a graph by ROY [19591 and

Linear Algebraic Optimization

174

WARSHALL [1962].

MOISIL [1960] discusses certain shortest

path problems, YOELI [1961] introduces Q-semirings and PANDIT

I19611 considers a matrix calculus. Further results and generalizations are due to GIFFLER ([19631, [19681). CRUON and HERVE [1965], PETEANU ([1967a],

[1967b],

[19691, [19701,

[1972]), TOMESCU [1968], ROBERT and FERLAND [19681, BENZAKEN

I19681 and DERNIAME and PAIR [19711. Methods based on fast -atrix multiplication are discussed by ARLAZAROV et al. [19701, FURMAN [1970], FISCHER and MCYER [19711 and MUNRO [19711. CARRE 119711 develops a systematic treatment of several linear algebra procedures. This approach is continued by BRUCKER

([19721,

[1974]), SHIER ([19731, [19761), MINIEKA and SHIER [19731, GONDRAN (119751, [1975a],

[1975bl, [1975c1, [1975d],

[19791

and [1980]), AHO, HOPCROFT and ULLMAN [19741, MARTELLI [1975]),

“19741,

ROY [1975], BACKHOUSE and CARRE [19751, TARJAN ([1975],

[1976]), FRATTA and MONTANARI

119751, WONGSEELASHOTE

[1979]), LEHMANN [19771, MAHR

(119791, (1980aI and [1980bl),

FLETCHER [1980], and ZIMMERMANN, U.

([1976],

[19811. GONDRAN and MINOUX

[1979b] a s well a s CARRE [1979] summarize many of these results.

The following proposition shows the different character of A k and A[k1.

T h e proof is drawn from WONGSEELASHOTE [1979].

(8.17) P r o p o s i t i o n L e t A be an n x n matrix over a semiring

for all i,] E N and all m E l N U { O ) .

(R,@,@).

Then

Algebraic Path Problem

Proof. For paths P E P

125

let p o q denote the path jk in Pik traversing at first p and then q. For Q C_ Pij and

S C_ P

jk

let

Q oT:=

o(Q

(3)

0

and q E P

ij

{PO q

T) =

O ( Q

p E Q , q E T 1 . Then 8 o(T)

r

provided that the weight function is well-defined for Q and T. Now for m

2

1

Pm = ij

(4)

u

1 CPir. 1

Pr

0

...

1 2

Therefore (1) is satisfied for m

0

P

jl

m-1

2

pairwlse different rlf...,r m

1. For m = 0 we find

Ao = A 101 = I. As Po contains the null-circuit with weight ii equal to the identity 1 of the semiring ( 3 ) and ( 4 ) show (1). AS

(2)

is Implied by ( 1 ) .

In our introductory example (IR U {-),min,+) A [sl = A [n-tl for all s

2

Am = A[m1 and

n-1. Such results do not hold in

the general case. If there exists r E N U ( 0 ) such that

then A is called s t a b l e . The minimum of the integers satisfying (8.18) I s called the s t a b i l i t y i n d e x of A. For stable matrices the values a* in (8.1) exist and from (8.17) we find ij

for a l l i , j E N . We remark that ( 8 . 1 8 ) all s

2

implies A[']

= A [rl

for

r. Sufficient conditions for a matrix to be stable

are conveniently formulated in terms of the circuits of the

I.

126

Linear Algebraic Optimization

network G. Let C denote the s e t of a l l e l e m e n t a r y non-null

c i r c u i t s and W:= {w(p) I p E

C ) .

1 a a a a2 a

(8.20)

Let

...

a am

for a E R and m E N U { O } and let (8.21)

a[ml:=

1

a

@ a

1

(a

1

@a2) B . .

.B

(al @ a 2

@...

@arn)

for a E RS, s z m EN U { O } . The network G is called a b s o r p t i v e if

(8.22)

p 1=

a [11

for all a E w

,

for all a E w

,

rn-regular if (8.23)

[ml = a [m+ll

and m - a b s o r p t i u e (8.24)

If

G

then

if

is absorptive then G

for all a E

a[ml = a[m+l] G

wm+l.

is m-absorptive. If G is m-absorptive

is m-regular. Absorptive networks over certain semirings

were introduced by CARRE 119711; the generalizations (8.23) and (8.24)

are due to GONDRAN [1975a] and ROY 119751.

We remark that a s e m i r i n g R is called a b s o r p t i v e

(m-regular,

m - a b s o r p t i v e ) if the respective property is satisfied for a l l elements of R. An absorptive semiring is idempotent.

(8.25) P r o p o s i t i o n If

G

(1)

(2)

is an m-regular network then for all a E w a[m] = a [Sl

a[ml

= a

for all s

2

m+l,

for all s 2 m+l;

@ as

if G is an m-absorptive network then for all a € W s !3)

a[ml

=

ark]

(4)

a[ml

= a[ml

(a1

...

@ as)

for all s

2

k

for all s

2

m+l.

m+l,

127

Algebraic Path Problems

Proof. We show ( 3 ) and ( 4 ) .

A

proof of (1) and (2) is similar.

( 3 ) . F o r s = k = m + l this is obvious. Let s = k > m + l and

assume B[s-21

for all 8 E

= S[s-lI

Then we get

=

1 @ a 1 fd (1 @ a2

= a[e-1]

Therefore a[s-l]

=

(B

...

(B

(a2 8

...

8

a 6-1 1 )

.

a[s]

for all a E Ws and for all s

2

m

+ 1.

This leads to ( 3 ) as a E W s implies (al,aS,...fak) E W k for s In particular

( 3 ) implies

arm] = a[sl = a[s-11 = a[ml

for all a E Wsf s

2

(B

(al 8 a2 8

(B

(a1

8

a2 8

... 8 ... 8

as) as)

m + 1. Thus ( 4 ) is satisfied.

In (5.14) we introduced the pseudoordering a c b

a ( B b = b

:c.,

of the semiring ( R f ( B f 8 ) . G is absorptive i f f

f o r all a E W ,

G i s m-regular iff

m+l < a [ml

0

for all a E w and G is m-absorptive i f f a

1

8 a2 8

...

8

a < a[ml m+l

-

for all a €

wm+l.

k.

12s

Linear Algebraic Optimization

Similarly, of

t h i s pseudoordering. P w i t h Q C_ T.

L e t Q,T b e s u b s e t s o f i f

c a n b e i n t e r p r e t e d i n terms

(8.25.1 - 4 )

properties

d e n s e i n 2'

Then Q i s c a l l e d

there exists a f i n i t e subset H of

for all pET\Q

Q such

5 o(H).

that w(p)

(8.26) P r o p o s i t i o n L e t Q b e a d e n s e s u b s e t o f T. Further, Proof.

B is f i n i t e ,

i f

Let pET\B.

I f Q C_ B

d e n s e i n T.

If

T t h e n B i s d e n s e i n T.

= o(Q).

then u(B)

and t h e r e f o r e t h e r e exists a

Then p E T \ Q

1. o ( H ) .

f i n i t e s u b s e t H o f Q w i t h w(P)

5

B the set B is

As H

B is f i n i t e t h e n B X Q i s f i n i t e .

a n d some H

Q w e know w ( p )

lead t o w(p)

5

U(H).

(5.14.2)

u(Q).

o(B) = o(BwQ) @

5

Now

for pEBwQ

Then 0 < U(QwH) and

< o(Q),

shows o ( B \ Q )

(5.14.1)

1.e.

= O(Q).

O(Q)

rn

t r a v e r s e an e l e m e n t a r y nonA m 1 (ml null c i r c u i t a t m o s t m t i m e s i s d e n o t e d b y P ij' S i m i l a r l y P ij The s e t o f

a l l paths P E P

i j

which

denotes t h e set of a l l paths P E P

i j

e l e m e n t a r y non-null circuits. L e t p (8.27)

therefore

l(p)

5

n

l(q)

5

n m

$:j

c_

IcI

P i[SI j

m + n

+

n

-

-

Egm i j

I

qE9:;).

Then

1 =:s,

1 =: t i

-

Pi,( m ) C_ P iIt1 j

and

t r a v e r s e a t most m

which

.

( 8 . 2 8 ) Proposition (1)

I f G is a b s o r p t i v e t h e n

(2)

i f G is m - a b s o r p t i v e dense i n Pi,

;

6"i j

is dense i n P

i j

;

and R is commutative then $ ( m ) i s

ij

Algebraic Path Problems

129

i f G is m - r e g u l a r and R i s c o m m u t a t i v e t h e n

(3)

dense in P

(1)

Proof.

3 m,

is

lj

-

ij

Let P E P

ij

\$:,.

L e t p' E

$yj

denote the elementary

p a t h d e r i v e d f r o m p by e l i m i n a t i n g a l l c i r c u i t s i n p. A s p c o n t a i n s a t l e a s t o n e e l e m e n t a r y n o n - n u l l c i r c u i t q and w ( q ) t h e m o n o t o n i c i t y c o n d i t i o n (5.2) (cf. 5.14) Thus (2)

Boij

implies w(p) zw(p').

is d e n s e in P

ij' F r o m (8.25.4) we k n o w t h a t f o r e a c h s - t u p l e q = ( q l l q 2 1 . .

2

E C S with s

..,q,) (8.29)

w(ql)

@

m+l t h e r e l a t i o n

w(q2)

Q

i s s a t i s f i e d . L e t p E P i j \$;;).

.. .

Q

5 q[ml

w(qs)

Then p traverses s elementary

non-null circuits q 11q21...lqs with s

2

m+l.

Successive elimi-

n a t i o n o f q s , q s ~ l l . . . , q l l e a d s t o t h e p a t h s pl,pz,...lps. particular p w(ps):=

In

i s an elementary path (or a null path with weight

1). A s R is c o m m u t a t i v e we f i n d

T h e m o n o t o n i c i t y c o n d i t i o n (5.2) t o g e t h e r w i t h

(8.29) s h o w s

' (m) is dense in P A s H C _ AP ( m ) t h i s s h o w s P ij' ij ij (3)

11

F r o m (8.25.2) w e k n o w

(8.31)

w(qIS

5

w(q) I m l

for all q E C and all s

2

m+l.

Linear Algebraic Optimization

130

Let p

E Pij

. Then

,$:

s times with some s

2

p t r a v e r s e s at l e a s t o n e g E C e x a c t l y m+l. L e t p = p l o q o p 2 0 q o

... ~ q o p , + ~ .

A g a i n t h e r e m a y be s o m e n u l l p a t h s i n t h i s r e p r e s e n t a t i o n w i t h w e i g h t 1.

Prom

Commutativity leads to

(8.31) a n d m o n o t o n i c i t y c o n d i t i o n

5

(8.32)

W(P)

Let

p ( l ) :=

W ( P 1 ) 8 w(p2) 8 Pi* P20

*..

( 2 ) := p l o q o p 2 0

p ( m ) : = PI. q.p2.

(5.2)

... 8

OPS+1

w e get

w(ps+l) 0 w(q)

[ml

.

’

... ~ P s + l ~ ’ ” ’ q....

.

oqopm-lD pmo . . . o p s + l

(8.32) together with commutativity shows w(p)

5

u ( { p ( l ) , p ( 2 )I

. . . ,

p(m)I).

F u r t h e r e a c h p ( i ) i s a s u b p a t h of p w h i c h d o e s t r a v e r s e q at most m times. If p contains further elementary non-null circuits t h e n t h e p r o c e d u r e i s a p p l i e d t o t h e p a t h s i n H = {p“’,p

(2)

,...I.

A s t h e r e is o n l y a f i n i t e n u m b e r o f s u c h c i r c u i t s t h e p r o c e s s t e r m i n a t e s a f t e r a f i n i t e n u m b e r o f steps. lead t o w(p) < in P

ij

u(g)

with finite

$m

Ij

(5.14.1)

. Thus

Bmi j

and

(5.14.2)

i s dense

-

T h e m a t r i x A is c a l l e d a b s o r p t i v e G is a b s o r p t i v e

f m - r e g u l a r , m - a b s o r p t i v e ) if

(m-regular, m-absorptive).

The following theorem

s u m m a r i z e s t h e r e s u l t s of C A R R g [ 1 9 7 1 1 , GONDRAN [ 1 9 7 5 a ] a n d R O Y 119751.

19791.

I n t h i s f o r m it c a n b e f o u n d i n W O N G S E E L A S H O T E

131

Algebraic Path Problems

( 8 . 3 3 ) Theorem Let

be a s e m i r i n g a n d l e t A b e an' n x n m a t r i x o v e r R.

(It,$,@)

Then A is s t a b l e w i t h s t a b i l i t y i n d e x r i f o n e o f t h e f o l l o w i n g conditions holds

5 n-1)

(I)

A is absorptive

(then r

(2)

A is m - a b s o r p t i v e

and R commutative

(3)

A

Proof.

5

(then r 5 , n m

is m - r e g u l a r a n d R c o m m u t a t i v e

nm

+

n-lIf

ICI + n

- 1).

F r o m ( 8 . 2 8 ) w e know t h a t

ST,)

(m) ij with szn-1 (;

(then r

is d e n s e i n P ( 8

z n m + n - l ,

( 1 ) ( ( Z ) , (3)) i m p l y t h a t $o ij [sl Therefore t h e f i n i t e set P

i,' s znmICl+n-l)

i j

is d e n s e i n P

ij

a n d f o r a l l s u c h s w e f i n d ( c f . 8 . 2 6 ) t h a t U ~ P ' ~is ~ e )q u a l to

if

( a ( $ ( m ) ) fu($y,)). ij

u(;y,)

(8.17)

Then

shows t h a t A i s s t a b l e

and h a s t h e r e s p e c t i v e s t a b i l i t y index.

Theorem

(8.33) g i v e s s u f f i c i e n t c o n d i t i o n s f o r a m a t r i x t o b e

stable.

For such m a t r i c e s t h e a l g e b r a i c p a t h problem

(8.1)

can be solved using t h e equations A* = A[r1

(8.34) for all

8

2

...

=

E

= A 181

r r t h e s t a b i l i t y index o f A.

--

...

By v i r t u e o f

(8.16)

t h e following r e c u r s i v e formula is v a l i d

(8.35) with A

A

Ik']

As A

Ikl

(8.37)

A)

@

a

I

,

k = 1,Zf...

= I where I d e n o t e s t h e n x n u n i t y m a t r i x o v e r t h e

s e m i r i n g (It,$,@).

(8.36)

= ( A Lk-l'

A*

From

= (A* 8 A )

8 A = A Q AIkl A*

(8.34) a n d (8.35) w e c o n c l u d e I

.

for all k

= (A @ A*)

@

I

.

Em

U {O} w e f i n d

132

Linear Algebraic Optimization

Equations (8.36) and (8.37) show that

A*

certain matrix equations over R. Let Y:=

is the solution of A*

0

B and

Z:=

B 0

A*

for a matrix B over R. Then (8.36) and (8.37) lead to Y = (A 0 Y )

(8.38)

B,

Z = ( Z O A ) $ B .

In particular, if B is the unit matrix then

is a solution

A*

of (8.3811 if B i s a unit vector, then a column resp. a row of A * are solutions of (8.38). The semiring of all n x n matrices over R is pseudoordered by

The solutions

A*

0 B resp. B 69 A * are minimum solutions of

(8.38) with respect to this ordering if R is idempotent. This

follows from

for all s = 0 , 1 , 2 , . . . , s

2

if Y is a solution of (8.38). Then for

r we find

(8.40)

Y

(B

(A*

0 B) =

(As

Thus, if R is idempotent, then

Y)

@

s

5 r

=

0 B

A*

graph contains no circuits then

As

n-1 and (8.39) shows Y =

(A*

(B

A*

0 B) @

1. Y.

(A*

0 B)

.

If the underlying

is the zero-matrix for 0 B is the unique solution

of (8.38). The same remarks hold for the second equation in (8.38).

The solution of such equations can be determined by methods similar to usual methods in linear algebra.

A

systematical

treatment o f such an approach was at first developed by CARRE

119711 for certain semirings. Then well-known methods

from network theory appear to be particular cases of such

Algebraic Path Problems

methods.

Several authors

133

( c f . remarks following 8.48) e x t e n -

ded t h i s approach t o o t h e r a l g e b r a i c s t r u c t u r e s o r d e r i v e d methods f o r n e t w o r k s o f s p e c i a l s t r u c t u r e .

BACKHOUSE a n d

CARRE [ 1 9 7 5 1 d i s c u s s e d t h e s i m i l a r i t y t o t h e a l g e b r a o f

regu-

l a r l a n g u a g e s , f o r which t h e m a t r i x A* a l w a y s e x i s t s due t o a s u i t a b l e axiomatic system. N e x t w e c o n s i d e r two s o l u t i o n m e t h o d s f o r

(8.1) which can be

i n t e r p r e t e d as c e r t a i n l i n e a r a l g e b r a p r o c e d u r e s .

A straight-

f o r w a r d f o r m u l a t i o n o f a g e n e r a t i z e d o r a t g e b r a i c JAGOBI-method f o r t h e second equation i n (8.41)

Z(k'l):=

with Z ( O ) : =

(Z(k)

( 8 . 3 8 ) is 8 A)

@ B

,

k = O,l,Z

,...

B.

Proposition L e t A be a n xn-matrix

i n d e x r . Then

(8.41) i s a f i n i t e recursion f o r t h e determina-

t i o n o f B @ A*; (k)

for a l l k

Proof. k

2

over the semiring (R,@,@) with s t a b i l i t y

we find = B @ A *

r.

A n i n d u c t i v e a r g u m e n t shows Z(k) = B Q A I k l

= 0,1,2,.

for

.. .

W e remark t h a t f o r t h e c l a s s i c a l s h o r t e s t p a t h problem o v e r

t h e s e m i r i n g (IR U { - ) , m i n , + )

a n d f o r B = (0-... - 1

t h i s method

r e d u c e s t o t h e m e t h o d o f BELLMAN [ 1 9 5 8 ] f o r t h e c a l c u l a t i o n of t h e w e i g h t s of a l l s h o r t e s t p a t h s from node 1 t o a l l o t h e r nodes.

I n t h e g e n e r a l c a s e with B = I a complete s o l u t i o n of

Linear Algebraic Optimzarion

134

(8.1) is d e t e r m i n e d ; t h e v a l u e s a* a r e n o t n e c e e s a r i l y ij

a t t a i n e d as weights of certain paths P E P O(r)

This method needs

ij’

2 i t e r a t i o n s and e a c h i t e r a t i o n r e q u i r e s O ( n 1 8 - c o m p o s i 3

t i o n s a n d ~ ( )n %-compositions.

r = nm I C l

+ n-1. T h e n u m b e r

~f A is o n l y m-regular

of e l e m e n t a r y n o n - n u l l c i r c u i t s

ICl is not b o u n d e d by a p o l y n o m i a l i n n. I f

A

is m-absorptive

t h e n r = O(n). In t h i s case t h e method r e q u i r e s O ( n s i t i o n s and O ( n

4

(m*O)

4

8-compo-

@-compositions.

)

T h e JACOBI-method is a so-called i t e r a t i v e method in l i n e a r algebra. T h e s e c o n d c l a s s of s o l u t i o n t e c h n i q u e s in l i n e a r a l g e b r a c o n t a i n s d i r e c t o r e l i m i n a t i o n m e t h o d s . We w i l l consider t h e g e n e r a l i z e d o r a l g e b r a i c GAUSS-JORDAN method. A(’):=

Let

A a n d d e f i n e r e c u r s i v e l y f o r k = 1,2,...,n

(8.43

for all i , j E N . T h i s r e c u r s i o n is well-defined, t i o n of (akk ( k - l ) ) * is well-defined. duce t h e set Ti:.’

if t h e computa-

F o r a d i s c u s s i o n we intro-

w h i c h c o n s i s t s i n all p a t h s P E P

ij

s u c h that

p d o e s n o t contain an i n t e r m e d i a t e v e r t e x from t h e s e t

{k+l,k+Z,

...,n ) .

For a c i r c u i t

PET'^) ii

with k < i this means

that p i s a non-null c i r c u i t t r a v e r s i n g i exactly once.

(8.44)

Proposition

(1

bO If A i s a b s o r p t i v e t h e n P kk

(2)

if A is m-regular a n d R is c o m m u t a t i v e t h e n

n

T R(k’ k is d e n s e in T k(k k

,

Akebmic Path Problems

(3)

135

if A is m-absorptive and R is commutative then

n T~~ (k)

i s dense in T (k)

kk

‘

Proof. Follows from the proof of ( 8 . 2 8 ) as in each case the constructed set H for p E T : o

(k) Tkk

.

satisfies H

This proposition is the essential technical tool in the proof of the next theorem.

( 8 . 4 5 ) Theorem Let (R,$,@)

be a semiring and let

A

be an n

x

n matrix over R.

Then A is stable with stability index r if one of the followihg conditions holds: (1)

A

is absorptive

(then r f n-11,

(2)

A

is m-absorptive and R commutative

(3)

A

is m-regular and R commutative

(then r (then r

1. n m + n

- 11,

5 nm I C I + n -

In each case A*

(4)

= A(n)

with respect to the recursive formula ( 8 . 4 3 )

for all k = 1 , Z ’ ,

and

. . . ,n.

Proof. We claim that (8.46)

for all i , j E N and all k E N U { O ) .

As T!o’

= { ( i , j ) } we see

13

that ( 8 . . 4 6 ) holds for k = 0. Now assume that ( 8 . 4 6 ) holds for k-1 with k

2

1 and for all i,j E N .

1).

136

Linew Algebmic Optimization

f i r s t we consider the case i = j = k.

A t

(1).

( 2 ) and

( 3 ) we f i n d

(k) Tkk f o r all s

1

( c f . 8.44) (k) Tkk

is d e n s e i n

r and t h e r e f o r e

In particular,

t h e e x i s t e n c e o f a(TL;))

h a s b e e n shown.

Now

c o n s i s t s i n t h e n u l l c i r c u i t and all c i r c u i t s of t h e

T

-

form p = P 1

u

Is]

'kk

I n e a c h of t h e c a s e s

=

P2

l , 2 , . ..,s, a n d S E N .

a* = 1 e x i s t s as

a s = o ( [ p l - pa

and

(k-1)

L e t a:= o(Tkk

a @ a'

@

E

-ps with c i r c u i t s p

- * - .

@

).

Tki-')

for

Then

...

- -.-

'p

o p S I

for u = 1 , 2 ,

(k-l) Tkk

. . . , sl)

(8.47) lead t o

a [rl = a [ r + l l T h u s we h a v e p r o v e d

( 5 ) f o r k and

Secondly w e consider

+

( i ,j )

(8.46) f o r i = j = k .

(k,k).

Then P E T . ( k ) 'Tij 11

v e r t e x k and h a s t h e form p = P I * p g - p 3 w i t h

p 2 E T k( k )

and p

3

E

T(k-l). kj

(8.43) t h a t

A n i n d u c t i v e a r g u m e n t s h o w s now t h a t

a

i j

for all i , j EN,

In particular

= 0fPij)

i.e.

contains

(k-1) Tik

,

Hence

which shows t o g e t h e r w i t h

a l l k = O,l,...,n.

'1

(k-1 f

A(")

= A*

.

(8.46)

is valid for k .

( 8 . 4 6 ) and

(5) hold for

Algebraic Path Problems

T h e o r e m (8.45) s h o w s t h a t t h e r e c u r s i o n

137

(8.43) c a n b e u s e d i f

some o f o u r s u f f i c i e n t c o n d i t i o n s f o r s t a b i l i t y o f A a r e fulfilled.

Another proof

i s d u e t o FAUCITANO a n d N I C A U D 1 1 9 7 5 1 a n d c a n b e

f o u n d i n GONDRAN a n d M I N O U X e xis ts then A

( n ) = A*.

[ 1 9 7 9 b ] . T h e y show t h a t i f A ( n )

I t i s claimed t h a t

(k-1) ) ( ak k

diagonal pivoting is incorporated i n recursion

*

exists i f

(8.43). Our

p r o o f shows t h a t p i v o t i n g is n o t n e c e s s a r y i f o n e o f t h e assumptions

(8.45.11,

case a proof

(8.45.2) o r

(8.45.3) h o l d s .

i s a l s o g i v e n by CARRg

"19711,

For t h e a b s o r p t i v e 119791).

R e c u r s i o n (8.43) c a n b e s i m p l i f i e d i n t h e f o l l o w i n g way.

I f a*

e x i s t s f o r a E R then a*

=

( a * @ a ) @ 1 = 1 IB ( a

i s s a t i s f i e d ( c f . 8.36 a n d 8.37).

o

a*)

Therefore, i n

(8.43) w e f i n d

f o r i * j = k

and s i m i l a r l y f o r j

If

and of

a*

*

i = k

e x i s t s f o r a E R t h e n a* @ a*

= a*

. Together

with

(8.43')

(8.43") t h i s l e a d s t o t h e f o l l o w i n g e f f i c i e n t f o r m u l a t i o n (8.43).

Linear Algebraic Optimization

138

(8.48) G A U S S - J O R D A N E L I M I N A T I O N Step 1.

Define the original matrix A; let k : = 1.

Step 2 .

akk:= (akk)

Step 3 .

a.

:=

a

:=

*.

ik

kj

a ik

@

a

@ a

kk

akk kj

Step 4 .

a. := a

Step 5 .

If k = n stop.

ij

(aik

@

ij

Otherwise k : = k

@

a

kj

)

+ 1 and

for all i

$

k;

for all j

9

k.

for all i,j

$

k.

go to step 2.

The method requires O(n) iterations. Each iteration needs O(n @-compositions, O(n (akk)*,

2

2

)

@-compositions and the computation of

)

i.e. (akk)* = 1

$

akk

@

...

(akklr

.

Here we need O(r) $-compositions and O(r) @-compositions. Again if

A

is only m-regular (m

$

0 ) then this does not yield

a polynomial bound. If A is m-absorptive then r = O(n). If A is absorptive then (akk)* = 1 and the method can be simplified. We may w.1.0.g.

assume that the main diagonal of A con-

tains only unities. Then step 2 and step 3 can be eliminated from the algorithm. If A is m-absorptive then the method requires O(n 3 ) 8-compositions and O(n

3

)

@-compositions. Therefore, in general, the

GAUSS-JORDAN-method is superior to the JACOBI-method. Nevertheless, for special structured networks (as for example in treenetworks, i.e.

iCl = 0 ) the JACOBI-method may turn out to be

better (cf. SHIER [ 1 9 7 3 ] ) .

139

Algebraic Path Roblems

Specific forms of this method have been developed by many authors. ROY [19591 and WARSHALL [1962] applied it to the transitive-closure problem in graphs, FLOYD [19621 to the classical shortest path problem, ROBERT and FERLAND [1968], CARRE 119711, BRUCKER ([19721,

[1974]),

BACKHOUSE and CARRE/

[19751, and GONDRAN [1975a] gave extensions to certain algebraic structures. MULLER-MERBACH 119691 used it for the inversion of Gozinto-matrices, MINIEKA and SHIER [1973], MINIEKA [1974] and SHIER 119761 applied it to the k-shortest path problem. MARTELLI

“19741,

[1976])

enumerates all mini-

mal cut sets by means of this method. Other solution methods, as the GAUSS-SEIDEL iteration method o r the ESCALATOR elimination method have been generalized also for certain semirings. The interested reader is referred to GONDRAN and MINOUX [1979b] and CARRE/ [1979]. We remind that stability of a matrix A implies the existence of the closure matrix A*. Theorem (8.45) shows that certain types o f stability imply the validity of the GAUSS-JORDAN method f o r the determination of A*. In general, it is not known whether the GAUSS-JORDAN method is valid for all stable matrices. Nevertheless, no counterexample is known and we conjecture the validity of the GAUSS-JORDAN method for all stable matrices. Instead of assuming that the matrix A satisfies certain conditions we may impose certain conditions on the underlying semiring. We remind that P

ij

is always a countable set. Thus

A* i s well defined for all matrices A over R if we assume that infinite sums of the form

I40

Linear Algebraic Optimization

are well defined in R for a . E R , i EN. We denote such a series by Z I a i for a countable set I. In arbitrary semirings Z

is well defined only if I is a finite s e t . Besides E

I

I

a

i

a . E R we 1

assume distributivity, i.e. b B (1 a , ) = ZI(b I 1 (Za I i

@

ai) ,

,

B b = Z (a. @ b ) I 1

for a l l b c R and associativity, i.e. Z

I

a

ZJ(ZI a . )

=

i

j

for all (countable) partitions

(I

j'

j E J ) of I. Then R is

called a ( c o u n t a b l y l c o m p l e t e semiring.

In a complete semiring the GAUSS-JORDAN method is well defined for all matrices

A

over R. Its validity is shown in the same

way as in the proof of theorem ( 8 . 4 5 ) by verifying

(8.46).

In

a complete semiring the proof becomes much easier since the occuring countably infinite sums 2

a * = l t 8 a @ a

$...=

i

are well defined.

( 8 . 4 9 ) Theorem Let ( R , C I , @ ) over R. Then

be a complete semiring and let A*

Theorem ( 8 . 4 9 )

A

be an n x n matrix

is well defined and A* = A ( n )

.

is proved by AHO, HOPCROET and ULLMAN [ 1 9 7 4 ]

f o r idempotent complete semirings. FLETCHER [I9801 remarks

141

Algebraic Path Problems

that idempotency i s not necessary if the GAUSS-JORDAN method is formulated in the right way (equivalent to (8.43)). We remind that for stable matrices A* is a solution of the matrix equation (8.50)

X = I e ( A Q X ) In a complete semiring

(cf. 8.37).

A*

=

Ai satisfies

z+ (8.501, too.

Next we discuss a weaker condition which does not necessarily imply the existence of

A*;

on the other side a solution of

(8.50) will exist and can easily be determined using a modifi-

cation of the GAUSS-JORDAN method ( 8 . 4 3 ) . We assume the existence of a unary c l o s u r e operation a

+

a

on the semiring R such that = 1

(8.51)

(a

a)

for all a E R . Then R i s called a c l o s e d S e m i r i n g .

A

complete

semiring is always a closed semiring with respect to the i a* = 1 a On the other z+ side, if R is a closed semiring and if R is m-absorptive for

closure operation defined by

a:=

.

-

some m E m , then a = a* for all a E R . In general, the closure operation does not necessarily coincide with the '*'-operation and is not uniquely determined. If a closure operation is defined only on a subset R' of R then LEAMANN [ 1 9 7 7 ] proposes to adjoin an element m a R , to add the definitions

and to extend the closure operation by

-

a:=

00

Linear Algebraic Opiinrizariun

142

for all a E ( R U (-1) a

Then R

X R ' .

u

I-1

satisfies all axioms of

closed semiring with the exception of

(a.52)

a B O = O B a = O

which is invalid for a =

m.

for all a E R , a c t o s u r e of R .

Ke call R U { - )

in this way an arbitrary semiring can be closed if we drop the assumption that R contains a zero. In particular, this approach is possible for the closure operation a + a* =

a

i

which is

z+ w e l l defined only on a subset of the underlying semiring.

Therefore we will avoid to use (8.52) in solving the matrix enuation

in closed semirings.

(8.50)

The following variant of the GAUSS-JORDAN method defined in a closed semiring. Let B ( O ) := sively for k = 1,2,

A

( 8 . 4 3 ) is well

and define recur-

..., n:

for all i , j E N . T h e following theorem is implicitly proved in LEHMANN [1977]. Its simple direct proof is drawn from MAHR [ 1980b].

(8.54) Theorem Let

(R,(B,@)

be a closed semiring and let

over R . Then I X = I

(B

(A

(B

A

be an n X n matrix

B (n) is a solution of the matrix equation

B X).

Proof. At first we show

for k = O , l , . . . , n. The case k = 0 is trivial. Let i,j E N and

Akebraic Path Problems

1

5

k

C

143

n. We assume that (1) is satisfied for k - 1 and we

denote B ( k - l ) and B(k) by B and B'. Since 1 =

(bkk 8

@

Ekk)

-

bkk we find b' = b kj kj

@

(bkk8gkk8b kj

=

1;

kk

B

b

kj

=:B

...,

Now b' = b @ (bvk 8 8 ) for all v = 1,2, k-1. Therefore vj vj the right-hand-side of equation (1) is equal to

Applying the inductive assumption to the terms in brackets we find equality with

by (8.53) this term i s equal to b!

,

11

NOW let B:= B (n), 6ij

= 0

c

C:=

I

@

.

B , and define 6

:=

1 if i = j and

ij

otherwise. Then

ij

= 6

ij

@ b

ij n

= 6 . . @ a . 11

lj

@

n X(a 8 b ) iv vj v=l

for all i,j E N . Therefore C is a solution of the matrix equation

X

=

I a ( A B X). rn

We remark that theorem (8.54) is proved without using (8.52). For k = n the equations ( 1 ) In the proof show that B(n) is a solution of the matrix equation (8.55)

X = A @ (A 0 X ) .

Linear Algebraic Oprimizatioti

144

In a closed semiring R satisfying the additional equation

for all a E R w e may use the G A U S S - J O R D A N method

for

(8.43)

the determination of I @ B(n) if we replace the ‘*‘-operation in ( 8 . 4 3 )

by the closure operation in R. Let

C ( O ) : = A

and

define recursively f o r k = l , 2 , . ..,n:

for all i,j E N .

(8.58) C o r o l l a r y Let (R,@,@) b e an n

x

b e a closed semiring satisfying ( 8 . 5 6 )

n matrix o v e r R. Then C(n) = I

of the matrix equation Proof.

A

X

= I @

comparison of ( 8 . 5 3 )

(A @

A

B (n) is a solution

@

X).

and ( 8 . 5 7 )

for all i ,j E N and for all k = 0,1,.

and let

shows that

. . ,n.

Therefore C(n) = I

We remark that Corollary (8.58) is proved without using

@

(8.52).

Therefore we may consider the ‘*‘-operation a s a particular well-defined example for a closure operation. Thus A always a solution of the matrix equation

X

=

I

@

(A

( a closure of) the semiring R. On the other side A *

exist, in general

(cf. 8 . 6 5 ) .

is @

B (n)

X ) in

does not

.

145

Algebraic Path Problems

I n t h e remaining p a r t of t h i s c h a p t e r w e d i s c u s s s e v e r a l e x a m p l e s f o r p a t h p r o b l e m s . The u n d e r l y i n g g r a p h i s a l w a y s d e n o t e d by G = ( N , E ) w i t h w e i g h t s a

i j

c h o s e n from t h e r e s p e c -

t i v e semirings.

(8.59)

C o n n e c t i v i t y i n graphs

I f p i s a p a t h i n G f r o m i and j t h e n t h e s e v e r t i c e s a r e c a l l e d c o n n e c t e d . The e x i s t e n c e o f s u c h p a t h s i n

(undirected)

g r a p h s h a s f i r s t b e e n d i s c u s s e d by LUNTS [ 1 9 5 0 ] a n d SHIMBEL [ 1 9 5 1 ] w i t h r e s p e c t t o c e r t a i n app i c a t i o n s . problem t o t h e d e t e r m i n a t i o n of t h e .-Ith

They r e d u c e t h e

power o f t h e

a d j a c e n c y m a t r i x A d e f i n e d by

a

ij

:=

{

1

if

(i,j) E E ,

0

otherwise

f o r a l l i,j E N o v e r t h e s e m i r i n g

({O,l),max,min).

T h i s is a

c o m m u t a t i v e , a b s o r p t i v e s e m i r i n g w i t h z e r o 1 and u n i t y 0. A b s o r p t i v e s e m i r i n g s h a v e b e e n d i s c u s s e d by Y O E L I c a l l e d them Q - s e m i r i n g s .

In p a r t i c u l a r ,

1 1 9 6 1 1 . He

YOELI proved t h a t

in

absorptive semirings

-

An 1 = A*

(8.60)

f o r a l l matrices A with A equivalent t o a

ii

(B

I = A.

The c o n d i t i o n A

= 1 ( u n i t y ) f o r all i E N .

n- 1

(B

I = A 'is

T h i s shows t h a t

9 QI. In the ij n- 1 c a s e of u n d i r e c t e d g r a p h s A and A a r e symmetric m a t r i c e s .

i n o u r above example A

i j

= 1 i f

and o n l y i f P

The g r a p h w i t h a d j a c e n c y m a t r i x A * = A n - '

sitive c l o s u r e o f problem.

ROY

G,

is c a l l e d t h e t r a n -

and w e s p e a k o f t h e t r a n s i t i v e c l o s u r e

119591 a n d WARSHALL 119621 d e v e l o p e d t h e

Linear Algebraic Optimizatioh

14tl

GAUSS-JORDAN method

(8.48) for the computation o f An-1. Methods

based on STRASSEN's fast matrix multiplication are proposed by ARLAZAROV et al. [1970], FURMAN [1970], FISCHER and MEYER [I9711 arid MUNRO [1971]. The average behavior of such methods i s discussed by BLONIARZ, FISCHER and MEYER [1976] and by SCHNORR [19781.

(8.61)

Shortest paths

The shortest path problem has at first been investigated by BOR6VKA 119261. SHIMBEL 119541 reduced this problem to the computation of the n-lth power of the real arc-value matrix A over the semiring ( W + U (-l,min,+). absorptive semiring with zero

a

ii

m

This is a commutative

and unity 0. SHIMBEL assumed

= 0 for all i E N . Therefore A @ I = I and

again that A* = A

n- 1

.

If negative weights a

ij

(8.60) implies are involved

then we have to assume that the network i s absorptive, i.e. G does not contain a non-null elementary circuit p with weight w(p) < 0 (with respect to the usual order relation of the extended real numbers). Then the semiring (IR U {m},min,+)

is the weight of a shortest path is considered. Hence Ak ij from i to j of length k and A i k l is the weight of a shortest ij path from i to j o f length not greater than k. In particular A* = A [n-11 (cf. 8 . 3 3 ) . Based on the paper of WARSHALL 119621 FLOYD [1962] proposed the application of the GAUSS-JORDANmethod for the determination o f A*. In particular, if all weiqhts are nonnegative then algorithm ( 8 . 4 ) (cf. DANTZIG [1960], DIJKSTRA [1959]).

can be used

Several other Solution

methods can be found in the standard literature (cf. LAWLER

Algebraic Path Problems

147

[19761). Recent developments can be found in WAGNER [19761, FREDMAN [1976l,,JOHNSON 119771, BLONIARZ [1980] and HANSEN [1980]. An excellent bibliography is given by DEO and PANG 119801.

If the weights a

are chosen in R:=

ij

IRk

u

{ (-,-,,,

then

the weight of a lexicographically shortest path may be determined. Let a

(B

b:= lex min(a,b)

for a,bEIRk with respect to the lexicographical order relation

6 of real vectors. Then (R,@,+) is an idempotent, commutative A* is ij the weight of a lexicographically shortest path p EPij. A* semiring with zero

(-,-I...I-)

and unity (O,O,...,O).

exists if and only if the network is absorptive, i.e.

if G

contains no circuit p with w(p) 4 (O,O,...,O). Such a problem was first considered by BRUCKER ([1972],

119741).

(8.62) M o s t r e l i a b l e p a t h s If the weights a

ij

are chosen from the semiring ([0,11,max,=)

with zero 0, unity 1 and usual multiplication of real numbers then the weight w(p) = a il,l=ai

. ...

2 2

8

of a path p = ( ( i ~ , j ~ ) , ( i 2 , j ~ ) , . . . , ( i SS,'1j) may be interpreted as the reliability of the path p . This semiring is again commutative and absorptive. A : j

is the weight of the most reliable

path from node i to node j. This problem was first considered by KALABA 119601.

148

(8.63)

Linear Algebraic Optimization

Maximum c a p a c i t y p a t h s

If the weights a .

are chosen from the semiring (lR+U{w),max,min)

lj

with zero

and unity 0 then the weight w ( p ) = min{a

.a. ,...,a. I i l j l 12j2 lsjs

o f a path P = ((il,jl),(i2,j2), . . . , (is,js)) may be interpreted

as the c a p a c i t y of the path p. This semiring is again commutative and absorptive. A* is the weight of a maximum capacity ij path from vertex i to vertex j. This problem was first considered by HU [1961].

(8.64)

A b s o r p t i v e semirings and a b s o r p t i v e networks

YOELI [1961] discussed absorptive semirings (Q-semirings) and the relevance of A

n- 1

and A*. The application of the GAUSS-

JORDAN-method in absorptive semirings was proposed independent-

ly by TOMESCU [I9681 and ROBERT and FERLAND 119681. CARRE 119711 considered such algorithms for absorptive network5 over idempotent semirings under the assumption that the composition

0

is cancellative. In particular, distributive lattices are discussed in BACKHOUSE and CARRd [19751. F o r Boolean lattices (Boolean algebras) we refer to HAMMER and RUDEANU [19681.

(8.65)

Inversion o f matrices over f i e l d s

A field (F,+,.) with zero 0 and unity 1 is, in particular, a semiring. If A is a stable matrix over F with stability index

r then

A

Irl

=

matrix f o r all

A [r+ll = A [rl s

+

Ar+'

implies that AS is the zero-

2 r+l. Let p be a circuit of length l(p). Then

Algebraic Path Problems

149

2 r+l for some m E N . Therefore w(p)

m.1 (p)

cuits of

= 0.

Thus all cir-

G have weight zero. If all circuits in G have weight

5 n-1. (8.36) and

zero then A is stable with stability index r (8.37) imply

In particular, if A is stable then I - A is a regular matrix over F. The algorithms for the computation of A * given in this section reduce to the conventional linear algebra methods for the inversion of I

Applications for networks without cir-

-A.

cuits have been given by MULLER-MERBACH 119701 (so-called Gozinto-graphs). For the closure IR U { = )

of the field of real numbers LEHMANN

[1977] introduces the closure operation a

-

otherwise

that

a

= 1

+

(a:)

= 1

+

(;a)

= a* = 1 + a + a 2 + . . .

The closure operation a if la1

2 1.

.

for all a EIR U (-1.

We remark

for all aEIR with la1 < 1 .

a* may be extended on I R U

+

n

a*:=

defined by

if aE{l,=),

m

a:=

Clearly

a

+

Since 1

+ a + a' +

...

{mf

by

is an oscillating di-

vergent series-for a e - 1 the algebraic path problem does not have a solution in IR U { - ) circuit p = (el,e2,...,e

P

if the underlying graph contains a )

with weight

For the different closure operations algorithm

(8.57) leads

to different matrices C(n) denoted by A- and A, which both solve the matrix equation X = I

+ (AX) in IR

U

I-) (cf.

Linear Algebmic Optimizatiun

150

C o r o l l a r y 8 . 5 8 ) . A, A_.

I f A-

w i l l c o n t a i n more i n f i n i t e e l e m e n t s t h a n

does n o t c o n t a i n an i n f i n i t e element t h e n A _ = ( I - A )

The same r e s u l t h o l d s f o r A * .

we find A-l

an i n f i n i t e e l e m e n t . (I -PA)-

For a given nonsingular matrix A

provided t h a t

(I -A)-

=

(I

- A) -

does not contain

I f P i s a permutation m a t r i x such t h a t

does n o t c o n t a i n an i n f i n i t e element t h e n A

U s i n g w e l l known p i v o t i n g r u l e s A

-1

-1

= (I-PA)-P.

can always be determined

i n t h i s way. [I9751 c o n s i d e r e d t h e n u m b e r o f p a t h s f r o m n o d e i t o

GONDRAN

n o d e j. F o r t h e a d j a c e n c y m a t r i x A w i t h r e s p e c t t o t h e f i e l d k o f r e a l numbers w e f i n d t h a t A . . i s t h e number o f p a t h s f r o m 13

i to j

of

of length k , A[k1 i j

i s t h e number o f p a t h s from i t o j

length not g r e a t e r than k and,

circuit, to j.

t h e n A*

i j

= A In-']

i j

I t i s always assumed

(8.66)

i f G does not contain a

i s t h e number o f a l l p a t h s f r o m i that A

ii

= 0 for

all i E N .

Regul a r a1 g e b r a

BACKHOUSE a n d CARRg c o n s i d e r e d c l o s e d

with a closure operation a +

-

a =

a.

idempotent semirings

satisfying

(1 @ a)

(8.67)

f o r a l l a,B,J,

E R and

f o r a l l a,B,$ E R such t h a t

for all X E R .

-1

They d i s c u s s o n l y e x a m p l e s i n which

coincides

.

Algebraic Path Ptoblems

with the usual closure operation a + a * ,

= 1

a*

a

(B

(B

a'

(B

151

i.e.

... .

The standard example of a regular algebra is the following.

1 b e a f i n i t e n o n e m p t y set. V i s c a l l e d a n

L e t V = {v1,v2,...,v

a l p h a b e t and its elements are called l e t t e r s .

A

word o v e r v i s

a f i n i t e s t r i n g o f z e r o or m o r e l e t t e r s f r o m V . , I n p a r t i c u l a r the empty string

is denoted by

c.

of t w o t a n g u a g e s t = t t

, . ..

i s c a l l e d e m p t y w o r d . T h e s e t of a l l words

E

a , B C_

ii

i s a U 5 . For w o r d s w = w l w2...w

r

and

t s we d e f i n e w o

w o

l a n g u a g e i s a n y s u b s e t o f ?. T h e sum a + 5

A

t:= w l w2...w

t t ,...ts

i

t is c a l l e d t h e c o n c a t e n a t i o n o f w a n d t. T h e c o m p l e x c o n c a -

Fenation of two languages a , B

(V,+,O)

C, 0 is d e n o t e d b y a

5 . Then

i s an i d e m p o t e n t s e m i r i n g w i t h zero # and u n i t y

{E}.

The closure a*

-

~ € B1 a

B a'

B

...

i s w e l l - d e f i n e d ; i n p a r t i c u l a r $* = ( € 1 . W e d e n o t e s i n g l e - l e t t e r l a n g u a g e s {a) s h o r t l y b y a. T h e n a r e g u l a r e x p r e c l e i o n o v e r V is any well-formed formula obtained from t h e elements V U operators

+ , a

and

*,

and the parentheses

# * + (v,

D

(v, + Vl)*'

(

and

).

{ # I t,h e

For example

v. 3 ) *

is a r e g u l a r e x p r e s s i o n . E a c h r e g u l a r e x p r e s s i o n d e n o t e s a l a n g u a g e o v e r V w h i c h is c a l l e d a r e g u l a r l a n g u a g e . T h e s e t of

a l l r e g u l a r l a n g u a g e s is d e n o t e d b y

-

s(v).

(S(V)

,+,a,*)

is a

regular algebra ( a = a * ) . The axiomatic system defining regular

Linear Algebmic Optimization

152

a l g e b r a s w a s p u t f o r w a r d f o r r e g u l a r l a n g u a g e s by SALOMAA [1969].

I n a p r o o f o f a t h e o r e m on l a n g u a g e s i n a u t o m a t a

t h e o r y K L E E N E [ 1 9 5 6 ] u s e s a GAUSS-JORDAN m e t h o d for S ( V ) . I n H c N A U G H T O N a n d YAMADA

[ 1 9 6 0 ] t h e same m e t h o d i s d e s c r i b e d

e x p l i c i t l y . For t h e t h e o r e t i c a l background i n automata t h e o r y we r e f e r t o E I L E N B E R G

([19741,

p.

175).

A n o t h e r e x a m p l e o f a r e g u l a r a l g e b r a is a n a b s o r p t i v e s e m i r i n g (R,$,B)

(cf. 8.64).

-

Then a = a* = 1

a = 1 f o r all a E R .

IB

T o g e t h e r w i t h i d e m p o t e n c y t h i s shows

(8.67). Distributive

l a t t i c e s are included as a particular case.

BACKROUSE a n d CARRE [ 1 9 7 5 ] d i s c u s s s e v e r a l l i n e a r a l g e b r a methods f o r s o l v i n g m a t r i x e q u a t i o n s i n r e g u l a r a l g e b r a . I n p a r t i c u l a r , theorem

(8.54)

t h e GAUSS-JORDAN m e t h o d

shows t h e v a l i d i t y o f a v a r i a n t o f

(cf. 8.53).

I f the considered closure

operation coincides with t h e usual *-operation ( 8 . 5 8 ) shows t h e v a l i d i t y o f t h e G A U S S - J O R D A N

(8.69)

then corollary method

(8.57).

Path and cut set enumeration

I n o r d e r t o d e t e r m i n e t h e set o f a l l p a t h s f r o m n o d e i t o n o d e j w e c o n s i d e r t h e r e g u l a r a l g e b r a S ( E ) where E d e n o t e s t h e s e t of

a l l a r c s of t h e u n d e r l y i n g g r a p h G.

Then t h e w e i g h t a

is i j

g i v e n by if

(i.j)

EE,

(8.70)

otherwise and t h e w e i g h t of a p a t h p = ( e l , e 2 ,

...,e s )

, is

Algebraic Path Roblem

w(p) = e l o e20

...

o

e

S

153

.

(Here we denote single-letter languages by ei rather than {el}.) For k E l N we get

for all i,j E N . An actual computation of A* is not possible in the general case, as the cardinality of a* may be infinite. ij If we are only interested in all elementary paths then a suitable modification of the underlying regular algebra is the following. A word w is called an a b b r e v i a t i o n of a word t if w can be obtained from t by deletion of some of the letters of t. For example, METAL is an abbreviation of MATHEMATICAL, or if an elementary path p is obtained from a nonelementary path p' by deletion of the elementary circuits in p' then w(p) is an abbreviation of w(p'

).

For a E S(E) let r ( a ) denote the language

containing all words of a for which no abbreviation exists in a.

Let N

s(E):=

{aES(E)I

a =

r(a)l

and define a @ E:=

r ( a +B )

a dD 8:-

r ( a o 8)

for all a , B Ez(E). with zero we find

0

Then ( z ( E ) , @ , @ I is an absorptive semiring

and unity

{E}.

Further, for A as defined by ( 8 . 7 0 1 , .

154

Linear Algebmic Optimization

for a l l k E N and

A*

= A In- 1

1

,

i.e.

a*

11

=

a In-'] ij

contains the

w e i g h t s o f a l l e l e m e n t a r y p a t h s f r o m i t o j. F o r t h e c o m p u t a t i o n of A * KAUFMANN and MALGRANGE I19631 proposed the sequence M,M w i t h M:=

2

,M

4 I . . .

I b A. T h e n M Z k = A* f o r 2 k

2

n - 1 shows that this

m e t h o d , c a l l e d Latin m u l t i p l i c a t i o n , n e e d s O ( l o g n ) m a t r i x 2

N

m u l t i p l i c a t i o n s o v e r t h e s e m i r i n g S ( E ) . M U R C H L A N D 119651 and B E N Z A K E N 119681 a p p l i e d t h e G A U S S - J O R D A N m e t h o d f o r a s o l u t i o n

of t h i s p r o b l e m .

Improvements can b e found in BACKHOUSE and

C A R R E 119751 a n d F R A T T A a n d M O N T A N A R I 119751.

A related problem is the enumeration o f all minimal cut sets o f t h e u n d e r l y i n g g r a p h G = (N,E). L e t

(S,T) denote a partition

of N. T h e n c = (S x T ) flE i s c a l l e d a cut s e t o f G. I f t h e a r c s of a c u t s e t c a r e r e m o v e d f r o m G t h e n t h e r e e x i s t s n o p a t h in G f r o m a n y i E S t o a n y j E T . T h e r e f o r e w e s a y t h a t c

separates i from j. I f a c u t s e t c d o e s n o t c o n t a i n a p r o p e r s u b s e t w h i c h i s a c u t s e t t h e n c is c a l l e d a minimal cut set. In o r d e r t o d e t e r m i n e t h e s e t of a l l m i n i m a l c u t s e t s s e p a r a t i n g I E N f r o m j E N t h e f o l l o w i n g a l g e b r a is i n t r o d u c e d by M A R T E L L I ([19741, [1976]). L e t P ( E ) d e n o t e t h e s e t o f a l l s u b s e t s of E. F o r a C_ P ( E ) l e t

r(a) denote the subset o f a containing all sets in a f o r which n o p r o p e r s u b s e t e x i s t s i n a.

Let

155

Algebraic Path Problems

and define

a @ B:=

r{aUbl

8 B:=

r(a U B)

Q

aEa, bEa)

f o r a l l a , B E C(E). T h e n ( C ( E ),@,a) i s an absorptive semiring with zero

and unity

a,,:=

0.

1 E C ( E ) where a i s defined by ij ij ( 8 . 6 0 ) . T h e weight w(p) of a p a t h p = ( e l , e 2 e ) with

We a s s i g n weights

{a

,...,

r e s p e c t t o t h e semiring C ( E ) is t h e set of its arcs, 1.e. w ( p ) = {e,l

k = 1,2,.

. . ,s}.

( 8 . 3 3 ) we find A* = A

Du e t o th e orem

In-1 1

. Each

element c E a

* i j

h a s t h e f o rm

w h e r e e ( p ) d e n o t e s an arbitrary arc of p. T h u s t h e r e exists n o p a t h from i t o j in t h e g r a p h G

C

=

(N,E'c).

Let S d e n o t e

t h e s e t of a l l vertices k s u c h t h a t t h e r e exists a p a t h from

i to j in G c

,

and let T:= . S ' N

Then c = ( S

x

T ) n E , 1.e.

c

is a cut set separating i and j in. G. T h u s a* is t h e set o f ij

a l l m i n i m a l cut sets separating i and j. T h e o r e m (8.45) shows that

R*

can be computed using t h e GAUSS-

J O R D A N method. T h i s approach i s discussed in d e t a i l by MARTELLI [ 1 9 7 6 ] . An arc

(p,v)

is called b a e i c with respect t o i , j E E i f { ( p , u ) }

is a m i n i m a l c u t separating i f r o m j.

CARRE 1 1 9 7 9 1

proposes

t h e f o l l o w ing semiring for t h e determination o f all basic a r c s in G. L e t P ( E ) d e n o t e t h e set o f all s u b s e t s o f E, and let UbE.

W e d efine

156

Linear Algebmic Optimization

f o r a l l a , B E P ( E ) a n d a d j o i n {a} a s a z e r o . L e t R = P ( E ) U {o}.

i s an absorptive semiring with zero ( w )

(R,@,@)

Then

and with

6. W e a s s i g n w e i g h t s

unity

A g a i n t h e w e i g h t o f a p a t h i s t h e s e t o f i t s arcs. D u e t o t h e o r e m (8.33)

-*

we find A

basic with respect to i and j provided

-*. a

ij

find

*: i j c o n t a i n s a l l P + 0; o t h e r w i s e ij

= A[n-ll. Then

arcs

= {a). S i n c e w e a s s u m e t h e e x i s t e n c e o f n u l l c i r c u i t s we

z:i

=

6 for all i E N .

(8.71)

Schedule algebra

GIFFLER

([1963]. [ 1 9 6 8 ] ) c o n s i d e r e d t h e d e t e r m i n a t i o n o f a l l

path weights in a network with nonnegative integer weights. H i s a p p r o a c h h a s b e e n e x t e n d e d by W O N G S E E L A S H O T E

Let ( H , + , I )

[1976].

be a totally-ordered commutative group with neutral

e l e m e n t 0. In p a r t i c u l a r , w e a r e i n t e r e s t e d i n t h e a d d i t i v e g r o u p o f r e a l numbers.

n

H =

@.F o r

Let

fi

=

x E H } be a copy of H with

convenience we define

we consider z +

x

= x f o r all

a s a u n i t a r y o p e r a t i o n o n Z = €3

;€HI

i.e.

u i.T h e n

if z E H .

I l z l l := if Z E H , for all z E Z . bY

We continue the internal composition

+

of H on Z

157

Algebraic Path hoblems

-

x x

-

+

y:= x

-

+

-

y:= x

+

y

+

y:=

I

x+y ,

for all x , y E H and for all ;,?Eli.

Then (Z,+) is a commutative

group with neutral element 0. The inverse elements are denoted by -z for z E 2. A

function a: Z

Z+ is called countable if

-B

is countable and a is called well-ordered if

is well-ordered (each subset of H(A) contains a minimum). Let

U denote the set of a l l countable and well-ordered functions a: z

-B

Z+.On U we define two internal compositions by

for a , b E U and z E Z and by (8.75)

E

(a o b) (z): =

a(x)b(y)

x+y=z for a , b E U and

~

€

2

The . right-hand sides of ( 8 . 7 4 )

and (8.75)

are defined with respect to the usual addition and multiplication of nonnegative integers. It is easy to show that a + b is a well-defined element of U. Now let a , b E U and ~ € 2 .Then we claim that S = { ( x , y ) E Z x + y = z, a(x)b(y) * O ) is a finite set. Suppose infinite. Then

{XI

S

2

I

is countably

(x,y) E S) contains an infinite sequence

IIxlII < IIx211 <

... <

lIXklI

<

... .

Let yk denote a corresponding infinite sequence in {yl with x k + y k = z for k = 1.2,

... .

Then

( x l y )E S )

158

Linear Algebraic Optimization

contrary to the assumption that b is well-ordered. Therefore S is finite which shows that (a. b)

(2)

E Z+. It is again easy to

show that a o b is countable and well-ordered. Therefore aobEU. Now we introduce a mapping r: U

for a E U and z E Z . r(a) (z) = 0

or

+

U defined by

This means that a function r(a) has either

r(a)

(i)

= 0 for each z

E

Z.

AS

in (8.69) we

consider only a subset V of the functions in U defined by

(8.77)

{ a E U I a = r(a)I.

v:=

Further we introduce internal compositions on V in the same manner as in (8.69) by

+ b)

a

@

b:= r(a

a

@

b:= r ( a o b)

f o r all a,b E V .

(V,@,@)

is a commutative ring which was dis-

cussed by GIFFLER in the particular case mentioned above. Verification of the ring axiomatics is pure routine. We only remark that the zero function 0

( O ( z ) a 0)

is the zero of V

and that the function 1 defined by i f z = O , l ( z ) :=

otherwise, is the unity of V. Further the additive inverse (-a) of a E V

-

satisfies (-a) = 1 8 a with

-

i f z = O ,

l ( z ) :=

otherwise. WONGSEELASHOTE

[1976]

gave the following nice result:

Algebraic Path Problems

159

(8.78) P r o p o s i t i o n i s an i n t e g r a l domain.

(V,@,@)

I t r e m a i n s t o s h o w t h a t a @ b = 0 i m p l i e s a = 0 o r b = 0.

Proof.

*

Suppose a , b

8 . 7 3 ) . We c l a i m t h a t

( c f . 8.72,

= x

Suppose

0 and l e t a E Z ( a ) ,

+

y

and a ( x ) b ( y )

/la11 + I I B I I = l l x l l

+ llyll

together with a ( x ) b ( y )

*

.

B E Z(b) such t h a t

(i) =

(a 0 b)

*

0.

= x

From

0 for

+y

The d e f i n i t i o n o f

0 implies

IIall 5

we find

Ilall

llxll

z = a + B.

IIBII

and

, IIBII 5

llyll

Therefore

IIBII

IIall = IlxlI,

=

IIyII.

But t h e n a ( x ) b ( y ) 9 0 i m p l i e s a = x and B = y c o n t r a r y t o a + B

z

=

* z-

Proposition field

= x+y.

( 8 . 7 8 ) shows t h a t

( F , @ , @ J ) (cf. 5 . 5 )

(V,@,@)

c a n b e embedded i n t o a

of quotients a/b with a , b E V ,

An e l e m e n t a o f V is i d e n t i f i e d w i t h a / l

b

*

0.

i n F.

Now w e c o n s i d e r t h e a p p l i c a t i o n t o t h e p r o b l e m o f f i n d i n g t h e w e i g h t s W(p) o f a l l p a t h s p i n a g i v e n n e t w o r k w i t h w e i g h t s i n H.

Let a

ij

denote t h e weight of

W i t h r e s p e c t t o t h e f i e l d F l e t now

(i,]) E E . denote the function

ij

d e f i n e d by i f z = a

N

(8.79)

ai,(z)

=

ij

otherwise

for all

(i,j) E E .

Let

2i

j

=

0 if

'

,

( i , j ) d E . Then t h e w e i g h t o f

.

Linear Algebraic Optimization

160

...,em

a path p = (el,e2, N

u

w(p) = a

IB

is given by

Z

el

IB

...

.

N

a a m

2

Therefore we get 1

N

W(P) ( 2 )

if z = w(p)

,

otherwise

.

=

With respect to the field F the entries

:Ik1ij

of

;Ik1

are func-

tions given by

-[kl a ( 2 ) is the number of paths pEP'k' with weight z . ij ij In the field F we know that A is stable if and only if G(p) 1 0 i.e.

for all circuits p in the network. This means that the network contains no circuit. Nevertheless the matrix

*;

may exist in a

network with circuits. I n order to characterize the existence of A * we consider the existence of a* for a E V . Here we assume that H is Archimedean. From ( 4 . 1 7 1

we know that in this case

H can be embedded into the additive group o f real numbers. The following result is due to WONGSEELASHOTE 119761.

(8.80) P r o p o s i t i o n Let H be the additive group of real numbers and let a E V . If a = min H(a) > 0 then a* exists i n V.

Proof. a = min H(a) implies k - a = min H(a

f o r all k E N . Let k E N . Then

k a (z) = 0

k

Alpebraic Path Problems

for a l l z

E

2 =

IIzII

for all

with

llzll C k-a.

Thus a * ( z ) E Z

C kma.

+

Therefore,

a*(z) = a r k ](2)

f o r a l l ~ € 2 a. * i s c o u n -

i s t h e u n i o n o f a c o u n t a b l e number o f c o u n t a b l e

table as Z(a*) sets.

IR U %

161

Suppose t h e e x i s t e n c e o f S

5

H(a*) which c o n t a i n s an i n -

f i n i t e s t r i c t l y d e c r e a s i n g sequence 8, Then B s E H ( a l k l ( z ) ) f o r s = 1 , 2 , . . . t o the fact t h a t aIkl

.

> 8, > . . .

.

L e t k*a

> 8,.

This i s a contradiction

i s well-ordered.

N

T h e e x i s t e n c e o f A*

c a n now b e shown u s i n g t h e GAUSS-JORDAN

m e t h o d f o r a c o n s t r u c t i v e proof. The n e x t t h e o r e m i s t a k e n f r o m WONGSEELASHOTE 1 1 9 7 6 1 .

(8.81 ) T h e o r e m L e t H be t h e a d d i t i v e group o f r e a l numbers and l e t f i n e d by

(8.79)

a l l non-null

w i t h r e s p e c t t o a network G.

I f w ( ~ )> 0 f o r

elementary c i r c u i t s i n G then N

= (I

A*

-

N

A)

-1 N

N

exists,

be de-

1 . e . A * is t h e i n v e r s e o f t h e m a t r i x I - A

i n the matrix

r i n g o f a l l n x n m a t r i c e s o v e r t h e f i e l d F. N

Proof.

L e t a b e d e f i n e d by

( 8 . 2 ) - w i t h r e s p e c t t o V and l e t . C

denote t h e s e t of a l l non-null w(p) > 0 f o r a l l p

E

e l e m e n t a r y c i r c u i t s . Then

C 'implies

L e t D be an a r b i t r a r y

s e t o f c i r c u i t s which c o n t a i n s a t l e a s t

one non-null

I f Y ( D ) e x i s t s then

circuit.

162

Linear Algebmic Optimization N

The existence of A* will be shown by proving the validity of (8.83)

for all i , j E

N

and k = 0,l

,.. . ,n.

Here,

z!k) J-I

is recursively

defined in the same way as, in the GAUSS-JORDAN method

(8.43),

.

a (k) Obviously (8.83) holds for k = 0. Then T ( O ) = I ( 1 , l ) I 11 ij and therefore (8.82) together with (8.80) implies the existence -(’)

of

(:::))*.

is well-defined. An argument similar as

Hence a ij

in the proof of (8.45) shows that (8.83) holds for k = 1. The inductive step from k

-

1 with k

5 1

to k is again quite N

(k-1)

similar. The inductive hypothesis yields U(T k k ( 8 . 8 2 ) and

(8.80) imply the existence of (:,(:-’)

to see that -a( k ) kk

-

N(k-1)

) = ak k

)*.

‘

It is easy

(k)

= u(Tkk 1.

The remaining part of the proof proceeds in the same way as the proof of ( 8 . 4 5 ) .

Theorem (8.81) shows that for the computation of

*;

we may N

use any linear algebra method for the inversion of I - A over the field F. In the GAUSS-JORDAN method the calculation of ( k - l ) ) * needs, in general, infinitely many steps since (akk a*

=

1 a a a a2 a

...

may be an infinite series and since the set H ( a ) may be infinite, too. If we are only interested in the path weights of -[n-1] elementary paths then we may calculate A

using the finite

JACOBI-iteration or a suitable modification of the GAUSS-JORDAN method.

Algebraic Path Fmblem

163

A particular example is the investigation of s i g n - b a l a n c e d

g r a p h s . A graph G is called s i g n e d if each weight is a

i j

= +1

= - 1 . The sign of a c h a i n is the usual product of its ij arc-weights. If the signs of all cycles of G are positive then

or a

G is called s i g n - b a t a n c e d .

Such properties have been studied

by HARARY, NORMAN and CARTWRIGHT [1965]. HAMMER [1969] developed a procedure for checking whether G is Sign-balanqed. BRUCICER [I9741 proposed the application of the GAUSS-JORDAN method.

Let H = {-1,+1).

Then (H,-,z) i s a totally ordered commutative

group with neutral element 1. Let i j

1

Al

b

for ( I , j ) E E.

i j

N

NOW A:=

A

(2)

l

B

=

{

F

0 if (1,j )

i f z = a

i j

else

d E and

' I

is symmetric. w.1.o.g.

we assume

(i,i) B E . As the neutral element is the maximum of H we know

that

i* will

exist if and only if G contains n o circuits. Now

G is sign-balanced iff all cycles in G have weight 1 or, e q u i -

valently, iff all elementary cycles in G have weight 1. Therefore G is sign-balanced if and only if

for all I E N . This is equivale,nt to (8.84)

for all i E N. For a computation of the sets Z(:;fn1)

lj

we may

simplify our method. We consider the semiring ( 2 { - I , ' } , @ , a ) with zero 2 , unity { l ) and X y E Y ) for all X,Y 5 {-l,l).

@

Y:= X U Y ,

X

IB

Y:= {xyl x E X ,

This semiring is idempotent, commu-

Linear Algebmic Optimization

I64

tative and contains no nontrivial zero-divisors. Furthermore

0,

{ l } are absorptive elements and {-l},

{ - l , l l are 1-regular

elements. Let

for all i,j E N . Then

aij 5

{-l,ll. Since all elements of the

semiring are at least 1-regular the matrix only Interested in

irnlas

G is balanced

A*

exists. We are

(cf. 8.84) iff

for all i E N . This semiring was Introduced in B R U C K E R [1974].

(8.85) k - s h o r t e s t p a t h s We consider a network G over the additive group of real numbers,

E W for (i,j) E E . We assume that G contains ij no negative circuits and w.1.0.g. (i,i) 6 E. In (8.84) we disi.e. all weights a

cussed methods for the determination of all path weights. Let F . .: = {w(p) 11

I

p

E Pij);

then F . , is countable. A s G contains no negative circuits 13

F , , is well-ordered. Let 13

F. 1j

with a

1

< a

<

... .

=:{U,,Cl,,

...I

Let U denote the set of all well-ordered

and countable subsets oflR. We introduce the internal composition X+Y:=

{ x + y l

X E X ,

Y E Y I

for X,Y C_ U . Then ( U , U , + ) is an idem?otent, commutative semiring with zero

0 and unity

{O}.

( U , U , + ) corresponds to

(V,@,@)

in

Algebraic Path Problem

(8.84).

165

For a E V with Z ( a ) E IR we find Z(a) E U. For such a the

mapping z : a

-v

Z(a) is a homomorphism. We define now

a

(8.86)

{aij 1 - = {

ij'

N

A

by

if (i,j) E E ,

. @

otherwise

I

for all i,j E N . The next theorem shows the existence of

-

A*

over U.

(8.87) Theorem If G contains no negative circuits (with respect to the usual sum of the arc weights a

ij

)

then

*;

exists over U.

Proof. We give a constructive proof using the G A U S S - J O R D A N method (8.43). The proof is quite similar to the proof of (8.81).

-

It should be clear that a * E U exists for all a E U with O z m i n a. Therefore with

U

defined by (8.2) with respect to U we can

prove (8.83) for all i , j E N and k = O,ll...ln. In particular, the proof uses 0 ; (kkk)

-

-(k-1) (akk )

5 min -a(kkk- 1 )

-

(k-1)

= min(o(Tkk

)

which implies

N

* =

o(T:L))

in the inductive step.

Theorem (8.87) implies that we may use the G A U S S - J O R D A N method for the calculation of the sets F ij ' as

for all i , j E N . In general, the calculation contains the dem(k-1)

termination of the possible infinite sets ( a t k

)

difference to theorem (8.81) lies in the fact that

*.

The main

'ii* exists

if w(p) = 0 for some circuits in G. Then the function

*; i j E

V

166

Linear Algebraic Optimization

N

in

f o r some z Em w h i c h i s f o r b i d d e n

( 8 . 8 1 ) may h a v e a * . ( z ) = 13

(resp. the f i e l d F).

f o r f u n c t i o n s i n t h e i n t e g r a l domain V

Over t h e s e m i r i n g U t h i s d o e s n o t l e a d t o a n y d i f f i c u l t i e s . Then z i s a n e l e m e n t o f t h e s e t

*; i j E U .

Now w e may b e i n t e r e s t e d i n f i n d i n g o n l y t h e k s m a l l e s t v a l u e s in F

T h i s p r o b l e m c a n be s o l v e d w i t h a f i n i t e m e t h o d .

ij'

,

{XEUl X = k - m i n ( X ) ]

V:=

Let

V c o n s i s t s i n t h e elements o f U with a t most k elements.

i.e.

The f u n c t i o n k - m i n :

U + U

was i n t r o d u c e d i n c h a p t e r 2

( c f . 2.11

and i s d e f i n e d by k -min(X):= Ixl,x 2,..., with x

< x2 <

1

...

and

1x1 2

xk'

k and by

k - m i n ( X ) := X if

1x1

C

k.

Now we define

for all X,YEV.

x

@

Y:=

k-min(XUY)

x

Q

Y:=

k-min(X+Y)

Then

semiring with zero i n Z+ in

then

(V,S,@)

b

i s an idempotent,

and u n i t y

(V,*,Z),

{O}.

I f we consider only weights

the lattice-ordered

( 2 . 1 1 ) , s a t i s f i e s X *Y:=

commutative

X 0 Y and X

c o m m u t a t i v e monoid

5

Y i f f

X @ Y = Y

for X,YEV. Let

b e d e f i n e d by

index r = n(k-1) + n

(8.86).

-

1 = n k

be c i r c u i t s of G w i t h w e i g h t

Then

-

i s s t a b l e and h a s s t a b i l i t y

1. F o r a p r o o f

l e t p 1'PZ'"''Pk

167

Algebraic Path Problems

for

u

= 1,2,

..., k .

w

u is the weight of p u with respect to

V.

Then

. . . + w k-1 1 ,w1+w2 + ... + w + Wk1 k-1

,w1+w2

k -min{0,wl,wl+w2,... =

k -min{0,wl,wl+w2,...

+

shows that G is (k-1)-absorptive (over V ) . Then theorem (8.33) N

shows the stability of A . Hence N

A*

for all s

2

n k

=

“sl A

- 1.

Thus the computation of

with the aid of the J A C O B I -

*;

is possible

as well as the G A U S S - J O R D A N - m e t h o d .

Then a* = Ealfa2,...,ak 1 ij contains the k smallest values of F

ij

( i f IF

ij

I < k

then

Such semirings as V were introduced by M I N I E K A and S H I E R [1973] and S H I E R [1976].

Methods for solving the k-shortest path

problem have also been discussed by HOFFMAN and P A V L E Y [1959], BELLMAN

and K A L A B A [1960], Y E N ([19711, [19751), F O X [19731,

WONGSEELASHOTE

[1976] and I S H I I 119781.

Some further examples for algebraic path problems are contained in GONDRAN and M I N O U X [1979b] and in C A R R E [1979].

9.

Eigenvalue Problems

In this chapter A denotes an n

x

( R , @ , @ ) with zero 0 and unity 1.

n matrix over a semiring From chapter 5 we know that

(Rn,@) is a right as well as a left semimodule over R. In particular, (A

Q

x)i:=

x

@

xi

for i E N defines the left (external) composition 9 : R x R n

Rn.

+

The right (external) composition coincides with the matrixcomposition of x and A .

e i g e n v e c t o r of (9.1)

n An n-vector x E R , x 8 0 is called an

if there exists A E R such that

A

A @ x = A Q x .

Then A is called an e i g e n v a z u e of A . tors with eigenvalue

A

The set of all eigenvec-

is denoted by V ( X ) .

V

0

(1) = V ( X ) U

{Oj

is a right semimodule over R; i.e. (x

@J

a) @

(y

@J

B) € V O W

for all x,y E V o ( A ) and for all a, f? E R .

No method is known for

the determination of the eigenvalues and eigenvectors of

A

in

the general case. The results in this chapter will focus on the case where the internal composition @ reflects more or less n . a linear order relation of R. The classical case ( R is a module over a field R) is not covered by the main results. We refer to BLYTH [I9771 for a discussion of the classical problem. Now we introduce a preorder 5 on R by a @ c = b

168

Eigenvalue Problems

169

for all a , b E R . Then (R,@,Z) i s a naturally ordered commutative monoid, (R,tD,z) is a preordered monoid and (R,$,@) is a preordered semiring with least element 0. If

5 is a partial

order then R is a partially ordered semiring and (9.3)

a @ b = O

*

a = b = O

for all a , b E R . If R is an idempotent semiring then R is

-

partially ordered and (9.4)

a z b

a @ b = b

for all a , b E R . Then (9.5)

a @ b = sup(a,b)

for all a , b E R and R is completely described by the partially ordered monoid

(R,@,f) and (9.5). If

for all a , b E R then R is linearly ordered and completely described by the linearly ordered monoid (9.7)

(R,@,Z) and

a @ b = max(a,b)

for all a , b E R .

The eigenvalue problem in such semirings is discussed by GONDRAN and MINOUX ([1977], [19781, [1979a]. They characterize V(A) in the particular cases that (R',@) with R':= R\{O] a group and that

@

is

is an idempotent composition. The group-

case is discussed in detail by CUNINGHAME-GREEN [19791. For the particular case (IR ,max,+) the eigenvalue problem'is solved in CUNINGHAME-GREEN [19621 and VOROBJEV ([19631, 119671 and [1970]).

Linear AIgebraic Opfimization

I70

The discussion o f the eigenvalue problem in such semirings will show its close relationship to certain path problems considered in chapter 8. F o r an n

X

n matrix

A

let G A = G(N,E)

denote the directed graph with set of vertices N = { l , 2 ,

...,n)

and set of arcs E defined by ( i , j )E E

a

:-

ij

* O

is similar as in

for i,j E N . The relationship between

A

chapter 8. Here G

and does not contain arcs

A

is derived from

with weight 0. The circuits of G

A

and

GA

play an important role in

A

the following discussion. Differently from chapter 0 we exclude circuits of length 0. Therefore the leading term in the matrix = I Q A Q A’

A*

...

Q

,

i.e. the unity matrix I , has no longer an interpretation by paths in G

A‘

Let P i j

(pf;])

denote t h e set of a l l p a t h s f r o m

i to j in G A ( w i t h t f p ) 5 k).

The length l(p) of a path p , the

weight w(p) and the function

are defined in the same way a s

U

at the beginning of chapter 0 . Similar to ( 8 . 1 7 )

we find that

the entry a ( k ) in the matrix ij A

(k)

:= A Q A

2

...

Q

fB A

k

is equal to u ( P [ ~ ] ) and that if i j

A+:=

A Q A’

Q A3

...

fB

exists then (9.8)

a

+ i j

= u(P. .

11

I

for all i,j E N . We define

IJ

j

:=

a

+ jj

for j E N. From

Eigenvalue hoblems

(9.9)

A * = I B A

we conclude that

+ ,

A

+

171

= A Q A *

exists if and only if

A*

A+

exists. There-

fore all results and methods of chapter 8 with respect to can easily be applied to

+ A .

we assume the existence of existence of

A*

Throughout this chapter either o r other assumptions imply the

A*

Only the latter case will explicitly be

A*.

mentioned.

We continue with the discussion of some sufficient conditions for the existence of solutions of the eigenvalue problem which are drawn from

and

GONDRAN

MINOUX

published in an English version

[1977]. This paper has been

(GONDRAN

and

[1979a]),

MINOUX

too.

(9.10) P r o p o s i t i o n Let

be a semiring; let V E R , k E N and p e p k k . Then

R

(1)

(1

@

(2)

if

( 1 I3

(3)

if I

Proof.

(1)

A:

Uk)

Q

p = Uk

w(p)) A = A

Q

P E

v0

Q

+

Ak

Q

ll

p =

then

iff

A;

w(p) @ p A;EV~(I)

Q

pEVo(l)

then

Q

Q

P EVo(l)

;

Q A:

Q

p = A* k

o

p

Therefore we find the equi-

= A.:

p = A:

A:

;

for all j E N .

( 1 ) is equivalent to A

F r o m (9.9) we know A Q A:

valent equation

Q

p.

(9.8) leads to the claimed

result. (2)

F r o m the definition of uk we know uk = w(p) I3 u(Pkk\{p)).

Therefore ( 1 Then

(1 @

CI

uk)

Q

p = [ ( l

w(p)) 0 p = w(p)

and ( 1 ) can be applied.

Q

I3 w(p)) Q

I3 [u(P

kk

\{PI)

Q

p1

p implies ( 1 I3 u k ) @ p = uk Q IJ

.

.

Linear Algebraic Optimization

172

I

(3)

@ A

= A shows a

j j

j

Therefore

Now 1 @ w ( j , j ) = w ( j , j ) a n d

( j , j ) E P . . for a l l j E N . I1 i m p l y A* E

for all j E N .

9 0

(2)

v0 ( 1 ) .

8

Proposition (9.9).

(9.10) shows t h e i m p o r t a n c e o f t h e e q u a t i o n s

The columns of A

*

and t h e columns of A

+

d i f f e r only

s l i g h t l y and t h e r e f o r e t h e y a r e c a n d i d a t e s f o r t h e e i g e n v e c t o r s of t h e m a t r i x A w i t h e i g e n v a l u e 1. W e remark t h a t A*

j

@

LI

E V ( 1 ) i m p l i e s A+ 0 p = A* 0 j j

E V ( 1 ) . T h i s v i e w is

q u i t e d i f f e r e n t from t h e a p p r o a c h t o t h e c l a s s i c a l p r o b l e m . W e know t h a t i n t h e c l a s s i c a l p r o b l e m A *

But i f I - A

=

(I- A )

-1

(cf.

8.55).

i s a r e g u l a r m a t r i x t h e n A cannot have t h e e i g e n -

value 1. N e x t we a p p l y

( 9 . 1 0 ) t o two p a r t i c u l a r c a s e s .

(R,@) i s a n i d e m p o t e n t m o n o i d , i n t h e second c a s e

(R',@)

i.e.

I n the f i r s t case

a 0 a = a f o r a l l a E R and

i s a group.

(9.11) P r o p o s i t i o n L e t R b e a n i d e m p o t e n t s e m i r i n g w i t h i d e m p o t e n t monoid

EVo(l)

( 11

A*

(2)

i f R i s c o m m u t a t i v e ( h e n c e A* e x i s t s ) t h e n A Q U

j

@ U

j

E Vo(X)

Proof.

for a l l j E N ;

find

and

(9.10.1)

8

u,

= a . 8 a

J

j

Q A*

j

h E R.

for a l l j E N ,

(1) A s b o t h i n t e r n a l c o m p o s i t i o n s a r e i d e m p o t e n t

(1 @ a ) j

(It,@).

j

leads t o t h e claimed r e s u l t ;

we

Eigenvalue Problems

(2)

173

In the commutative case the existence of A

follows from

n

the fact that R is 1-regular (1 @ a @ a L = 1

(B

a; cf. 8.33).

Clearly (1) implies ( 2 ) .

Proposition (9.11) shows that if both internal compositions are idempotent then at least 1 i s an eigenvalue and that atZ elements of R may be eigenvalues. In the second case it i s possible to reduce the eigenvalue problem (9.1) to the solution of (9.12)

(A-1

8 A)

B x = x

provided that we are only interested in eigenvalues A > 0.

( 9 . 1 3 ) Proposition Let R be a semiring with group (R',8) and let R be partially ordered by (9.2). If GA i s strongly connected and x E V ( 1 ) then A > 0

x

and

> 0.

Proof. If X = 0 then A B x = 0. (9.3) shows a all i,j EN. A 6

(R',b) is a group we get a i j

all i,j E N . A s x E V ( A ) there exists x all j EN,contrary to G

A

I

*

ij

= 0

8 x

or x

0. Then a

for

= 0

j

j

i j

= 0

= 0

for

for

strongly connected. Therefore A > 0.

Now suppose xk = 0 and let M:= { j l x

j

*

0). Then M , N X M 9 $

and

for all i EN'M. contrary to G

Therefore a A

ij

= 0

strongly connected.

for all ( i , j ) E (N'M)

x M,

Linear Algebraic Optimization

174

I n t h e d i s c u s s i o n of t h e e i g e n v a l u e problem i n t h e c a s e t h a t

(R',@)

i s a group w e w i l l only consider solutions of

w i t h A > 0, x > 0.

Such s o l u t i o n s a r e c a l l e d finite

t19791). P r o p o s i t i o n (9.13)

CUNINGHAME-GREEN

c i e n t condition such t h a t every solution of

Now we a p p l y

(9.10) t o

N

(9.12).

L e t A:=

h

-1

A

and A-'

(cf.

gives a suffi(9.1)

0 A.

is finite.

Then G

d i f f e r only i n the weights assigned t o the arcs

G-

(9.1)

A

and

(i.e.

a. lj

The c o r r e s p o n d i n g w e i g h t f u n c t i o n a n d o t h e r 11 N N N d e n o t a t i o n s i n v o l v i n g w e i g h t s a r e l a b e l e d by ' I - " (U,W,Uj I . . 0 a. .).

.

1.

(9.1-4) P r o p o s i t i o n L e t R be a s e m i r i n g w i t h g r o u p N

(R',@).

L e t k E N.

N

N

(1)

If

1 tB u k = u k

(2)

if

R i s commutative and 1 @ pk =

Proof.

( 1 ) Here

then

A:EVo(X);

vk

for

N

fore

;:

(9.10.1)

i s f u l f i l l e d w i t h LI = 1 for A .

i s an e i g e n v e c t o r o f

There-

( 9 . 1 2 ) w i t h e i g e n v a l u e 1 . Hence

N

A:

E Vo(h).

( 2 ) I n t h e commutative c a s e t h e weights o f a p a t h p w i t h respect t o G

A

a n d GX a r e g i v e n b y N

w(p) = x - l ( p )

Therefore

k

= pk

and

(D

w(p).

(1) is applicable.

These s u f f i c i e n t c o n d i t i o n s should be seen t o g e t h e r w i t h t h e following necessary conditions.

Eigenvalue Problems

175

(9.15) Proposition Let R be a semiring with group ( R ' , B ) and let R be partially ordered by (9.2). Let x E V ( h ) . b e a finite solution. w

(If R is commutative then)

(1)

for all circuits p in

G

w(p)

5 1

(w(p)

5 h 1 (PI)

.

A

'

if R is idempotent then

(3)

if R is linearly ordered (and commutative) then there

exists a circuit p in G Proof. -

is absorptive (hence

'ii* exists);

(2)

with ~ ( p )= 1

A

(1) Let p be a circuit of (A-'

a a,,)

B

xj

(w(p) = ~'(p)).

Then (with respect to 9 . 2 )

GA.

5 xi

for all arcs (i,j) of p. Together these inequalities imply N

W(P)

@

Xk

5

Xk

f o r each vertex k of p. Then xk > 0 shows N

W(P) 5 1.

(4)

In the commutative case y(p) = w(p) (2)

If R is idempotent then ( 4 ) and (9.4) lead to N

w(p) 13 1 = 1

N

i.e.

@

A

is absorptive and

*;

f o r all circuits p in G

ij

B x.1

3

'

exists (cf. 8 . 3 3 ) .

( 3 ) In the linearly ordered case

maxis

A

A B

x = h

x means

j E N ) = A B xi

f o r all i E N . This maximum is attained for at least one j ( i )

in row i . The sequence k , j ( k ) , j ' ( k )

...

contains a circuit p

Similarly to ( 4 ) we find N

W(Pk) = 1 1 (P,)

which in the commutative case means w(pk) = h

k'

Linear Algebraic Optimization

176

If R is linearly ordered, commutative and finite solution then propositions

contains a

and (9.15) show in

(9.14)

particular that at least some columns

V(A)

-* Ak

are elements of

V0(A).

Now we consider again the more general situation without specific assumption on the internal composition 0.

(9.16) P r o p o s i t i o n Let R be a semiring. If R is idempotent then x E V ( 1 )

(1)

implies

0 x = x and

A*

A + @ J x = x ;

(2)

if R is absorptive and commutative (hence

A*

exists) then

implies A * 0 x = x ;

xEV(X)

if R is absorptive and aii = 1 for all i E N then V ( X ) C _ V ( 1 )

(3)

for all A E R. Proof. ( 1 ) Let x E V ( 1 ) . Then A k

8 x

= x

for all k E N . There-

fore Ark]

0 x = x B

for all k E R . Then Let x E V ( A ) .

(2)

Ak

A*

( A @ X ) 8)

8 x

=

B

An-'

x. Similarly

A

+

B

...

8 x

= x

= x.

Then commutativity implies

0 x = Ak

8 x

for all k Em. Due to ( 8 . 3 3 )

..

(A2 0 x )

E R shows

A*

= A

rn-'1.

1 B a = 1. Therefore A *

Now a = A

(B

...

h2

0 x = A[n-~l ~

= (1 @ a ) 0 x = x . ( 3 ) Obviously I =

(1 B A )

0

(B

A = A.

Let x E V ( h ) . Then

A 0 x

=

(I @ A) 0 x

x = 1 0 x = x. rn

177

Eigenvalue Problems

We remark that, if R is absorptive and aii = 1 for all i E N , then A

*

exists and (9.10.3) shows that all A

*

j

are elements of

Vo(l) which, by (9.16.31, contains all eigenvectors of A .

(9.17) T h e o r e m Let R be a linearly ordered semiring. If x E V ( 1 ) then x = A* @ x M

with

A*

V

8 x

V

M

E V ( 1 ) for all v E M

5

N.

Proof. From (9.16.1) we know x = A*

0

x. Thus it suffices to

find a reduced representation by terms A* (9.16.1) we also know

+ A

V

@

Q

x V E V ( l ) . From

x = x . We consider the partial di-

graph G' = ( N , E ' ) with A

with respect to an eigenvalue X

(here A = 1). Then we introduce

an ordering on N by 1 5 j if and only if there exists a path from i to j in G'

A'

On the set of maximal vertices this induces

an equivalence relation. This fo1,lows from the fact that for all i E N there exists at least one j E N such that i A 4D x = A

@

x and R linearly ordered). Let Nk

, k

5

j

(as

= 1,2,...,s

denote the corresponding equivalence classes. Then a similar argument shows that the subgraphs G;

generated by Nk in G A are

strongly connected and contain at least one circuit of G' A' Now let p denote a path from v to l~ in GA. Using the equations aiJ

8 x, = xi along p we find

(9.18)

w(p) 8 x u = x

V

.

178

Linear Algebraic Optirhization

Obviously x of

(B

P

> 0 iff x V > 0. From (9.8) and the idempotency

we get

+

(AV 8

w(p))

(A:

xV) @

A

(B

+

= A

v

+

.

U

Therefore (9.19)

Q

0 x

(A:

u)

= A

+ u

8 x

U

which leads to x = A

for M : =

{v,[

A:

k

+

O

xV

@

x

=

A

+

M 8 xM k = 1.2,

9 0,

u k E N k for all k = 1,2,. ..,s. p of G h

...,s )

provided that

k As

each v E M lies on some circuit

(9.18) yields

w(p)

.

xu = x

8

Idempotency of R implies (w(p) for u E M .

8 X")

Then (9.10.2)

A ; @ x

for v E M . Thus x =

leads to

= A : B x A:

B

=

@ XV

x

(B

V

x u = w(p) B X"

*

AV 8 x

E V o ( l ) for v E M .

Hence

V

x M is a representation with

A*

V

8 xVEV(l)

for all v E M .

This theorem shows that all eigenvectors with eigenvalue 1 are

*.

The corresponding

linear combinations of certain columns of

A

vertices lie on circuits of the graph

If R is furthermore

absorptive, i.e.

GA.

1 is the maximum of R, then

(9.16.3) implies

that we can determine all eigenvectors in this way provided that the main diagonal of

A

contains only neutral elements.

179

Eigenwlue Problems

Next we consider the spec$al case of a linearly ordered, absorptive (commutative) semiring with idempotent monoid

(R,@).

We may as well consider a linearly ordered (commutative) monoid with minimum 0 and maximum 1 which has an idempotent

(R,B,()

internal composition

The eigenvalue problem ( 9 . 1 )

8.

is thus

of the particular form

for i E N .

(9.21) T h e o r e m Let R be a linearly ordered, absorptive semiring with idempotent (R,O). Hence A* exists. Let B := A* 8 u, j j

monoid

J:= {j E N 1 u

j

*

for j E N and

0). Then

is the right semimodule generated by { B . I j E J);

(1)

Vo(l)

(2)

if R is commutative then A 0 V o ( l ) = A 8 V o ( h ) ;

(3)

if R is commutative and a

I

A 8 V0(l)

= V0(A).

Proof. ( 1 ) From ( 9 . 1 1 . 1 ) we know B absorptive semiring

* a

@

p.

for all j E N . I n an

*

for all k E M which shows x = B M 8 xM

-

c A

Q

= A 8

Vo(A).

and ( 1 ) imply A 8 V o ( l ) If x E V o ( A )

( A @ x), i.e.

A

which implies A b V o ( A )

*

0 implies

iff

*

*

(9.11.2)

*j

x E V ( 1 ) .has a representation x = AM 8 xM

Due to ( 9 . 1 7 )

with Ak 0 x k E V ( 1 ) for all k E M. Therefore Ak

(2)

U

show that Ak @ p E V o ( l )

(9.10.1)

1

uk

EVo(l)

j

= 1 for all j E N . Thus

jj

B . E V ( ~ ) .Absorptivity and =

= 1 for all I E N then

ii

then A

@ xI E

-

0

Vo(l).

x =

8 U

k 8 xk

. Vo(A). A 8 x.

Then A 8 V o ( l )

Thus A B x = A b ( h e x )

Therefore A

A 8 Vo(l).

xk = :A

8

@

Vo(A)

Vo(l)

180

Linear Algebraic Optimization

(9.16.3) implies A b x = x for x E V ( A ) . Therefore

(3)

A B Vo(A)

Together with ( 2 ) we get A

= Vo(A).

Q Vo(l)

=Vo(A).

Now we consider the special case of a linearly ordered semiring with group (R',b). We may as well consider a linearly ordered group (R',O) with an adjoined least element 0 acting as a zero. The neutral element of R' is denoted by 1 . CUNINGHAME-GREEN ment

[I9791

additionally adjoins a greatest ele-

but mainly considers finite solutions of the eigenvalue

problem (0 < A <

-,

0 < xi < - 1 .

A

vertex v E N is called an

e i g e n u e r t e x of A if v is a node on a circuit of G

A

with weight 1.

Two eigenvertices are called e q u i v a l e n t if they belong to a

common circuit in G

A

with weight 1. Otherwise two eigenvertices

are called n o n e q u i v a t e n t . T h i s denotation is motivated by the observation that " i equiva-

lent j " is an equivalence relation

-

on the set of all eigen-

vertices. Reflexivity and symmetry are obvious. Let i

3

N

j and

k. Let the corresponding circuits be p = a o b , q = c o d

N

with paths a from i to j, b from j to i, c from k to j and d from j to k. Then a o d o c o b is a circuit with vertices i and k. w(a) = w(b)

As

i.e.

i

-

-1

and w(c) = w(d)

-1

we find w(a - d o c ~ b =) 1 ,

k.

(9.22) T h e o r e m Let R be a linearly ordered semiring with group (R',Q). (1)

Vo(l)

i s the right semimodule generated by { A * ( 3

A*EV(l)}. 3

If the eigenvalue problem has a finite solution with eigenvalue A

(hence ( A

-1

0 A)*

exists) then

Eigenvalue h b l e m s

(2)

181

every finite eigenvector of A has the same finite eigenvalue 1,

v0 ( A )

(3)

is the right' semimodule generated by

S:=

{ (A-1 0 A)

j E J ) where J denotes a maximal set of nonequivalent -1

N

eigenvertices of A : =

IB A ,

S i s a minimal generating set of V o ( X )

(4)

all such

S

(S is a b a s e ) and

have the same cardinality.

Proof. ( 1 ) From ( 9 . 1 7 )

we know that x E V ( 1 ) has a representation

,

x = A * IB x with A * B x E V ( 1 ) for all j E M . M M j -1 therefore x exists. Thus A * E V ( l ) . j

Then x . > 0 and 7

j

Let A , B be two finite eigenvalues of A for finite solutions

(2)

of the eigenvalue problem. Then ( 9 . 1 5 . 2 ) and

-1

i:= B @

X < B . From

A

implies that

are absorptive. Suppose 1

9

x:=

A

-1

@ A

B and w.1.o.g.

we know that there exists a circuit p in

(9.15.3)

N

N

such that w(p) = 1. Then w(p) > w(p) = 1 contrary to A ab-

GA

sorptive. Thus A = B . ( 3 ) Let J denote a maximal set of nonequivalent eigenvertices

of G-. A

Let j E J. Clearly, ( 9 . 1 5 ) N

1 CB

j

= u j and, by

get x * E V ( A ) . j

(9.10.1),

-* A

j

implies j Eyo(l)

From the proof of ( 9 . 1 7 )

a representation x =

-* A

M

-

@

A

x.

= Vo(h).

As

*: j j

= 1 we

we know that x E y ( 1 ) has

xM such that all vertices j E M lie

on a circuit p of GA and w(p) * x j is an eigenvertex of

= 1. Therefore

,

- x,. -

N

Thus w(p) = 1 , i.e.

The elements of

M

were chosen in such

a way that any two of them do not belong to the same circuit

of

Gi.

in

G i

of

x.

As

any circuit p in G-

A

with weight z(p) = 1 is a circuit

we find that M consists in nonequivalent eigenvertices

Linear Algebraic Optimization

182

N

for an arbitrary k E J . Suppose : A

Let K:= J\(k)

(4)

for some y e R

K

.

N

= A;

B y

Then N

N

1 = a*

= a*

kj

kk

yj

QD

N

for some j E K and as a? = 1 ij N

2

a;k N

N

Now a*

(a*

ik

kj

.

1 @ y . = yj

1

is the weight of a shortest path p (9) from j to

)

q) = a* Q *: 2 1 and p kj jk Absorptivity implies w ( p - q) = 1. Then k N

k ( k to j) in GN and therefore w(p A

is a circuit in

GN.

A

D

q

and j are equivalent vertices contrary to the choice of k and j. Thus

S

is a minimal generating set of V o ( A ) . As

N

is an equi-

valence relation on the set of all eigennodes all maximal sets of nonequivalent eigenvertices have the same cardinality. 8

Theorem ( 9 . 2 2 ) Vo(x),

i.e.

shows how we can compute the finite eigenvectors,

if the unique finite eigenvalue N

--1

N

we determine A* for A:= h

C3

x

is known. Then

A and select columns

*; j E

(R'In

N

with a ? . = 1. Let J denote the set of the indices of such finite 33

eigenvectors and choose a maximal set of nonequivalent eigenvertices from J. Such a set can easily be found by comparing the N

columns A* j

, j E J , pairwise as N

A* = 3

*; k

(follows from ( 9 . 1 9 ) x:=

-* A , ) .

Q

j

-

k implies that

*; k j

on the corresponding circuit in G' with A

I

The remaining problem is the determination of

x

(if it exists).

We assume now that R is commutative. Then we may assume w.1.o.g. that ( R l . 8 ) is a divisible group (cf. 3.2). F o r b E R '

let

183

Eigenvalue Problems

x = bnlm denote the solution of the equation xm = bn with n/m E Q. From (9.15.3) we know that if

exists then there

exists a circuit p with

-a -

2

and A

= w(p) l/l(P) = :

;(p)

w(p') for all circuits p' in G

...

tary then p = p l e p2. k = 1,2,.. . , s .

p, with elementary circuits p

-

, k

A = W(Pk)

A ( A ) :=

(9.23)

exists then

-

B:=

A

Then

=

1,2,...,s.

max{w(p)

I

= A(A).

In any case the matrix

x

A(A)

i s absorptive. Hence

of

k '

contains at least one circuit then let

GA

If

If p is not elemen-

The cancellation rule in groups implies that

If

A'

-1

p elementary circuit in G

A

I.

8 A

i * exists.

i * contains

A(A)

is the finite eigenvalue

i* >

0 with g* = 1. j jj is a finite eigenvector. Otherwise the eigenvalue

if and only if

;*

a column

j problem has no finite solution. A

direct calculation of

A(A)

from (9.23) by enumeration of all

elementary circuits does not look very promising. For the special case of the semiring (IR

u

{--I,max,+)

LAWLER

1 1 9 6 7 1 pro-

posed a method which has a str.aightforward generalization to semirings derived from linearly ordered, divisible and commutative groups

with

A

k

=

(R',@).

k (a ij A(A)

Let

A'k'

be defined by

for all i,j ,k € N (cf. chapter 8 ) . Then = maxEa jk'l

ii

i,kEN).

Linear Algebraic Optimization

184

The computation of Ak for k E N needs O(n 4 ) @-compositions and O(n

4

)

( 5 comparisons). Then aii 'k'

@-compositions

can be determined by solving O(n 1

5

2

for i , k E N

equations of the form ak = 6 ,

)

5 n. Further O(n 2 ) @-compositions ( 5 comparisons) lead

k

to the value

h ( A ) .

For the same special case another method was proposed by DANTZIG, BLATTNER and RAO 119671. Its generalization is an algebraic linear program over a certain module (cf. chapter 11). Its solution

can be found by a generalization of the usual

simplex method (cf. chapter 11). that

x

It can be shown (cf. 11.62) N

X(A) and, in particular, h =

determination of

h(A)

if

1

exists. After

using methods discussed in chapter 1 1 we

-- 1

compute the matrix ( h

@

A)* which exists since

x

2 X(A). If

-

this matrix contains a finite column with main diagonal entry 1 then h = X(A) and the column is a finite eigenvector. Otherwise the finite eigenvalue problem has no solution. In particular, if

7

> h(A) then all main diagonal elements of (x-'

@

A)*

are strictly less than 1. For the above mentioned special case this method is discussed in CUNINGHAME-GREEN 119791.

We remark that the above results for linearly ordered, commutative groups have some implications for lattice-ordered commutative groups. From theorem ( 3 . 4 )

we know that a lattice-

ordered, commutative group (R',@,z) can be embedded in the lattice-ordered, commutative and divisible group V root system

T.

(x) with

a

Two different finite eigenvalues with finite

eigenvectors are incomparable and therefore the number of such finite eigenvalues is bounded by the number of mutually in-

185

EigenvafueProblems

comparable w(p) f o r elementary circuits p of GA ' This number

-

is bounded by the number of the elementary circuits in the number of maximal linearly ordered subsets in A . lar, if

G

A

and

In particu-

consists in a finite number of mutually disjoint

maximal linearly ordered subsets C

,C g ,. .. ,C

then we can

solve the finite eigenvalue problem by solving the finite eigenvalue problems restricted to V(C ) for k = l,Z,...,s. k

In the remaining part of this chapter we discuss some examples for eigenvalue problems. A s mentioned previously the approach described in this chapter has been developed in view of B-compositions which correspond to linear ( o r , at least partial) order relations. Therefore only such examples will be considered although the eigenvalue problem may be formulated for all the examples given in chapter 8 .

(9.24) P e r i o d i c s c h e d u l e s CUNINGHAME-GREEN [1960] considers the semiring ( I R U (-=),max,+) and proves the existence of a unique eigenvalue ces A over lR

.

for matri-

In this example X(A) is the maximum of the

circuit m e a m

-

w(p) : = W(P)/l.(P) for elementary circuits p in G

A'

An interpretation of the

eigenvalue problem as described by CUNINGHAME-GREEN ([1960], 119791) is the following. Let x ( r ) denote the starting-time j

for machine j in some cyclic process i n the rth cycle and let a

i j

(r) denote the time that this machine j has to work

Linear Algebmic Optimization

186

until machine i can start its r+lst cycle. Then

for all i E N . Hence the fastest possible cyclic process is described by

for r = 1,2,...

.

The finite eigenvalue A is then a constant

time that passes between starting two subsequent cycles, i.e.

In this way the cyclic process moves forward in regular steps, i.e. the cycle time for each machine is constant and minimal with respect to the whole process. The right semimodule Vo(A) contains the finite eigenvectors which define possible starting times for such regular cyclic processes. An example is given in CUNINGHAME-GREEN [1979]. The solution of the equivalent eigenvalue problem in the semiring (IR+,max,.) is developed in VOROBJEV (119631, [1967] and

.

[ 19701)

Methods for the determination of the finite eigenvalue A(A) and further applications have been discussed in DANTZIG, BLATTNER and RAO 119671, and LAWLER 119671.

( 9 . 2 6 ) Maximum c a o a c i t y c i r c u i t s GONDRAN [1977] considers the semiring (IR A

be an n

x

n matrix with a . . = 11

m

U {m],max,min).

Let

for all i E N . Let a . denote 1

Eigenvalue Roblems

187

the maximum capacity of a circuit with node j . Then ( 9 . 2 1 ) Implies that the vector B

j

with components

for i E N is an eigenvector for each A E l R + U ( - 1 {B

j

and that

I j E N ) generates Vo(x). An application to hierarchical

classifications are given in GONDRAN 119771.

(9.27) Connectivity GONDRAN and MINOUX [1977] consider the semiring ({O,l),max,min) with the only possible finite eigenvalue 1. If GA is strongly connected then a

* i j

= 1 for all i , j E N and the only finite eigen-

vector is x with x i = 1 for i E N . In general, if G into

s

A

decomposes

strongly connected components with vertex sets 11,12,..

..,Is then the eigenvectors with eigenvalue 1 are x 1 ,x2

,...,x s

defined by xi = 1 if j E Ii and xi = 0 otherwise. j

1

More examples are considered by GONDRAN and MINOUX [1977] and can easily be derived from the examples in chapter 8. F o r a discussion of linear independence of vectors over certain

semirings and related topics we refer to GONDRAN and MINOUX 119781 and CUNINGHAME-GREEN [1979]. CUNINGHAME-GREEN [1979] considers further related eigenvalue problems.

10.

Extremal Linear Programs

In this chapter we consider certain linear optimization problems over semirings (R',$,O) with minimum 0 and maximum

m

derived from residuated, lattice-ordered, commutative monoids (R,O,L)

(cf. 5 . 1 5 ) .

AS

in chapter 9 the correspondence is

given by a z b

a @ b = b

es

o r , equivalently, by a

(10.1)

Due to 12.13) tive over

(D.

@

.

b = sup(a,bl

we know that the least upper bound is distribuThe e x t r e m a l linear program is defined by

z:= inf{c

(10.2)

T

8

XI

A 8

2

x

with given coefficient v e c t o r c E R n l m

x

b,

xERn)

n matrix

A

over R and

m-vector b over R. Extremal linear programs are linear algebraic optimization problems (cf. chapter 7 ) .

The adjective

'extremal' is motivated by the linearly ordered case. Then the internal composition a

(B

bE{a,b).

(B

attains only extremal values, i.e.

In general, linear functions of the form c

'I'

8

x = supfc

j

Q

x

j

I

x . > 01 l

are considered. We define a d u a t e x t r e m a l linear program by (10.3) As

t:= sup{y

T

8 bl yT

(D

A

5 cTI

yERm).

we assume that the reader is familiar with the usual linear

programming concepts we may remark that this kind of dualization is obviously motivated by the classical formulation of a

188

189

Extremal Linear Programs

d u a l p r o g r a m . I t is, i n fact, a s p e c i a l c a s e o f t h e a b s t r a c t dual program which HOFFMAN [I3631 considered for partially o r d e r e d s e m i r i n g s . He s h o w e d t h a t t h e w e a k d u a l i t y t h e o r e m o f linear programming can easily be extended t o such programs. On t h e o t h e r h a n d i t i s v e r y d i f f i c u l t t o p r o v e s t r o n g d u a l i t y results. At f i r s t w e w i l l g e n e r a l i z e t h i s w e a k d u a l i t y t h e o r e m t o a r b i t r a r y o r d e r e d s e m i m o d u l e s ( R , ~ , 0 , ~ ; n ; H , * , ~ ) ( c f . c h a p t e r 5). T h e a l g e b r a i c l i n e a r program (10.4)

z

= inf(x

T

OcI A 0 x

2

b, x

2

0, x E R n )

5

c, y

2

e, Y E H ~ I

h a s t h e dual

(10.5)

t = sup{b

T

nyl A

The feasible solutions of

T

oy

(10.4) a n d

(10.5) a r e c a l l e d p r i m a l

and d u a l f e a s i b l e .

( 1 0 . 6 ) Weak d u a l i t y theorem L e t H b e a n o r d e r e d s e m i m o d u l e o v e r R. L e t x b e a p r i m a l feasible solution o f

(10.4) and l e t y b e a d u a l f e a s i b l e s o l u -

t i o n o f (10.5). T h e n bToy

<

x T O c .

Proof. Using some properties f r o m proposition

-

b T D y < (A @ x

)

~ =O x ~T O ( A T m y )

(5.21) w e € i n d

-C x T O c .

For semimodules over real numbers such weak duality theorems h a v e b e e n c o n s i d e r e d by B U R K A R D [1975a], D E R I G S a n d Z I M M E R M A N N , U .

Linear Algebraic Optimization

190

and Z I M M E R M A N N , U. f 1 9 8 0 a I . The value of the weak

[197Ea]

duality theorem lies in the fact that, if we succeed in finding a primal feasible solution x and a dual feasible solution y with b

T

O y = x T o c r then

x

is an optimal solution

of (10.41, y an optimal solution of (10.5) and we get t = b i.e

T

o y = xToc = z

,

a strong duality result. H O F F M A N [ 1 9 6 3 ] gives examples

f o r strong duality results which will be covered by the

fol owing discussion. With respect to semimodules over real numbers many examples are known and wi 1 be considered in chapters 1 1

-

13.

The weak duality theorem is valid for in the case O @ R. Next we will solve

10.2) 10.3)

and ( 1 0 . 3 ) , even explicitly.

is a residuated semigroup (cf. 2 1 2 ) we know that f o r

As R

all a , b f R there exists c E R with

-

a b x f b

x < c

f o r all x E R . This uniquely determined element c is denoted by b : a and is called residual. This result can be generalized to

matrix inequalities. In the following let N:= {1,2, M:=

11,2,

..., n]

and

. . . ,m).

(10.7) P r o p o s i t i o n Let R be a residuated, lattice-ordered commutative monoid. Let A and B be m x n and m x r matrices over R and define B:A

(B:A)

jk

: = inf{bik:a..I

i EM).

Then

11

A @ X L B

C.

X i B : A

by

191

Extremal Linear Programs

for all n

x

r matrices X over R.

Proof. A 8 X f B is equivalent to a

B xjk

ij

5 bik for all

i,j,k. These inequalities have the residuals b fore we find equivalence to X

ik:aij'

There-

5 B:A.

Proposition (10.7) shows that the dual problem (10.3) has a T Hence y is the optimal solution maximum solution y:= c : A

.

N

N

of ( 1 0 . 3 ) and t = b

(10.8)

Let

@

T (c:A )

.

denote the maximum solution of

N

i.e.

T

x = t:c

T

.

N

If x is primal feasible then

N

implies z = t and therefore x is optimal. We can prove feasihiN

lity of x for lattice-ordered groups, Boolean lattices and bounded linearly ordered sets (cf. chapter 1 ) . For the discussion we introduce the set R ( v , w ) with respect to v , w E R k which consists of all solutions b E R of the inequality (10.9)

SUP

...,k

~=1,2,

v

8

IJ

[(b

@

inf (wA:vA)):w 1 u A=l,2,...,k

2

The relevance of this strange-looking inequality is shown by the following result.

( 1 0 . 1 0 ) Theorem Let R be a residuated lattice-ordered commutative monoid. Let Ai denote the rows of the matrix A. If

b.

192

Linear Algebraic Optimiiation

b i E R(Ai,c) for all i E H then

is an optimal solution of (10.2).

Proof. It suffices to prove primal feasibility of x. As e implies e:g

5 f

5 f:g for all e , f , g E R we find N

N

(bk

Q

yk) : c

j -< t:cj = xj

for all k,j. For i E M this leads to

N

As

y. = inf{ch:aiAl h E N ) the assumption b i E R ( A i , c ) implies 1

bi

5 Ai

N

Q

x.

We do not know how to characterize the sets R(Ai,c) in the general case. Aowever, for lattice-ordered groups (R=R'\{O,-}), Boolean lattices (R' = R ) and bounded linearly ordered sets (R' = R ) the following result gives a sufficient description.

(10.11) Proposition (1)

Let R be a lattice-ordered commutative group. Then R(v,w)

(2)

=

.

R for all v , w E R k

Let R be a Boolean lattice and let a t3 b:= inf a,b). Then R(v):=

{bl b <

v

SUP

I J = 1 , 2 l...,k

(3)

IJ

C_ R(v,w)

for

11 v , w E R k

Let R be a bounded linearly ordered set with maximum 1 and let a 8 b:= min(a,b). B:=

min{w

a:-

maxiv

Then R(v,w) = {bl b

C

For v , w E R k let

I

:v

P I J

I

w

a )

.

IJ

IJ = l,21...,k)

< 6,

P -

p =

1,2,

...,k).

.

Extreml Linear Programs

193

Proof.

We will denote the left-hand-side of the inquality

(10.9)

as a function f of the parameter b E R .

(1)

For lattice-ordered groups b:a = b 49 a

-1

and the greatest

lower bound is also distributive over 49 (cf. 2 . 1 4 ) .

Therefore

f has the form f(b) = sup,, v

Q

[(b

-1

infX(wX 49 v h

Q

-1

-1

= b @ sup (v Q w ) 49 infX(vX P I J P As

in lattice-ordered groups sup f(b)

=

a

=

P

(inf

P

a

Q

B

w-'1 P

wh).

- 1 ) - 1 we find P

b

-

which implies R(V,W) (2)

U

) )

R.

F o r calculations in Boolean lattices it is more convenient

to use the denotations fl and U instead of sup and inf. Let

B:=

flP

(w

u

P

v;)

where v* denotes the complement of v lJ

b:a = b U a*

(cf. 1 . 1 6 ) . B*

-

UP

P

.

In a Boolean lattice

We remark that

(WE f l

v

P

).

Then f has the form f(b) = U A s a Boolean lattice

P

v

n

P

[ (b fl 6 ) U ] : w

i s in particular a distributive lattice

we find

Now let bER(v1. Then b

5

U

P

v

P

implies

Linear Alpebraic Optimization

IY4

i.e. (3)

b E R(v,w). Bounded linearly ordered sets are introduced in an example

for pseudo-Boolean lattices after proposition (1.19). Here 1

b:a

if b z a ,

=

{b

i f b < a .

With B and a as defined in (3),f has the form f(b) = max

min[v

lJ

P

,min(b,B):w

lJ

1.

If b E R ( v , w ) and b' < b then b ' E R ( v , w ) . Therefore it suffices to show f(b) = a for all b E R with b K-:=

Assume K- = $.

{ul

w

w

lJ

=

B,

*

uEK-1

< v 1 P

Then B = 1 and for b f(b) = max

Otherwise K-

u

P

a . Let

min[v

$ and 8 = min{w

2 a we find

lJ

, b : w 1 = max v = a . P l J P

I

w

< v 1 . Further a = maxiv P

l J ) !

and thus a > 6 . For b f(b) = max

lJ

2

min[vp,B:wP1

lJ

I

a we find =

a

. m

This characterization of R(v,w) shows immediately that for N

lattice-ordered commutative groups x is the optimal solution of ( 1 0 . 2 )

and that a strong duality result holds for (10.2)

and (10.3).

5 sup{a

I j E N ) is ij necessary for the existence of primal feasible solutions. ThereF o r Boolean lattices the assumption bi

fore

;is

an optimal solution provided that there exists any

feasible solution of ( 1 0 . 2 ) .

Extreml Linear Programs

195

For bounded linearly ordered sets the situation is not so transparent. We will transform the coefficients of such a problem in order to sim'pllfy the problem. In a first step let

:=

a'

(10.12)

ij

.

otherwise

Then the set of primal feasible solutions remains unchanged.

(10.13) P r o p o s i t l o n Let R be a bounded linearly ordered set with minimum 0 and maximum 1 and a

@

b:= min(a,b) for all a , b E R . Then

for all ~ E R " . Pioof. Let

Ai

(A;)

denote the i-th row of A

following statements are equivalent to Ai

and A; @ x

2

3 jEN:

min(a

3 j EN:

a

3 j EN:

a' = 1 ij

bi

ij

ij

->

bi

bi

and

x

and

x, >_ bi

for I E M . The

2 bi

0 x

2

,x ) j

(A')

:

I

> bi j -

I

,

. 8

Due to the reduction step ( 1 0 . 1 2 )

and proposition ( 1 0 . 1 3 ) we

may assume in the following w.1.o.g.

that A is a matrix over

{ o , l ] . Further we exclude trivial problems and nonbinding constraints. If bi = 0 then we delete the i-th constraint. Thus w.1.o.g.

let b > 0. Then Ai = ( O , O , . . . , O ) implies that no

feasible solution exists. Therefore we assume in the following

Linear Algebraic Optimization

196

w.1.0.g.

that A contains no zero-rows. If A contains a zero-

= 0 for some j E N then an optimal value of the j corresponding variable is x . = 0. Thus we assume w.1.o.g.

column j or c

1

that A contains no zero-columns and c > 0. Summarizing our reductions we have to consider in the following only problems with 0 - 1 matrix A containing no 0 - c o l u m n or 0 - r o w and with b,c > 0. Such a problem is called r e d u c e d .

(10.14) P r o p o s i t i on Let ( 1 0 . 2 ) be a reduced problem over the bounded linearly ordered set R with minimum 0 and maximum 1 and a

0

b:= min(a,b)

for all a , b E R . Then R(Ai,c)

=

R

for all i E M . proof. Due -

to proposition

R(Ai,c) = {bl 0

5

b

Bi:=

(10.11.3)

we know that for I E M

5 ail with min{cj:aij

ai:= max{a

ij

I

I

cj

j EN)

I

5 Bi ,

j EN}.

F o r a reduced problem this ieads to

Bi

=

maxfc.1 a , . = 1 , I 13

EN}

and therefore to a i

-

Proposition ( 1 0 . 1 4 )

shows that for a reduced problem over a

1.

bounded linearly ordered set

is an optimal solution of (10.2).

We summarize these explicit solutions of ( 1 0 . 2 ) theorem.

in the following

Extremal Linear h g m m

197

(10.15) T h e o r e m (1)

Let (R,@,Z) be a lattice-ordered commutative group. Then the optimal value 0.f (10.2) is

and an optimal solution

T

is given by

N

xj:=

2

@ c-l

j

for all j E N . (2)

Let (R,Z) be a Boolean lattice with join (meet) U (n), complement a* of a, and a 0 b = a n b

for all a , b E R .

Then the value

and the solution N

xj:=

2

u

T

is given by

c* j '

for all j E N are optimal for (10.2) provided that

is

primal feasible. Otherwise there exists no feasible solution of ( 1 0 . 2 ) . (3)

Let (R,I) be a bounded linearly ordered set with minimum 0, maximum 1 and a 0 b = min(a,b)

f o r . a l l a , b E R . If (10.2)

is a reduced problem then its optimal value is

z

=

max

iEM

min[bi,min{cjI

and an optimal solution

xj:=

N

{

1 2

for all j E N .

i j

=

is given by

< z , j otherwise ,

ifc

a'

111

Linear Algebraic Optimization

198

It should be noted that HOFFMAN 119631 gave the optimal solution in (10.15.1) for the additive group of real numbers ( l R , + , z )and a solution o f (10.2) in Boolean lattices. Theorems

(10.10) and

(10.15) show that such problems can be treated and

solved in a unified way. Further they describe in axiomatic terms a class of optimization problems for which strong duality results hold. On the other hand we have to admit that for general residuated,

lattice-ordered, commutative monoids no solution method is known for (10.2). A s we did not assume that R is conditionally complete it is even unknown whether an optimal value exists. A solution o f (10.16)

z:= inf{cT b

XI

A t9 x 5 b,

x E Rn)

is trivial if we assume O E R , a solution of (10.17)

z:= sup{cT

b

XI

A t9 x

5 b,

x E Rn)

is obviously x:= b:A which is the maximum solution of A b x f b. The solution of (10.16)

z:= sup{c T b

XI

A

(D

x

2

b,

x E Rn)

is trivial if we assume m E R .

If we consider such problems as (10.2) or (10.17) with equality constraints then we can only derive some lower and upper bounds on the optimal value. Let (10.19)

z:= inf{cT

t9

XI

A (D

and assume the existence of z.

x = b,

xERn}

Extremal Linear Programs

199

(10.20) P r o p o s i t i o n Let R be a residuated, lattice-ordered, commutative monoid. If S:= { x E R n I A B x = b) 4 @ then ;:=

b:A is feasible and

a maximum element of S. Proof.

is the greatest solution of A 8 x f b. In particular, N

5 x. This implies b

if A 0 x = b then x

=

A 8 x 5 A

N

@

x. Thus

N

A B x = b .

m Obviously

;yields

an upper bound on z. On the other hand

(10.3) leads to a lower bound. We find

If the greatest lower bound i s distributive over B then these bounds are certain row- and column-infima of matrices with 8 (bi:a ). j ij As in chapter 9 we have to assume that R is linearly ordered

entries bi

8

(c.:a. . ) resp. c

I

11

if we want to develop a finite and efficient solution method. Thus in the following R is a residuated linearly ordered commutative monoid. Further we will assume that (10.2) and (10.19) have optimal solutions if a feasible solution exists. In particular, for one-dimensional inequations (equations) this assumption means for all a,b E R : (10.24)

if a

B

x

2

b

(a

0

x = b)

has a feasible solu-

tion x then it has a minimum solution. Such minimum solutions are uniquely determined and will be denoted by b/a

(b//a)

.

and we find b/a = b//a.

If b//a

exists then b/a exists, too,

If b//a

exists for all a,b then R is

200

Linear Akebmic Optimization

a r e s i d u a t e d a n d d u a l l y r e s i d u a t e d d-monoid. In a l i n e a r l y o r d e r e d m o n o i d l i n e a r a l g e b r a i c i n e q u a t i o n s ( e q u a t i o n s ) can b e i n t e r p r e t e d i n t h e f o l l o w i n g way. A B x z b m e a n s t h a t f o r a l l r o w s i E M t h e r e e x i s t s at l e a s t o n e c o l u m n j (i) such t h a t

(10.25)

aij (i)

o x

j(i)

> bi -

.

A B x = b m e a n s t h a t for a l l i E M t h e r e e x i s t s at l e a s t o n e

column j (i) such that (10.26)

aij (i)

B x

j (i)

= bi

and that (10.27)

a

ij

o x

< b i 1 -

f o r a l l i E M , j E N . T h i s i n t e r p r e t a t i o n is e x t e n s i v e l y u s e d in c h a p t e r 9 f o r t h e s o l u t i o n of e i g e n v a l u e p r o b l e m s in c e r t a i n linearly ordered semirings. Let G (GI) denote the set of all (i,j) E M x N s u c h t h a t t h e s e t

ordered pairs

({XI a i j 0 x . = bill i s n o n e m p t y

I

(and a

kj

Q

(XI

a

ij

B

(bi//aij)

x

> bi)

j -

5

bk

f o r a l l k E M ) . T h e n w e m a y r e d u c e t h e s e t of f e a s i b l e s o l u t i o n s t o a set containing only solutions x with (10.28)

x j E {bi/a

ij

I

(1,j) € G I

in t h e c a s e of i n e q u a t i o n s and w i t h (10.28'

x

j

ECbi//aij

I

(i,j) E G ' )

in t h e c a s e o f e q u a t i o n s . T h i s l e a d s t o t h e f o l l o w i n g t h e o r e m s .

20 1

Extremal Linear Pmgnzmr

( 1 0 . 2 9 ) Theorem L e t R b e a r e s i d u a t e d l i n e a r l y o r d e r e d c o m m u t a t i v e monoid s a t i s f y i n g ( 1 0 . 2 4 ) . With r e s p e c t t o t h e e x t r e m a l l i n e a r p r o g r a m (10.21, l e t x ( a ) : = j

max{bi/ai,[

(i,j)EG,

f o r a l l j E N and f o r a l l a E R .

If

a

2

c

(bi/ai,)}

8

j

(10.2) has an optimal s o l u -

t i o n with optimal value z then (1)

x ( z ) is an o p t i m a l s o l u t i o n o f

(2)

z E Z:= ( c , 8 ( b i / a i , )

Proof.

(1)

cT Q x ( a )

5

I

( i , j ) €GI.

a f o r a l l a E R and t h e r e f o r e it s u f f i c e s

t o show t h a t x ( z ) i s f e a s i b l e . let i E M .

fore y

,

-> b i / a i , (2)

( 1 0 . 2 ) and

L e t y b e an o p t i m a l s o l u t i o n and

Then t h e r e e x i s t s j = j ( i ) w i t h a

2

bi/aij

and z

1. c,

which i m p l i e s a

i,

Q y,

2

2

8 x,

c,

bi.

0

i j

2

8 y,

bi.

There-

,

T h u s x (z)

( b i / a i,).

Eence x ( z ) is f e a s i b l e .

F o l l o w s from z E ( c T 8 x ( a ) l a E R 1 .

( 1 0 . 3 0 ) Theorem L e t R b e a r e s i d u a t e d l i n e a r l y o r d e r e d c o m m u t a t i v e monoid s a t i e -

With r e s p e c t t o t h e e x t r e m a l l i n e a r program

fying (10.24).

,

,

( 1 0 . 1 9 ) l e t x ' ( a ) : = max{bi//ai,

I

f o r a l l j E N and f o r a l l a E R .

I f t h e r e e x i s t s an o p t i m a l s o l u -

( i , j ) EG', a

2

c

8 (bi//ai,)}

t i o n with optimal value z then (1)

x ' ( z ) is an o p t i m a l s o l u t i o n o f

(2)

zEZ':=

Proof.

( 1 ) W e g e t cT 0 x ' ( a )

A

Q

x ' ( a ) 5 b.

fie6 A Q x(z)

1

{ c 8 (bi//ai,) j

5

( 1 0 . 1 9 ) and

(i,j) EG').

a a n d f r o m t h e d e f i n i t i o n o f G'

T h e r e f o r e i t s u f f i c e s t o show t h a t x ' ( z ) s a t i s -

2

b.

L e t y b e an o p t i m a l s o l u t i o n and l e t I E M .

202

Linear Algebraic Optimization

Then there exists j = j(i) E N with a . . 13

fore y

> bi//a.,

j -

Further a

17

@

PV

Therefore x l

I

(2)

yv

( 2 )

.

1. b

for all

v

2 bi//aij

.

? . I E

Thus

Follows from z E {cT 8 x(a)I

heo or ems

(10.29)

2

This implies z

@

y . = bi and there3

c . @ yj 3

2

cj 8 (bi//ai,)

.

M , V E N shows ( i , j ) E G ' .

A

(D

b.

x'(z)

aER1.

and ( 1 0 . 3 0 ) reduce the extremal linear programs

(10.2) and (10.19) to min(aEZ1

(10.31)

A

x(a)

(D

2 b)

and to (10.32)

min(aEZ'1

A

x'(a) = b].

(D

For the determination of 2

( 2 ' )

minimal solutions of O(nm)

a.

lj

@

B = b i and O(n m

@-compositions.

As

2

2

it is necessary to compute the

inequations a

ij

8

B 2

bi (equations

comparisons) and to perform

and

2'

O(n m )

are linearly ordered an optimal

solution can be determined using a binary search strategy together with the threshold method of EDMONDS and F U L K E R S O N 119701. Here we need O(log

2

(n m ) )

x(a) consists in O(n m ) check consists in O ( n m )

steps in which the determination of

comparisons and in which the feasibility @-compositions and O(n m )

comparisons.

This approach can easily be extended to residuated latticeordered commutative monoids satisfying ( 1 0 . 2 4 ) which are finite products of linearly ordered ones. Thus, in particular, extrema1 linear programs over the lattice-ordered group of real k-vectors (IRk K:=

{1,2,

,+,C),finite

..., k),

Boolean lattices ( 2 K

,fl,z)

for

and of finite products of bounded linearly

E x t r e m l Linear Programs

203

ordered sets (R,min,L) can be solved using a similar reduction. As

for such monoids the resulting sets Z resp. Z' are in gene-

ral not totally ordered it will be necessary to perform enumeration. A complexity bound for the solution of such problems will contain the cardinality of 'the finite product considered (in the above examples: k).

Similar results as given in this chapter for residuated, lattice-ordered, commutative monoids can be given for dually residuated, lattice-ordered, commutative monoids by exchanging the role of sup and inf. These results follow from the given results by replacing (R,@,z) by its lattice-dual ( R , @ J , ~ )We . remark in this context that lattice-ordered commutative groups are residuated and dually residuated.

Many authors have considered extremal linear equations, inequations and extremal linear programs. The solution of linear equations and inequations is closely related to the eigenvalue problem discussed in chapter 9. The solution of such equations and inequations has been considered by VOROBJEV (119631, 119671) in the special case (IR+,.

):,;

in a series of papers

ZIMMERMANN, K. ([1973al, [1973b], [1974a1, 11974131, [1976a] and [1976b])

discussed solution methods for extremal equations, .in-

equations and extremal linear programs in ( I R + , . , z ) .A compact solution of all different possible extremal linear programs in this monoid is given in ZIMMERMANN, U. [1979c] (cf. 1 0 . 3 4 ) . Extremal linear equations and inequations in linearly ordered commutative monoids are discussed by ZIMMERMANN, K. and JUHNKE [ 19791.

Linear Algebmic Optimizarion

204

For lattice-ordered groups and related algebraic structures such equations and inequations are considered in detail in the b o o k o f C U N I N G H A M E - G R E E N [19791. F o r B o o l e a n l a t t i c e s a d i s c u s sion i s given in the book of HAMMER and RUDEANU [1968] and in t h e s u r v e y a r t i c l e o f R U D E A N U [1972]. F o r r e s i d u a t e d l a t t i c e ordered commutative monoids the solution o f extremal linear p r o g r a m s is p r e v i o u s l y d e s c r i b e d i n an e x t e n d e d a b s t r a c t o f Z I M M E R M A N N , U.

[198Oc].

T h e d i s c u s s i o n o f s u c h p r o b l e m s i n (IR+

leads in a natural

way t o a g e n e r a l i z e d c o n v e x i t y c o n c e p t w h i c h e n a b l e s a g e o m e t r i c a l d e s c r i p t i o n . S u c h c o n c e p t s w e r e i n v e s t i g a t e d by Z I M M E R M A N N , K.

(119771, [1979a], [1979b]) and in a more general

c o n t e x t by K O R B U T 1 1 9 6 5 1 , T E S C H K E a n d H E I D E K R t k E R ( [ 1 9 7 6 a ] , [ 1 9 7 6 b ] ) , a n d T E S C H K E a n d Z I M M E R M A N N , K.

( 1 9 7 9 1 . Z I M M E R M A N N , K.

( 1 9 8 0 1 d i s c u s s e s t h e r e l a t i o n s h i p of c e r t a i n s e m i m o d u l e s and j o i n g e o m e t r i e s a s d e f i n e d by J A N T O S C I A K a n d P R E N O W I T Z [1979]. A u n i f i e d t r e a t m e n t of t h e l i n e a r a l g e b r a v i e w p o i n t i s d e v e l o p e d in C U N I N G H A M E - G R E E N

([1976], [1979]) for lattice-ordered

groups and related algebraic structures. Further, CUNINGHAMEG R E E N 119791 d e s c r i b e d s e v e r a l a p p l i c a t i o n s . A s a n e x a m a l e w e consider a machine scheduling problem.

(10.33) Machine Scheduling We assume that a production process i s described as

L a

(9.24

by s t a r t i n g t i m e s x w h i c h l e a d t o f i n i s h i n g t i m e s A 0 x o v e r the monoid

(IR U I - - } , + , : ) .

If we have prescribed finishing

times b then a solution of the system of extremal equations A 8 x = b i s a vector x o f feasible starting times. Assume -L -L

-

-------&aA-

.-.----A..-..

4 -

..arroccmrv + n

ptarf

maphinp

4

Extremal Linear Programs

205

which takes time c

This means that after starting machine j j' the next start of machine j is possible not before time cj + xj. In order to be able to start the process as early as possible a solution of

min{c

T

8

XI

A 8 x = b)

is required. If the time constraint is not absolutely binding then situations may occur in which we consider inequations instead of equations. It is also possible that A 8 x = b does not have any feasible solution. Then a reasonable solution is given by minimizing max{bi

-

(A 8

1 E MI

x),l

N

subject to A 8 x 5 b. Its solution is x = b:A since we know that A @ x

5 A

for all x with

0

A Q

x

-

5 b. Here x is given

by N

xj = min{bi-a

for j E N

ij

I

i E M )

(cf. 1 0 . 7 ) .

( 1 0 . 3 4 ) Bounded L i n e a r l y Ordered Commutative Groups In the following we assume that ( R , @ , L ) is a linearly ordered commutative group with I R l > 1. R is residuated and dually residuated; the residual b:a, the dual residual b/a, and the minimum solution b/a

(b//a)

in ( 1 0 . 2 4 )

coincide with b 8 a - l .

We consider several different types of extremal linear programs denoted by

(O,O,A,p)

with

An extremal linear program

O,O,A E

{max,min) and with

(O,O,A,p)

is given by

p€{z,z,=).

Linear Algebraic Uptimizarion

206

(10.35)

o{z(x)

I

x €PI

with objective function defined by Z(X) : = o I c . 7

and with the set P

C -

B X.

1

I

jE

NI

Rn o f feasible solutions defined by

AIa.. B x.1 ] E N ) P b i 1

11

for all i E I f .

For example, ( 1 0 . 1 ) corresponds to (min,max,max,z)

Since (R,@,z) is a linearly ordered group, too, each program has a dual version in the order-theoretic sense which is obtained by exchanging max w min and

5

CI

2 . Thus it suffices to solve

one problem from each of these pairs. For example, the dual version of

10.1)

is (max,min,min,<). -

we assume b . = 1 for all i E M . In table 1 we list the

W.1.o.g.

explicit so utions to all different programs with the exception of two programs with equality constraints which have to be solved by a threshold method

(indicated by "T"). Some tri-

vial cases are excluded by stating an assumption which can easily be checked for given coefficients. For the special group

lR+

[1979c

.

b;

*,z) such

{O],

a table is given in ZIMMERMANN, U.

A l l nontrivial exp icit solutions can be described

-

x = ;(A)

(10.36)

with

;,:=

[A{aij I i E M 1 1 - l

for all j E N or by x* j E N and with z *

(10.37)

z*:=

x*(o,A) with x*:= z*

=

B

1

= z*(o,A)

A{O{C.

3

B

-1 for all 7

c.

defined by a;?/ j E N ] [ i E M ) .

Then the optimal values are ;:=

z(;)

and z *

=

z(x*), respec-

tively. At first, w e verify the results in table 1 for inequality constraints:

Extremal Linear Pmgram

111

207

I

rnin

~~

rnax

-

max

min

max

rnin

-

z

Z

max

rnin

max

0

0

min

0

0

max

min

m

.

.>

max

min

m

m

max

z*

0

(n>l)

min

m

max

z

-

OD

-

min

z

max

T

-

z

-

z

0

(IPI > 1 )

rnin

w

T

min

-

z

(IPI>l)

Table 1 :

O p t i m a l v a l u e of e x t r e m a l l i n e a r programs

-

z

(o,o,A,p).

208

Linear Algebraic Optimization

For each problem of the form f m < n , o , A , z )

no comaonent x

j

(j E N ) of a feasible solution is bounded from below. Clearly,

*

P

0. We admit x

0 as a n optimal solution. Similarly,

x = = is considered as an optimal solution of problems of the form ( m a x , o , A , ~ ) . We remind that 0 and

-

are the adjoined

minimum and maximum.

f?%ax,max,max,g) has the optimal solution x

=

b:A (cf. 10.17),

i.e. =

miniayl I i E M I

=

[maxia

x. 3

lj

ij

I iEn11-'

=

j

(max)

for all j E N . Clearly, the dual of this problem in the ordertheoretic sense is f m i n , r n i n , m i n , ~ ) and has the optimal solution &(min). Similarly, we find that ;(max)

and x(min) are

the optimal solutions of f r n a x , m i n , r n a x , ~ ) and f r n i n , m a x , m i n , ~ ) . For ( m i n , m i n , r n a x , ~ ) the case n = 1 has to be treated separately. Then

a,

=

minib

i

a;:/

@

i E M ) = b:A is the optimal solu-

tion. Otherwise the objective function is not bounded from be ow We admit the optimal solution x defined by x l : =

*

x . : = 0 for all j 7

1.

MI

(n = 1 )

1

-1

max{bi@ail

=

and the optimal solution x with x : = 1

with x.:= = for all j 3

and by

Then z(x) = 0. F o r the dual version

f m a x , m a x , m i n , ~ ) we find the optimal solution 2 iE

GI ,

*

2,

and

1.

f m i n , m a x , r n a x , ~ ) is a special instance of the problem (10.1) which is solved in theorem ( 1 0 . 1 5 . 1 )

for lattice-ordered

groups. Thus, its optimal value is

z

=

max iEM

min ( c . a a-') jEN

1

ij

= z*(min,max),

and an optimal solution is x*(min,max).

Then the dual version

Extremal Linear Rogram

209

f m a x , r n i n , m i n , ~ ) has the optimal value z*(max,min) and the optimal solution x*(max,min). Secondly, we discuss.problems with equality constraints. Then the set P(A) of feasible solutions is defined by b x 1 = bi AjEN(aij 9

j

for all i E M. Proposition (10.20) shows that x(max) i s the maximum feasible solution of P(max) i f and only, if P(max)

9

@.

Similarly, x(min) is the minimum feasible solution of P(min)

* @.In

if and only if P(min) P(A)

*

0. This assumption can easily be checked for a given E P(A).

problem by testing ;(A) A

the following we assume that

maximum (minimum) feasible solution i s optimal for maximiza-

tion (minimization) problems. Therefore, ;(A)

is an optimal

solution for each of the problems fmax,max,max,=) fmax, m i n , mar, =)

that P(A)

*

, ( m i n , m i n , rnin, =) ,

,

and f m i n,max, m i n , =) provided

@.

Next we consider f m i n , m i n , max, =)

*

is optimal. Otherwise, let x

.

If

P(max) I = 1 then x(max)

x(max) be another feasible

so,lution. Since s(max) is the maximum feasible solution there exists some l~ E N with x Then

with

lJ

z j := x j (max)

solution for a'l1 a 5

- . lJ

< x

Thus a

lJ

0

for j * p and with

; (max). lJ

x

< 1 for all i E M .

lJ lJ

: = a i s a feasible

Since a is not bounded from below

in R , the objective function is not bounded from below in I?. N

admit a = 0. Then x i s an optimal solution with optimal value N

z(x) = 0. Similarly, the optimal solution of the dual version

fmaz,max,min,=)

i s x(min),

an optimal solution

with

if IP(min) I = 1 , or there exists j

=

j

(min) for j

*

l~ and with

We

Linear Algebmic Optimization

2 10

N

x

P

=

QD

for some

u

E N .

Obviously, the condition I P ( A )

> 1 can

easily be checked. I f P(max)

@ then the optimal value z o f

4

fmin,max,max,=)

satisfies the inequalities z*(rnin,max) 5 z max min ( c . iEM J E N

-1

a.. ) 11

@

x

(10.301. With

=

5

-1

max rnin ( c . 19 a ) ] E N IEM ij

x(max) we find

otherwise,

J

a E

z

*

z(max), i.e.

~f z*(rnin,max) < i(max) then we apply theorem

(cf. 10.23).

for all

5

5

R. Again, x! ( a ) 1

=

0 expresses the fact that this

component of x' is not bounded from below. Theorem (10.30) implies that an optimal solution can be determined using one of the following threshold methods. W.1.o.g. c

1

e x1

c c2 8 -

X 2 5 ... <

c

we assume

exn.

(10.38) T h r e s h o l d method f o r (min,max,max,=) Step 1

k : = min{jl x!:= 3

step 2

If x '

-

x

j

c. 3

B

x 1 >-

for j 5 k

z*(min,max);

and

x!:= 0

is feasible then stop (x' optimal);

k:= k + 1. Step 3

otherwise.

1

x'k.- = xk

and go to step 2 .

Extremal Linear Programs

21 1

( 1 0 . 3 9 ) P r i m a l t h r e s h o l d method f o r (min,max,max,=)

-

Step 1

X I : =

x; k:= n.

Step 2

x i : = 0;

if x' is not feasible then go to step 4 . Step 3

k:= k

Step 4

X I : =

-

1

-

xk

Since x' in method

and go to step 3 . and stop (XI optimal).

(10.39) is feasible throughout the perfor-

mance of the algorithm, we call this method primal. To improve the complexity bound o f threshold methods it is possib'le to combine (10.38) and (10.39) and to use a binary search strategy. For example for assignment problems such an approach is described in S L O M I N S K I [19791. Then O(logz n) steps are

sufficient. The dual version frnax,min,min,=)

can be solved in

the same manner. We remark that the literature on extremal linear programs over the multiplication group of positive real numbers is discussed above (preceding 10.33).

11:

Algebraic Linear Programs

In this chapter (H,*,I) denotes a (weakly cancellative) dmonoid which is a linearly ordered or an extended semimodule over the positive cone R+ of a subring (R,+,*,i) of the linearly ordered field of real numbers. Nevertheless, we will mention some results which hold in more general semimodules over real numbers. For a discussion of the relationsh P of the different types of semimodules and for properties which hold in such semimodules we refer to chapters 5 and 6

We

assume that the reader is familiar with basic results for the usual linear programming problem (cf. for example chapter 2 in BLUM and OETTLI [ 1 9 7 5 ] ) .

We consider the algebraic linear

program N

z : = inf x T O a

(11.1)

XEP for given coefficients a

j

E H , j EJ:=

{1,2,

...,n j

and

(11.2) with an m x n matrix

C

and a vector c with entries in R. We

N

assume z E H and for simplicity of the discussion that P is nonempty and bounded. For constraints with positive entries such a problem was discussed in DERIGS and ZIMMERMANN, U. [1978a]; a great part of this chapter can be found in ZIMMERMANN, U. [1980al. The case R =

is also discussed in

FRIEZE [1978]. It is possible to introduce slack variables with cost coefficients e (neutral element of H) to transform inequalities to

212

Algebraic Linear Programs

equations.

213

W e p r e f e r t o d e v e l o p r e s u l t s explicitly o n l y f o r

i n e q u a l i t i e s o f t h e form C x

2

c. T h i s c h o i c e y i e l d s t h e s h o r -

t e s t f o r m u l a t i o n , and a s in c l a s s i c a l l i n e a r , p r o g r a m m i n g , it i s s u f f i c i e n t t o c o n s i d e r c o n s t r a i n t s o f t h i s type. E q u a l i t y

c o n s t r a i n t s Bx = b a r e r e p l a c e d by Bx

i s r e p l a c e d b y -Dx

2

2

b, -Bx

2

-b; and D x z d

-d. T h e n all r e s u l t s o f t h i s c h a p t e r c a n

easily b e c a r r i e d over. S o l u t i o n p r o c e d u r e s wil;

b e g i v e n in

t e r m s o f equality c o n s t r a i n t s w h i c h a d m i t a c o n v e n i e n t u s e o f b a s i c s o l u t i o n s w i t h i n t h e s e methods.

On t h e o t h e r h a n d , t h e p r o b l e m N

(11.3)

q:=

xT o a

sup XEP

i s n o t a s p e c i a l c a s e o f (11.1). T h i s i s d u e t o t h e fact t h a t w e c a n n o t r e p l a c e t h e c o s t c o e f f i c i e n t s by t h e i r inverses in a semigroup. I f H i s a w e a k l y c a n c e l l a t i v e linearly o r d e r e d c o m m u t a t i v e s e m i g r o u p in w h i c h axiom a > b

*

3 cEH:

(4.1) i s r e p l a c e d by a * c = b

f o r a l l a , b E H t h e n a s i m i l a r t h e o r y a s g i v e n in t h i s c h a p t e r f o r (11.1) c a n b e g i v e n f o r (11.3) s i m p l y b y r e v e r s i n g t h e o r d e r r e l a t i o n in H. A s o l u t i o n of

(11.3) in a s e m i m o d u l e a s

c o n s i d e r e d i n ' t h i s c h a p t e r can b e found u s i n g a r e d u c t i o n d i s c u s s e d at t h e end o f t h i s chapter. I n fact, i t w i l l t u r n o u t t h a t i n a c e r t a i n s e n s e (11.3) i s a s i m p l e r p r o b l e m t h a n (11.1). W e r e m a r k t h a t for t h e c l a s s i c a l c a s e s p a c e (IR

,+,z) o v e r

, 1.e. f o r t h e vector-

t h e field I R , a r e c e n t result of

HAEI J A N

119791 s h o w s t h a t (11.1) can b e solved by a method w t h

214

Linear Algebraic Optinuzation

polynomial complexity bound.

(ll.l),

Such a result is not known for

in general. We will not try to generalize the

ellipsoid algorithm described in HAEIJAN [1979] which i s based on results of SHOR ([19701, [19771). We prefer to develop a duality theory which leads to a generalization of the simplex method for (11.1) and which is applicable to combinatorial optimization problems (cf. subsequent chapters).

The first results on dual linear problems due to HOFFMAN [1963] and BURKARD [1975a] have been mentioned in chapter 10 as special cases of the weak duality theorem (10.6) which holds in ordered semimodules.

If H is a linearly ordered commutative group then we assume w.1.o.g.

that H is a module over R (via embedding as described

in proposition 5.11.3).

Then a reasonable dualization of (11.1)

is N

(11.4)

g:= sup

CTOS

sED

with

This is the same dualization as used in the weak duality theorem (10.6). Unfortunately such a dualization fails even for simple examples if H is not a module. Then C T o s is not defined if negative entries occur in C (or c ) .

Such difficulties will

appear in the discussion of equality constraints even if all entries in the original constraints are positive. In order to find a proper dualization it is helpful to split

215

Algebraic Linear Pmgram

m a t r i c e s a n d v e c t o r s i n t o " p o s i t i v e " a n d " n e g a t i v e " parts. F o r a E R let a + : =

-a .

a _ : = a+

a if a

2

For matrices

0 a n d a+ = 0 o t h e r w i s e .

(vectors) A over R we define A+

in the same manner componentwise.

and

A-

NOW

m we call S € H + d u a l f e a s i b l e if

5

c T+ O s

(11.6) If ( H , * , z ) As

Then

a*cTOs

.

i s a m o d u l e t h e n (11.6) is e q u i v a l e n t t o C T D s

5

a.

u s u a l a s o l u t i o n x E P i s c a l l e d p r i m a l f e a s i b l e and a p a i r

( x i s ) o f a f e a s i b l e p r i m a l and a f e a s i b l e d u a l s o l u t i o n i s called a f e a s i b l e p a i r . Now let a(x;s) := :=

(c-xlT

s

(C+X)TOS

r. F u r t h e r w e d e f i n e r e d u c e d c o s t c o e f f i c i e n t s

with

a f :=

e i f e q u a l i t y h o l d s in t h e j-th r o w o f (11.6). C o l l e c -

ting all equations after composition with x (11.7) As

a E 8:

(11.8)

a(x;s)

*

xToa = x

T

o a

*

j

yields

~(x;s).

w e f i n d t h e weak d u a l i t y t h e o r e m a(x;s)

*

x

T

D a

5

B(x;s).

We remark that if C does not contain negative entries then C- = 0 and t h e n a ( x ; s ) = e w h i c h i m p l i e s t h e u a u a l w e a k d u a l i t y

-

-

t h e o r e m , a s x T 0 a > ( C X ) ~ O S> c T 0 s .

I f H is a m o d u l e t h e n (11.7) i s e q u i v a l e n t t o

Linear Algebraic Optimization

216

which yields x T

0

-

a > cT

0

s,and xT

0

a = cT

0

s if and only if

In the general case, we consider the ordinal decomposition of the d-monoid H , i.e.

the family ( H A ; h E A )

as described in

chapter 4 . Such a decomposition leads to a decomposition of the semimodule (cf. chapter 6 ) . Then the i n d e x condition

plays an important role. If a(x;s) = e then ( 1 1 . 1 1 ) fied as X(e) = X o = min A . fied as A = { A o }

.

In a group ( 1 1 . 1 1 )

The importance o f ( 1 1 . 1 1 )

is satis-

is always satiscan be seen in the

following optimality criterion.

(11.12) T h e o r e m

(Dual o p t i m a l i t y c r i t e r i o n )

Let H be a weakly cancellative d-monoid which is a linearly ordered semimodule over R+. Let s € :H a(y):=

i(n(x))

dual feasible and let

( ~ - y ) ~ ~ ~s(,y ) : = ( ~ + y ) ~ n for s all ~ E R ; . If X E P ,

5

and

then x is an optimal solution o f ( 1 1 . 1 )

and

217

Algebmic Linear Rogmms

Proof. L e t Y E P . T h e n

*

a(x)

x

T

o a = B(x)

*

x

T

n a = B(x)

c -

~ ( y )* y T o a = a ( y )

< -

a(x)

*

T

*

yToa

yTna.

5

D u e t o t h e i n d e x c o n d i t i o n (11.11) and A(:) X(y

1. B(y)

T min()r(x O a ) ,

a)), cancellation o f a(x) yields optimality.

follows from

(11.12.3)

and

(11.12.4)

(11.7).

T h e o r e m (11.12) i s a g e n e r a l i z a t i o n of t h e u s u a l c o m p l e m e n t a rity criterion.

(11.12.1)

-

(11.12.3) c o r r e s p o n d t o (11.10) and

t h e i n d e x c o n d i t i o n (11.11) is a d d e d t o a s s u r e t h a t n o " d o m i n a t i n g " e l e m e n t s o c c u r in t h e i n e q u a l i t i e s .

( 1 1 . 1 3 ) Corol 1 a r y Let A be a d-monoid which i s a linearly ordered semimodule over

R,.

L e t C,c h a v e o n l y e n t r i e s i n R,.

(1)

cTos =

C T O S

*

I f x E P and

(Cx-c)Tns

*

xToa

t h e n x is a n o p t i m a l s o l u t i o n and (21

c T 0 s = xToa.

(11.14) C o r o l l a r y L e t H b e a l i n e a r l y o r d e r e d m o d u l e o v e r R. I f x E P and (1)

(cx-c)Tos = xToi = e

t h e n x is a n o p t i m a l s o l u t i o n and (2

1

C T o s = xTOa.

Linear Algebraic Optimization

218

Special cases of theorem ( 1 1 . 1 2 )

and its corollaries are used

in solution methods for combinatorial optimization problems in subsequent chapters.

A

sequence of pairs (x;s) is construc-

ted in such a manner that each pair fulfills some of the primal feasibility conditions, some of the "weak complementarity" conditions ( 1 1 . 1 2 . 1 )

-

(11.12.3)

and the index condition ( 1 1 . 1 1 )

In each step a new dual feasible solution is determined. Therefore it is possible to define reduced cost coefficients

:

(sometimes called "transformed costs"). Then a "transformation" a

+

a

is considered with respect to the respective class of

pairs.

A

sequence of such pairs is determined in order to find

a pair fulfilling all these conditions. The construction of such a sequence is rather involved for combinatorial and linear integer problems. This reflects the complicated combinatorial (integer) structure of the underlying primal problems. On the other hand optimality criteria, i.e. feasibility and "weak complementarity" are comparably easy to check for the problems considered. The most difficult condition is the index condition (11.11).

This difficulty is due to the occurence of the unknown N

optimal value z in ( 1 1 . 1 1 ) .

N

Therefore lower bounds on z play

an important role in these primal-dual or dual methods.

In as much as we can interpret "transformation" methods by certain duality principles, it is possible to use theorem (11.12)

without an explicit knowledge of duality. Let T: H n + H n

denote a "transformation" called admissible with respect to (11.1)

if there exist functions a: P

such that

x

Hn

+

H,

B: P

x

Hn

+

H

219

Algebraic Linear Programs

(11.15)

a(y,a)

*

yToa = yToi

Then x E P is optimal if (11.12.1)

-

~(y,a)

v

YEP.

(11.12.3) and the index

condition in theorem (11.12) are satisfied by a ( y ) : = a(y,a) and B(y):=

B(y,a) for all Y E P . In the same way corollaries

(11.13) and (11.14) can be viewed in terns of a transformation

concept. Such an approach was necessary since in the early discussion of algebraic optimization problems a duality theory as described in this chapter was not known. Nevertheless, the transformations were always based on the solution of certain inequalities which could be derived from the primal linear descriptions of the respective problem. Till now it has not been possible to develop transformations for problems the linear description of whiCh is not known. In an extended semimodule the "weak complementarity" conditions in theorem (11.12) can be simplified. If the index condition (11.11) is fulfilled then (11.7) is equivalent to (11.16)

* xToa = xTni ~ T o s

*

( ~ x - c ) ~ o* sc+T o s

(11.16) follows from the cancellation rule (6.17.7).

.

This is

quite similar to the case that H is a module over R. The diffeT

rence lies only in the splitting of the dual objective c 0 s into positive and negative parts. In order to obtain the equation (11.16) for all x E P we assume the s t r o n g i n d e x c o n d i t i o n (11.17)

max(;X(s

k

)I

k = 1,2

,...,m )

5 A(;).

Clearly, the strong index condition implies x(Q(Y;s)) for all Y E P . We call

s

5 a(;) m E H+ s t r o n g l y d u a l f e a s i b l e i f s is

220

Linear Algebraic Optimization

dual feasible and satisfies the strong index condition. The N

set o f all strongly dual feasible vectors is denoted by D. Then the equation (11 -16) holds for all x E P and s E E.

(11.18) Theorem

(Dual o p t i m a l i t y c r i t e r i o n )

Let H be a weakly cancellative d-monoid which is an extended 4. .

semimodule over R+. Let s E D. If x E P and (1)

CTOS = CTOS

*

(Cx-c)Tos

*

xTna

then x is an optimal solution of (11.1) and (2)

cT0s = xTOa

*

.:as.

Proof. In the same way as in the proof of theorem (11.12) we find cTos

*

xT o a

5 cT0s

*

yToa

for all y E P. Cancellation of cT 0 s is possible due to the strong index condition (11.17). (11.18.2) is a direct consequence of equation (11.18.1) and equation (11.16).

Similar remarks as for theorem (11.12) hold for theorem (11.18). The usual complementarity condition (11.14.1) in corollary (11.14) is replaced by the weak complementarity condition (11.18.1). The complementarity condition is only a sufficient

condition, but the weak complementarity condition is necessary and sufficient for the existence of a pair (xis) fulfilling a "duality theorem" (11.18.2). Till now we have described the use of certain duality principles

for optimality criteria. In order to define a dual program with respect to (11.1) we have to introduce a dual objective. If

22 1

Algebraic Linear Programs *

c

$

0 then cT 0 s is not defined. Therefore let f: D

-B

H be

defined by (11.19)

*

.To.

f(s):=

for all s EE with cIf 0 s

1. :C

C T O S 0

s or A

*

Otherwise let f(s):= e . As c T O S

(CT 0 S ) =

xT O a

2

A(cT

0

=

s)

Ao.

c T O s for all x E P

and s E 5 , we find N

f:=

(11.20)

sup f(s) 5 i n f x T n a s€8 XEP

. N

Equality in (11.20) can occur only if there are s E D with A(f ( s ) ) =

x:= A (;I.

Therefore it seems to be useful to restrict

consideration to values of the objective function which are elements of Hy. This yields another approach. It is easy to s e e is the optimal value of the index b o t t l e n e c k p r o b l e m

that

N

(11.21)

A = inf ~

(

n a) x

.~

XEP Its denotation is motivated by i ( x T o a ) = max({A(a

J

) I xJ

>

01 u

and was introduced in ZIMMERMANN, U. ([1976],

{ A ~ I )

[1979a]) for the

solution of algebraic matroid intersection problems (cf. chapter 13). N

A determination of

is possdble via a combination of the

threshold method of EDMONDS and FULKERSON [19701 and usual linear programs. In fact, (11.21) is a linear bottleneck program as considered by FRIEZE 119751 and GARFINKEL and RAO 119761 and

-

thus is a particular example of an algebraic linear program. If A is known then the algebraic linear program (11.1) reduced to an equivalent problem in a group. Let M:=

< y}. -

can be

Ijl

A(aj)

For a vector u E H m and X E A let u ( x ) denote the vector with

222

Linear Algebraic Optimization

(11.22)

if h(ui)

1,

U(ui:=

,

otherwise

..., m.

for i = 1 , 2 ,

We consider the r e d u c e d a l g e b r a i c l i n e a r

program T

N

t:= inf y YEPM

(11.23)

"'1

N

oa()OM

CMy 2 c}. This is an optimization problem

with P M'. = { y E R +

over the module G3; over R (cf. chapter 6 ) . Thus its dual is N

g:= sup

(11.24)

cTos

SED

with D:= { s c ( G Y ) ; ~ C

T M 0 s

N

N

N

5 a(h)M}. Weak duality shows g 5 t.

The relationship of these different algebraic linear programs is given in the following theorem.

(11.25)

Theorem

Let H be a weakly cancellative d-monoid which i s an extended semimodule over R,.

Let P

$

N

(a.

Hence h exists. N

(1)

(11.1)

and ( 1 1 . 2 3 )

optimal solution for ( 1 1 . 1 ) for ( 1 1 . 2 3 1 -

N

are equivalent. z = t. If x E P is an then x

is an optimal solution

M

If y i s an optimal solution for ( 1 1 . 2 3 ) then

(y,O) is an optimal solution for ( 1 1 . 1 ) . If the weak complementarity condition ( 1 1 . 1 8 . 1 )

(2)

fied for some x E P , W

s

E

'i; then

N

N

f = g = cTns(T) = x T o a =

Proof.

(1)

3

N

I

= h.

2.

If y c P M then x defined by xj:= y . for j E M and x

otherwise is primal feasible. Therefore X ( a ,)

i s satis-

Thus yT

0

a

(Y) =

x

T

x

j

= O

> 0 for some j with N

0

j

a which leads to t

2

-

z.

If x E P

223

Algebraic Linear hgmm

N

then X (xT 0 a) = X implies

is an optimal solution of ( 1 1 . 1 )

xJ

= 0 for all j d M . Thus x

Let x E P. If x NT

N

x E P with h(x Therefore

j

E P

N

M

and ~ , o a ( h ) =~ x T D a .

> 0 for some j d M then xT n a > N

m a ) = A.

7 5 y.

M

Otherwise x M E PM and

X

Together with ( 1 ) we find

an optimal solution of ( 1 1 . 2 3 )

=

zT

0

T

a for some N

M

n a ( h ) M = ~ Tm a .

y.

I f y E PM is

then x as defined in the proof

of ( 1 ) is an optimal solution of ( 1 1 . 1 1 . ( 2 ) Due to theorem ( 1 1 . 1 8 )

f(s)

=

xT

N

0

a = z. Then

s(y)

we know c:Os E D and c

T

0

=

s(7)

xTUa =

*

c T o s . Thus

complete the

proof. w

Naturally we cannot prove an existence theorem for optimal pairs or show the necessity of the assumptions in theorem ( 1 1 . 1 8 ) , as a duality gap can occur in combinatorial and integer problems even in the classical case. On the other hand, we will now develop a primal-dual simplex method in order to determine such a pair provided that R is a subfield of the field of real numbers. Then

€3

is an extended semivectorspace over R + .

For the development of a solution method we consider the algebraic linear program N

( 11 . 2 6 )

z:= inf x T n a XEP

with P:= { x E R y I Ax = bl with m

X

n matrix A of rank m over R

and b E R y . Every linear algebraic program of the form ( 1 1 . 1 ) can be transformed to such a linear algebraic program. We have chosen this form for convenience in the description of basic solutions. Another convenient assumption 1s the non-

Linear Algebmic Optimization

114

negativity of the cost coefficients a . I

, i.e. a > e for all

j EJ.

j-

This allows an obvious starting solution. Self-negative cost coefficients will play a role in the solution of ( 1 1 . 2 6 ) only N

if h ( z ) = X o ;

then it suffices to solve a reduced problem

similar to (11.23) in the vectorspace

over the field R.

G xO

Such a problem can be solved by the usual primal-dual

(or other

variants of the) simplex method which can easily be seen to be valid in vectorspaces. In the following method a sequence of strongly feasible vectors u,v E :H Ax

is determined corresponding to the inequality constraints

2 b and

(11.27)

-Ax A

2 -b. From T

+

Ou

*

(11.6) we derive

ATOv

5 a

*

T A-OU

*

T

A+Ov

and the strong index condition (11.17) reads h(ui) ,h(vi)

...,m}.

for all i E I:= { l r 2 ,

5

A(;)

n

Reduced cost coefficients a E H + are

n defined accordingly. A corresponding sequence of vectors x E R + will satisfy x T o a = e. The method terminates when the current vector x is primal feasible. Then the complementarity condition (11.10) is fulfilled which implies (11.18.1)

and hence x is

optimal, Partitions of vectors and matrices are denoted in the usual way. In particular A

j

is the j-th column of A . E denotes the m x m unit

matrix and ( 0 1 ) is a row vector with n zero-entries followed by m one-entries.

(11.28) P r i m a l d u a l s o l u t i o n m e t h o d Step

1

Let u . = vi = e for all i E I, x = -j

Step

2

Let K:= {j E J I

a I.

0

for all j E J .

= el; determine an optimal basis B

225

Algebraic Linear Programs

of the usual linear program

I

min{Zy.

E:=

A K x K + y = b; x

> 0; y

K -

2

0);

let n E R m be defined by T ~ : =

Step

3

If

Step

4

Let K : =

N

if

Step

= 0

E

z

then

{j

ElB -1

(0 1)[A

stop

EJ\KI

= @ then

(x is optimal).

T

n A

j

T

j

>

01;

a).

(P =

stop

Let d:= min { (l/n A

5

.

) 0

let ui:= ui

*

(n+)iOd

vi:= vi

*

(n-Ii 0 d

aj I

j

€;I;

for all i E I and go to step 2. Provided that (11.27) is always fulfilled we see that all steps of the method are well-defined in the underlying algebraic structure. In particular in the definition of n in step 2 and the definition of d in step 5 we observe the necessity of the assumption that ( R , + , - ) sitions

"*"

and

"0"

is a field. If we replace the compo-

by the usual addition and multiplication

of real numbers then the above method is the usual primal-dual one for linear programming. For the minimization of b o t t l e n e c k

objectives z(x):= maxja I x j

j

> 03

with real cost coefficients a E I R , j

(x E P) j E J , the method reduces

to a method proposed by FRIEZE [1975] and GARFINKEL and

RAO

19761. (11.28) is closely related to the primal-dual method in FRIEZE [1978]. Both methods reduce to the same methods for the

226

Linear AIgebmic Optimization

usual linear program and the bottleneck p r o g r a m u

z:=

min max{a.I x . > 0). 1 1 XEP

The main difference lies in the introduction of a further vector w E Hm in the constraints replacing ( 1 1 . 2 7 ) .

This leads to

a more complicated optimality criterion and a more complicated method in FRIEZE 1 1 9 7 8 1 .

In the following we give a proof of the validity and finiteness of ( 1 1 . 2 8 ) .

Due to the assumption a . > e for all j E J the 1 -

objective function of the ALP is bounded from below by e. Therefore there are only two possible terminations of the method. In the same way as for usual linear programming we see N

that K =

0 in step

4

means that the current value

f

> 0 is the

optimal value of the linear program minizy. I Ax + y = b; x

(11.29)

0, y

2

0) ;

this implies P = 0. Finiteness will follow from the fact that in step

2

a sequence of bases B for ( 1 1 . 2 9 )

It suffices to show that the value

f

is considered.

- * 0.

strictly decreases in an

iteration. For an iteration we always assume K tioned after ( 1 1 . 2 8 )

As men-

we have at first to show that ( 1 1 . 2 7 )

holds after an iteration. We denote the redefined vectors in step

2 by

i,;.

(11.30) P r o p o s i t i o n m If u , v E H + satisfy ( 1 1 . 2 7 ) ATn;

+

then ; , ; E H T

A ~ O ;

-c ~-

~

and

<,;

*0 ATO; ; *

satisfy a

.

221

Algebraic Linear Program

Proof. F o r convenience denote the left-hand-side of (11.27) by B(u,v) and the respective part on the right-hand-side

+ T T by a(u,v). Furthermore let 6 : = A+n+ + A-nAs

a E :H

in step

i,; E H+rn

implies d E H+ we find

.

5

, 6

-

T

T

: = A-r+ + A + a - .

from the definition

This definition also leads to

for all j E J . Here the distributivity axioms in the semivectorspace play an important role. At first we consider j E J T with n A

< 0. Then 6'

1 -

C

j -

As 5 ( u , v )

6-.

j

j

1. a

j

(u,v)

a

j

is

assumed we find 5 (;,;I 5 5.(;,;) * a using the monotonicity j I j of the external composition and (11.31). Secondly, let j E J with

TI

T A

j

> 0. Then j B K follows from the optimality of B in

step 2 . Therefore d f (l/TTA

j

) 0 :

.

j

Using distributivity and monotonicity this leads to

Due to o u r assumptions we know

-

5 . ( u , v ) * a . = a (u,v) * a 3

l

j

j

and therefore using (11.31) we find

In particular the proof of (11.30) shows that for a defining row k E ? of d:=

T ( l / n Ak) o a

k

we find equality in (11.32). As

228

Linear Algebmic Optimization T

>

k a K and n Ak

step

2

0 we

t h e value of

know t h a t i n t h e n e x t p e r f o r m a n c e o f E

c a n s t r i c t l y be d e c r e a s e d by i n t r o -

d u c i n g xk i n t o t h e b a s i s p r o v i d e d t h a t t h e c o n s i d e r e d p o l y hedron i n

(11.29)

is n o n d e g e n e r a t e .

As

(R,+,-) i s a s u b f i e l d

o f t h e r e a l f i e l d t h e s e t R i s d e n s e i n IR. T h e r e f o r e a n y p e r t u r b a t i o n m e t h o d c a n b e u s e d a n d w e may a s s u m e n o n d e g e n e A s t h e p r i m a l d u a l method c a n b e i n t e r p r e t e d

racy w.1.o.g..

as s p e c i a l method f o r s o l v i n g

( 1 1 . 2 9 ) we c a n o f c o u r s e a p p l y

o t h e r f i n i t e n e s s r u l e s a s f o r example t h e u s u a l criterion

( c f . BLUM a n d OETTLI

lexicographic

[ 1 9 7 5 1 ) o r t h e r u l e o f BLAND

[19771. Now i f

E

= 0 in

step

3

then x i s primal feasible. Further-

m o r e we know t h a t x T n a = e . T h e r e f o r e w e h a v e o n l y t o p r o v e

i s s t r o n g l y f e a s i b l e . Again t h e i t e r a t e d v e c t o r s

t h a t u,vEH: in step

5

a r e d e n o t e d by

i,;.

(11.33) P r o p o s i t i o n m

If u , v E H + are s t r o n g l y d u a l f e a s i b l e t h e n

i , i E

HY a r e strongly

dual f e a s i b l e . Proof.

T h e f i r s t p a r t h a s b e e n shown i n p r o p o s i t i o n

T h e r e f o r e i t s u f f i c e s t o show A d e f i n i t i o n of d i n s t e p

5

( i i ,)A ( ; i )

5

A

N

( 2 ) .

(11.30).

Now f r o m t h e

w e c o n c l u d e t h a t A ( d ) = A(:,)

for

some k E F . From t h e d e f i n i t i o n o f t h e r e d u c e d c o s t c o e f f i c i e n t s

5

we find

C

A(;).

As the

p l i e s A(d) step

5

5

A(;).

max{h(ui),X(vi),X(ak)

I

i E I).

Clearly,

h(ak)

strong index condition holds f o r u,v t h i s i m F i n a l l y t h i s and t h e d e f i n i t i o n o f

leads t o A ( G i ) , A ( i i )

5

in

w

A(z). w

2 29

Algebmic Linear Programs

As t h e

s t a r t i n g s o l u t i o n f u l f i l l s t h e index c o n d i t i o n w e con-

c l u d e from p r o p o s i t i o n

(11.33) t h a t i n t h e p r i m a l d u a l method

a sequence o f s t r o n g l y d u a l f e a s i b l e s o l u t i o n s u , v is d e t e r mined.

T h e r e f o r e i f t h e method t e r m i n a t e s i n s t e p

w e have

3

found an o p t i m a l s o l u t i o n x .

The p r i m a l d u a l s o l u t i o n method y i e l d s a c o n s t r u c t i v e p r o o f f o r t h e f o l l o w i n g s t r o n g d u a l i t y theorem.

( 1 1 . 3 4 ) Theorem

( L i n e a r programming d u a l i t y theorem)

L e t H b e a w e a k l y c a n c e l l a t i v e d-monoid

w h i c h is a n e x t e n d e d

I f P is bounded and nonempty t h e n t h e r e

s e m i v e c t o r s p a c e o v e r R+.

e x i s t s an optimal b a s i c s o l u t i o n Y E P with b a s i s B f o r t h e l i n e a r a l g e b r a i c program

(11.26)

-T

d u a l f e a s i b l e s o l u t i o n €:,; (1) (2)

bT

0 ;

= YToa

bT

0';

*

(-b)

and t h e r e e x i s t s a s t r o n g l y

HY s u c h t h a t x

*

0

-a

= e and

b T o y ,

'oG(X)

= GToa

for

X

= ~

-T ( n xa).

N

We r e m a r k t h a t t h e d u a l o b j e c t i v e f u n c t i o n f : D + H

Proof.

on t h e s e t o f a l l s t r o n g l y d u a l f e a s i b l e s o l u t i o n s

( u , v ) €;

is

g i v e n by

i n t h e same way a s f ( s ) i n f o l l o w s from t h e o r e m s

(11.19).

(11.18) and

T h e c l a i m e d r e s u l t now

(11.25)

interpreted for linear

programs w i t h e q u a l i t y c o n s t r a i n t s and a p p l i e d t o t h e f e a s i b l e pair

-

-

N

( x ; u , v ) w h i c h is d e t e r m i n e d b y t h e p r i m a l d u a l s o l u t i o n

method

(11.28).

Linear Algebraic Optimization

230 N

We r e m a r k t h a t t i o n s of

N

N

( x ; u I v ) i n theorem

( 1 ) and

(2)

i n theorem

(11.34)

s a t i s f i e s t h e assump-

(11.25); therefore the corres-

ponding a l g e b r a i c l i n e a r programs have i d e n t i c a l optimal v a l u e s N

(t =

W

2

N

N

= f = 9)

-

Next w e w i l l s t a t e a s t r o n g d u a l i t y theorem from matching t h e o r y which h a s b e e n g i v e n i n ZIMMERMANN, U .

[ 1 9 8 0 a ] b a s e d on t h e

r e s u l t s i n D E R I G S [ 1 9 7 9 a ] a n d E D M O N D S a n d PULLEYBLANK 1 1 9 7 4 1 .

Let G =

(V,N)

d e n o t e a c o m p l e t e g r a p h w i t h e v e n number

Then a s u b s e t M

vertices.

c a l l e d a p e r f e c t matching only one edge e

IrV

N:=

U {eijl

i , j EV,

jl is

E M w i t h l~ = i o r v = i . W e c o n s i d e r s u b s e t s tElN.

of t h e s e s u b s e t s .

I

L e t NT:=

U {eij

a n d d e n o t e t h e s e t o f a l l N T b y FN.

L e t FT d e n o t e t h e s e t

i , j ET,

i

E F N with

t h e e d g e s of G.

of C corresponding t o edge e

ij'

j ) for TEFT

$

Now l e t B denote t h e vertex-

e d g e i n c i d e n c e m a t r i x of G and C t h e s e t - e d g e T

*

of

i f for a l l i E V t h e r e e x i s t s o n e a n d

T C V w i t h odd c a r d i n a l i t y 2 t + l ,

of sets N

1

IVI

Let C

i j

incidence matrix

d e n o t e t h e column

Then t h e v e c t o r w i t h components

c := t f o r T E F T i s d e n o t e d b y c . T

EDMONDS a n d PULLEYBLANK 1 1 9 7 4 1 h a v e shown t h a t t h e v e r t i c e s o f (11.35) a r e i n one-to-one

correspondence with t h e incidence vectors of

t h e p e r f e c t matchings o f G. respect t o

Thus w e c o n s i d e r

(11.1) with

( 1 1 . 3 5 ) . T h i s problem is s o l v e d i n DERIGS

f o r c o e f f i c i e n t s w i t h ai, E H + f o r a l l

(i,j) E N .

[1979a]

An i n c i d e n c e

v e c t o r x of a p e r f e c t matching and a d u a l f e a s i b l e s o l u t i o n ( u , v , s ) i n t h e s e n s e of satisfies

(11.6) a r e determined;

1.e.

(u,v,s)

23 1

Algebraic Linear hograms

u

i

*u

< aij j -

* Vi * v, * CTi j

0 8

for all ( i , j ) E N , u i , v i E H + for all i E V and s T E H for all

TEFT.

Furthermore (u,v,s) is strongly dual feasible and

satisfies (11.36)

(Cx), < dT l = x i j 'O

*

-

sT = e

ui*'j

- aij

-

* vi * vj * cTi j 0 s .

(11.36) implies the complementarity conditions. Application of theorems (11.18)

and (11.25) yields the following results

on duality.

( 1 1 . 3 7 ) Theorem Let

H

( P e r f e c t matching d u a l i t y theorem)

be a weakly cancellative d-monoid which is an extended

semimodule over R+. Then there exists an optimal feasible pair '(x;urvrs)with (11.36) and

(2)

l T o u * (-l)Tov(?)

*

(-d)T

Os('xy)

N

A(x

for A : =

= xToa

T

ma).

The solution method for such a combinatorial optimization problem is developed similar to the classical case (the vectorspace ( I R , + , f )

over IR

).

Since feasibility and complementarity

known from classical linear programming is used in such methods, the development of similar criteria w a s made at first in the form of a transformation concept. The duality concept described in this chapter shows a way to interpret these transformation concepts again by certain generalizations of the classical duality concept. It should be emphasized that the construction

232

Linear Algebraic Optimization

of optimal pairs as in theorem (11.37) fulfilling the generalized feasibility and complementarity criteria makes full use of the special combinatorial structure of the respective problems. In subsequent chapters we will discuss methods in network flow theory and matroid intersection theory. Matching theory can be investigated using the same concepts. The main difference lies in the combinatorial structure considered for the respective problems. Therefore for results in matching theory we refer to a series of papers by DERIGS ([1978a], [1978bl, [1979a],

[1979b]).

In the development of primal dual methods for the solution of combinatorial optimization problems it turned out to be

very important to use lower bounds on the index of the optima1 value f o r a reduction of the problem. Let p

N

1. X ( z )

be a

lower bound of the index o f the optimal value of (11.26). Then ( 1 1.26' 1

N

z = inf XEP

xToa(p)

N

and x E P is an optimal solution of (11.26) if and only if it is an optimal solution of (11.26'). During the performance

of the primal dual solution method a strongly dual feasible solution u,v is known. Then u(u).v(v)

is strongly dual feasible

with respect to (11.26'). Further, if u,v is strongly dual feasible then 1.1:"

max{h(ui) ,A(vi) I

is a suitable lower bound of A(;).

i E 11

Thus we may modify step 5

Algebraic Linear Rograms

233

in the primal dual solution method by an additional revision of the dual variables.

(11.28') Modification o f s t e p 5 i n ( 1 1 . 2 8 )

Further the reduced cost coefficients (11.27' )

A

T

+

nu

*

*

A T ~ V

i

=

a(p)

a

are determined f r o m ,

*

T A-nu

*

T ~ + o v .

,In each iteration all dual variables have index 1.1 or are equal to the neutral element e. As A(d) = A ( : . ) j €;

and a s

aj

1

for some

is determined from (11.27') the index A(d)

satisfies A(d) 5 max{A(ui,X(vi) I i E I} with respect to the current dual variables b e f o r e the performance of step 5 .

u:= A(d) yields the same index as

p : = max{A(ui) ,A(vi)

1

Thus

i E I]

a f t e r the performance of step 5 . The modification results in a possibly enlarged se,t K in step 2 ; therefore the number of iterations may be reduced. The fact that all dual variables which are not equal to e have the same index is very helpful in proving finiteness of special primal dual methods for combinatorial optimization problems.

234

Linear Algebraic Optiniimtion

Next we consider a solution o f algebraic linear programs with primal methods. We develop primal optimality criteria for (11.26). In the discussion of the primal dual method we assumed that u , v E H m were dual feasible. Now we assume only that the given vector x is primal feasible. The respective dual solutions are not necessarily dual feasible. Therefore optimality criteria as

in theorems (11.12) and (11.18) cannot be applied. This is

essentially due to the fact that a definition o f reduced cost coefficients

a

is possible only if a dual feasible solution is

known. In particular, a primal method will not be covered by the transformation approach. Primal methods for algebraic linear programs are considered in ZIMMERMANN, U .

[1980a].

In the following discussion we assume that P (as defined for (11.26)) is nonempty and bounded. H is a weakly cancellative

.

6-monoid which is an extended semimodule over R

N o w let

x EP

with respect to

(11.38)

i:= X(iT

and

x

a). w E

Gm

x

is called i - c o m p t e m e n t a r y

if

x . > o 3

*

(cf. 11.22). Further W E

(11.39)

0

ow 5

T

A . 0

7

-

w = a(i)

t/

j

G X ) m+ is called

j E J

i - d u a t f e a s i b l e if

a(i)

I f H is a module over R then x-complementarity and i-dual feasibility coincide with dual feasibility (cf. 11.5) and complementarity (cf. 11.10).

235

Algebraic Linear Program

Let D ( X ) denote the set of all i-dual feasible vectors. Then yields bT

composition of (11.39) with N

w E D(h) and therefore with A:=

0

w

5

-T x 0 a for all

X ( y ) we find

This is another dual program f o r A L P . F o r a !-complementary -T vector w we find bT O w = x O a . This concept leads to the

following optimality criterion.

(11.41) Theorem

(Prima

optimality criterion)

Let H be a weakly cance lative d-monoid which is an extended

x E P and x and 1-dual

semimodule over R+. Let

!:=

mentary with respect to

optimal solution o f (11.26) and N

h = b

(1)

T

-T O W = X

7

implies optimality of

oa). If w is i-comple-

feasible, then

x

is an

-

O a = z .

Proof. Due to the assumptions we find for all y E P. If 1 =

-T

X(x

GT 0 a

= b

T

0

w

5 yT

0

a (1)

then y T O a ( X ) = y T o a for all Y E P . This

x.

Otherwise let Y E P denote an optimal

-T -T solution. Then y O a ( 1 ) = e < x O a yields a contradiction. Now

=

follows f r o m (11.40).

Theorem (11.41) shows that a special type of complementarity and dual feasibility implies optimality without an additional

136

Linear Algebraic Optimization

index condition. Thus ( 1 1 . 3 8 )

can replace ( 1 1 . 1 8 )

if complemen-

tarity is used as optimality criterion. Furthermore it can be seen that if (11.41.1)

is an optimal solution, w 1-dual feasible and

holds then w is 1-complementary. Otherwise if

optimal, w is i-complementary and ( 1 1 . 4 1 . 1 )

x

is

holds then in gene-

ral it is not guaranteed that w is h-dual feasible. In this sense ( 1 1 . 4 1 )

describes only a sufficient optimality criterion

even if a result on strong duality holds. A comparison of ( 1 1 . 4 0 )

with the other duals ( 1 1 . 2 0 )

shows that if the assumptions o f ( 1 1 . 4 1 )

and ( 1 1 . 2 4 )

hold then all optimal

objective values are equal. In this case all three approaches are equivalent. If X-dual feasibility is used as optimality criterion and w

is 1-complementary then it suffices to check (11.42)

-

for x

j

= 0 and A(a.1 1

To use theorem

1. 1. Then

(11.41)

(11.42)

is equivalent to ( 1 1 . 3 9 ) .

in a primal method we have to consider

conditions for the existence of 1-complementary solutions. In primal methods basic solutions play an important r o l e . If B is a feasible basis for P then x (11.43)

B

=

AilbER:.

If we assume

det A - 1 E R B

for all feasible bases of P then the existence of a X-complementary solution with respect to a feasible basic solution

x

is

guaranteed. The existence of an optimal basic solution for bounded, nonempty P follows from theorem ( 1 1 . 3 4 )

in the case

that R is a subfield of I R , i.e. provided that H is a weakly

231

Algebmic Linear Program

cancellative d-monoid which is an extended semivectorspace over R+. Clearly, (11.43) is valid in this case and we can N

assume that P is nondegenerate. Let P denote the non-

w.1.0.g.

empty set of all feasible basic solutions. In the following we develop a primal simplex method in such semivectorspaces.

Let G E v with basis bor of

;if

neighbors of -T x oa

IB

n iI

i . Then

i.

Then

= m -1.

xEF

x E F with

basis B is called a neigh-

Let N ( i ) denote the set of all

is called l o c a t l y o p t i m a l if

T

5 x m a for all XEN(;).

(11.44) Proposition Let H be a weakly cancellative d-monoid which is an extended N

semivectorspace over R+. Then G E P is an optimal solution of the algebraic linear program ( 1 1 . 2 6 )

if and only if

x

is locally

optimal. Proof. We only show the nontrivial if-part. Let

G

be locally

n d

optimal and let x E P denote an optimal solution. Then we know from linear programming that

with uxEIR+. A s (R,+,.)

is a subfield of (lR,+,-) we can show

p x E R + for all xEN(x).

Therefore

NT x oa

with

u:= 1

pX.

*

u

(iTna)

2

( i + u )0 ( i T o a )

Cancellation due to ( 6 . 1 7 . 7 )

yields NT x oa

1. x-T m a . a

From the proof we see that proposition (11.44) may be invalid if the semimodule is not extended. In particular this may happen

Linear A lgebmic Optimization

238

for the linear bottleneck program. Proposition

(11.44)

again

is attained at a

shows that the optimal value of ( 1 1 . 2 6 )

feasible basic solution. We can state a necessary and sufficient optimality conditi.on and another strong duality theorem.

(11.45) Theorem

( L i n e a r programming d u a l i t y theorem)

Let H be a weakly cancellative d-monoid which is an extended -

N

semivectorspace over R+. Let x E P with basis

and

x:=

h(XOa).

Then the following statements hold:

x

(1)

is an optimal solution of ( 1 1 . 2 6 )

w:=

0

a(X),

if and only if

is x-dual feasible:

(2)

there exists an optimal basic solution:

(3)

the optimal values of the considered dual programs ( f l . 2 0 1 , (11.24)

and ( 1 1 . 4 0 )

Proof. Due to ( 1 1 . 4 1 )

coincide with

T.

it suffices to show for ( 1 ) that

x

opti-

mal implies that w is 1-dual feasible. If w is not X-dual feasible then there exists a E Hi

,

ci

B = g\{r}

s

6

> e. Then bT

U

{ s }

with A(a 0

w = xT

0

)

5

x

T

0

w = a

(x) * a

with

a(1) t (xs 0 a ) for x E 'i; with basis

(Existence o f such a basic solution x follows

from linear programming and ( 1 1 . 4 3 ) ) . x

and A

Since b

T

-T

o w = x

m a and

-T T > 0 we find x m a > x O a ( X ) which implies the contradiction

xTOa

> xTOa.

(11.25)

(2)

follows from ( 1 1 . 4 4 )

and ( 3 ) follows from

if the equality constraints of ( 1 1 . 2 6 )

are replaced by

inequality constraints in the usual manner.

Similar to the classical case theorem ( 1 1 . 4 5 ) (11.44)

and proposition

imply the validity of the following primal simplex method

Algcbmic Linear Programs

239

for algebraic programming problems. We give an outline of the method, showing the differences to the classical case.

(11.46) P r i m a l s i m p l e x m e t h o d

€y

Step 1

Find

Step 2

Calculate w:= if A

Step 3

T 1

w

5

Choose s d

6

1 T ( -~ i oa(h)E

determine xEN(;) redefine

x,

and

If the method stops in step

5

x

a).

;

a . (1) for all j d 1 with h (as)

-T h(x

h:=

E;

with basis

i

and

with h(a.) 1 Af 0

5

then stop.

w > as(:);

with basis B containing s ;

1

and return to step 2 .

2

then w is h-dual feasible and

theorem (11.41) yields optimality. Finiteness of (11.46) can be seen from the proof o f (11.4s). A basis exchange step due to step tion. A s

3

decreases strictly the value of the objective funcis a finite set this implies finiteness. Primal

degeneracy can be handled with some of the usual finiteness rules (cf. BLUM and O E T T L I 119751 or B L A N D [1977]).

If H is a weakly cancellative d-monoid which is an extended semimodule over R + then the primal simplex method can be applied provided that (11.43) is satisfied. In particular, (11.43) holds if the constraint matrix A is t o t a l l y u n i m o d u -

l a r , i.e. or

+1.

A

contains only submatrices o f determinant 0

Linear Algebraic Optimization

240

The primal simplex method is applied to algebraic transportation problems in chapter 12. The optimality criterion (11.42) can in this special case be improved such that it suffices to check a less number of feasibility conditions.

We have developed two solution methods f o r algebraic linear programs which simultaneously solve the corresponding dual programs (11.20), (11.24) and (11.40). These duals have objective function and constraints with values in the semimodule H. Therefore, in general, such problems seem to have a structure differing from algebraic linear programs. We will now consider the solution of such problems. The dual (11.20) of ( 1 1 . 2 6 ) has an objective function f: H Y b

if b

T

ov

5

b

T

T

*

Ov

ou

:H

f(u,v):= b A(b

01

X

T

T

+

H defined by

Ou

o v ) = X(b

T

Ou) =

Xo

and

f (u,v):= e

otherwise. The set of all strongly dual feasible solutions consists in all (u,v) EX:

x

5

HY with

(11.46) (11.47)

f o r all i E I. If H is a vectorspace over R then

is equivalent to

(11.48)

sue WED

with 6:= { W E Hml

A

T

b

T

O w

ow

5

a}. This form l o o k s much more familiar

Algebmic Linear Fmgram

24 1

and does not contain an explicit index condition with respect to the optimal value of the original primal problem. Such a problem can be solved by the application of the generalized simplex methods to ( 1 1 . 2 6 ) . the dual of ( 1 1 . 4 8 ) .

Here we may interpret ( 1 1 . 2 6 )

as

In the general case of a semivectorspace

difficulties arise from the fact that the external composition (otherwise

a nontrivial extension to R

implies that H is a modul over R

and that variables in H are

is only defined on R +

more o r less positive ( A ( a ) > A.

-D

a

-1

does not exist). There-

fore, splitting of variables and matrices leads to constraints of the form ( 1 1 . 4 6 ) .

If the strong index condition ( 1 1 . 4 7 )

be assumed without changing the problem then ( 1 1 . 2 6 ) be interpreted as a suitable dual. If ( 1 1 . 4 7 )

I s

can

again may

not added to

the constraints ( 1 1 . 4 6 ) and is not implicitly implied by these constraints then even weak duality may fall f o r the pair of programs ( 1 1 . 2 6 )

and

SUP f (u.v) (u,v) E D '

(11.48)

with D':={(u,v) E H ~ x+ H A ~ TIO A solution of

(11.48)

u

*

T A-OV

5 A T- O U

*

ATOV

*

a).

is not known in general.

Based on the max-matroid flow min-cocircuit theorem for algebraic flows in HAMACHER [1980a] we consider an example for a problem of the form ( 1 1 . 4 8 ) .

For fundamental definitions and

properties o f matroids we refer to chapter 1 . Let M denote a regular matroid defined on the finite set

I = {0,1,2,

...,m}.

SC resp. SB denotes the set of all circuits

resp. cocircuits. Let A denote the o r i e n t e d i n c i d e n c e matrix of elements and cocircuits, 1.e.

242

Linear Algebraic Optimization

i E B+

1

i E B-

i 6 B m for all i E I and all B E SB. For a given capacity-vector c E H+ the elements of D:= {(t,u)I

T

A+

t o( ) = U

T

t

A-• (u), u z c ,

m u E H + , tEH+j

are called a L g e b r a i c m a t r o i d fZolJs in HAMACHER [1980a].

The

m a s - m a t r o i d flo1.7 p r o b l e m is N

(11.49)

t:=

max (t,u) E D

t .

Now we consider its formal dualization (11.50)

N

z : = min XEP

x

T

O c

with the set of feasible solutions

HAMACHER [1980a] develops an algorithm which determines an optimal solution

for a l l i = 1 , 2 ,

(7,;)

...,m.

of (11.49) with the property

Furthermore the algorithm yields a

constructive proof for the existence of a cocircuit

with O E Z -

such that N

(11.52)

t = (alg,a2;,...,am;)+

T

oc

and N

t

5 (a1B.a2B,...,amB)+

for all B E S B w i t h . O E €3-.

T

o c

This implies directly the max-matroid

f l o w - m i n cocircuit theorem in HAMACHER [1980a].

Furthermore it

Algebraic Linear Programs

243

leads to the following strong duality result for (11.49) and (11.50).

(11.53) Theorem

(Matroid f l o w d u a l i t y theorem)

Let H be a weakly cancellative d-monoid which is an extended semimodule over R+. Then there exists an optimal pair of a N

N

feasible primal solution (t,u) E D and a feasible dual solution

-

-

NT

x E P such that t = x

oc.

Proof. At first we show weak duality, i.e. t N

(t,u) E D , x E P . Clearly t of (11.49). W.1.o.g.

5 xT O c for all -

N

5 t for all optimal solutions (t,u)

we assume that (11.51) holds. x E P implies

the existence of y E R

SB

with Ay

1.

[-:I.

Using distributivity

and monotonicity in the semimodule we find

(11.51) shows that we can cancel the first terms on each side (cf. 6.17.7) which are equal due to the equality constraints N

in the definition of D. Now u semimodule imply

75

Secondly cocircuits

xT

0;

G E SB

5 c and monotonicity in the

5 xT

0

c.

with 0 E Z- correspond to certain

elements of P defined by

x := i

0

N

otherwise.

-

N

Indeed, yB:= - 1 if B = B and y := 0 otherwise, shows feasibility B

of

F.

-T Clearly x o c = (al;,a2;,...,am;)+

implies

T

=

T o c . Therefore (11.52)

YToc. rn

244

Linear Algebraic Optimization

In fact the proof shows that (11.50) has always a zero-one solution which is the incidence vector of the positively directed part of a cocircuit. As in particular we may consider the usual vectorspace

( l R , + , z )over IR

a direct conse-

quence is that all vertices of the polyhedron

are in one-to-one correspondence with the set of all cocircuits B E S B with O E B - for a given regular matroid M.

A detailed study of the matroid flow problem is given in XAMACHER [1980b].

Related problems are discussed in BURKARD

and HAMACHER 119791 and HAMACXER [198Oc].

At the beginning of this section we gave some remarks on problem (11.3) which i s also not equivalent to an algebraic linear program of the form (11.11, in general. (11.3) may be solved via the solution of an index problem of the form N

a : = sup X(xTna)

(1 1 . 5 4 )

XEP similar to (11.21). Its objective is A(x

If

T

n a ) = max({a(a,)l

-

xJ

.

> 01 u ( ~ ~ 1 )

i s known then ( 1 . 3 ) can be reduced to a problem over the

module

GY. A

Let M:= {j

A(aj)

1. A } .

Then we consider the reduced

problem (11.55)

- 1

q

with PM:= { y E R

I

CMy

2 c}. This is an optimization problem

245

Algebmic Linear h a r a m

over the module

GT

which is equivalent to the algebraic

linear program inf YEPM

y

T

ob

for all j E M. The relationship between ( 1 1 . 3 )

with b := j

and ( 1 1 . 5 5 ) is described by the following theorem.

(11.56) T h e o r e m Let H be a weakly cancellative d-monoid which is an extended semimodule over R+. Let P

* @.Hence x N

exists. Then ( 1 1 . 3 )

N

and ( 1 1 . 5 5 ) are equivalent (q = q').

If x E P is an optimal solu-

tion for ( 1 1 . 3 ) then xM is an optimal solution for ( 1 1 . 5 5 1 .

If

y E P M is an optimal solution of ( 1 1 . 5 5 ) then (y,O) is an optimal solution of ( 1 1 . 3 ) .

-

Thus g: P P M with NT is a bijective mapping. Let x E P with A(x m a ) = A .

Proof. If x E P then x

j

= 0 for all j b M .

N

#u

g(x):- x

M

NT T NT Then x T O a < x O a and x ~ O ~ ( =A e) <~ x O a for all x E P with T A (xT a) < On the other hand x 0 a = xT 0 a(A), for all x E P M T with A(x 0 a) = A . Therefore q = q'. N

x.

N

N

N

N

We remark that a solution of the index problem ( 1 1 . 5 4 ) is trivial if there exists x E P such that xK > 0 for A(ak) = max{A(a

j

1I j EJ)

.

This condition is fulfilled if there is no trivial variable x 1.e.

j'

x, = 0 for all x E P . Such a reasonable assumption will

often hold in the application of our duality theory to combinatorial optimization problems in subsequent chapters. In general,

Linear Akebmic Optimization

246

the determination of such trivial variables can be performed via suitable usual linear programs.

Finally we consider some examples for algebraic linear programs. As

the difference to the usual linear program lies only in the

choice of other objective functions which are linear with respect to the semimodule considered it suffices to discuss such objectives and the corresponding semimodule. All d-monoids at the end of chapter 4 lead to such examples as every d-monoid i.e. a discrete semimodule. is a semimodule over Z + ,

(11.57)

D-monoids

i n t h e r e a l a d d i t i v e group

The linearly ordered modules (IR

,+,z), (Q,+,z)and

,+,z) over

(Z

R E {IR , Q , Z 1 . The objective function is the classical one, i.e. x T o a = ~ xa j

(sum o b j e c t i v e

j ’

We remark that the choice of R is restricted (cf. 6 . 6 ) . Similar modules (cf. 4 . 2 0 )

lead to the objective functions X.

xToa =

n

j

xT o a = (1 x with 0 c p <

m.

(retiabiZity objective)

a 3

j

aPll’P j

The case p =

-

(p-norm o b j e c t i v e )

may be interpreted as bottleneck

objective (cf. 1 1 . 5 9 ) .

(11.58) D-monoids

i n Hahn-groups

k The linearly ordered modules (IR ,+, 4 1 ,

k ( Q ,+, 4 1 ,

over R E { W , Q , Z 1. The objective function is

k

(Z

,+,d

1

Algebraic Linear Programs

with n x k matrix A = (a

x

T

X

n a = a

ji

l

1

r a t h e r g e n e r a l o b j e c t i v e is

):A

X2 *a2

247

X

(group-objective)

* . . . * an (H,*,L)

f o r l i n e a r l y o r d e r e d modules

o v e r 22. S u c h a m o d u l e

c a n a l w a y s b e embedded i n a Hahn-group a linearly ordered vectorspace over

(cf. 4.21),

and t h u s i n

IR ( c f . d i s c u s s i o n a t t h e

end o f c h a p t e r 6 ) .

(11.59) Weakly c a n c e l l a t i v e d - m o n o i d s The e x a m p l e s ([0,11,*,5) and a * b : =

given i n

(4.22)

with a * b : = a + b

-

ab

( a + b ) / ( l + a b ) lead t o objective functions of t h e

form

xT n a = 1

w i t h Aa : = j

T[

j

( 1 - a

)xj

j

I

for j EJ.

(1 - a , ) / ( l + a j )

The n a t u r a l l y , max,l)

-

l i n e a r l y o r d e r e d b o t t l e n e c k monoid

i s a l i n e a r l y o r d e r e d s e m i m o d u l e o v e r R,

(IR

u {--I,

with R E {IR,Q,ZZ

1.

Its objective function reads

z ( x ) = x T o a = max{a

j

I x

j

> 01.

(bottleneck objectioe)

T h e c o r r e s p o n d i n g e x t e n d e d s e m i m o d u l e is d e r i v e d f r o m t h e time-cost

semigroup ( c f . 4 . 2 2 ) .

x

T

0

(a,b) = ( z ( x )

I

I t s o b j e c t i v e f u n c t i o n is

1 aj=z (x)

x b 1. j j

(time-cost o b j e c t i v e )

248

Linear Algebraic Optimization

Linear optimization problems with bottleneck objectives have been discussed by F R I E Z E [19751 and G A R F I N K E L and R A O 119761. Bottleneck- and time-cost objectives with one n o n l i n e a r p a r a -

m e t e r are considered in Z I M M E R M A N N , U. [1980b]. In general, A-dual feasibility characterizes the set of parameter values for which a given basis is optimal in a similar manner a s in the classical case (cf. theorem 11.45.1).

A

particular case of a homogeneous weakly cancellative d-monoid

(cf. 4 . 2 2 )

H:=

is the set { (0,a)

I

n aElR

k O)

U

U {(u,bl

I

n bEIR+’\{O)).

u=l An internal commutative composition on

is defined by

Together with the lexicographical order relation this leads to an extended semivectorspace with objective function

the time-cost objective is a special case o f this objective.

(11.60) C o n d i t i o n a l l y c o m p l e t e d - m o n o i d s The d-monoid (IR

x

[0,1],*,*)

as defined in ( 4 . 2 3 )

is an example

for a discrete semimodule which is not weakly cancellative. The corresponding objective function is similar to the time-cost objective (cf. 11.59)

249

Algebraic LinearprOgmtm

x

T

1

( a r b ) = (z(x),min(

a =z(x) j

x b 1)) j j'

.

We r e m a r k t h a t t h i s s e m i m o d u l e c a n n o t be e x t e n d e d t o a s e m i module over R+ with

R

+

7 =+

(cf. c h a p t e r 6). F r o m t h e s t r u c -

ture of the conditionally complete d-monoids described in

(4.23) w e s e e t h a t t h e c o r r e s p o n d i n g o b j e c t i v e s h a v e a s i m i l a r structure.

(11.61) G e n e r a l d - m o n o i d s CLIFFORD'S

e x a m p l e i n (4.24) y i e l d s a d i s c r e t e s e m i m o d u l e w h i c h

c a n n o t b e s u b s u m e d u n d e r p r e v i o u s l y d i s c u s s e d classes.

An interpretation o f the results in this chapter f o r the differ e n t t y p e s of o b j e c t i v e f u n c t i o n s m e n t i o n e d is l e f t t o t h e reader. In particular, t h e strong duality theorems in this chapt e r a s w e l l a s in t h e f o l l o w i n g c h a p t e r s s h o w t h e c l o s e r e l a tionship o f all these optimizhtion problems which are treated in a unifying way using an algebraic approach.

T h e f o l l o w i n g e x a m p l e c o n t a i n 8 a n a p p l i c a t i o n of t h e r e s u l t s i n t h i s c h a p t e r t o an e i g e n v a l u e p r o b l e m d i s c u s s e d i n c h a p t e r 9.

(11.62) A f i n i t e e i g e n v a l u e p r o b l e m Let

(R',@,Z)

be a linearly ordered, divisible and commutative

group. T h e n R' is a v e c t o r s p a c e o v e r t h e f i e l d Q o f a l l r a t i o n a l numbers with external composition defined by (m/n) ma:=

a m/n

f o r a l l m / n E Q a n d for a l l a E R ' . T h e o r e m

(6.5)

s h o w s t h a t R'

Lineat Algebraic Optimization

250

> 1 then R' i s

is a linearly ordered vectorspace. If 1R'I

an extended vectorspace over Q; this will be assumed w.1.0.g. in the following. The finite eigenvalue problem consists in the determination of h E R', y E (R')n such that max{a

(11.63)

ij

@ y

for all i E N : = {l,2,

j

I

...,n]

j = 1,2,

...,nI

h

=

yi

with respect to a given n x n matrix

(a ) over R' u ( 0 ) (0 denotes an adjoined zero). The soluij tions of such a problem are characterized in theorem ( 9 . 2 2 ) and

A =

are discussed subsequent to this theorem.

-

GA

denotes the di-

graph with set of vertices N and arc set E defined by (i,j)E E

:

a

i j

* O

-

for i,j E N . If a finite eigenvalue A E R ' exists then

G

A

contains

an elementary circuit p with length l ( p ) and weight w(p) such that

1

=

(l/l(p)) nw(p). Then further

We assume that

G

A

contains at least one circuit. Otherwise the

finite eigenvalue problem has no solution.

Let a denote the vector with components a i j

,

(i,j)E E .

We

consider the algebraic linear program d .,

(11.64)

w:= max XEP

xTna

with set P of feasible solutions described by the linear constraints

(11.65)

25 1

Algebraic Linenr Proflams E

a n d x E Q + . I n m a t r i x f o r m w e d e n o t e t h e s e e q u a t i o n s by C x = c. T h e n t h e d u a l o f (11.64) i s N

(11.66)

min sED

A:=

cTus

w i t h D = { s E (R')n+ll C C

T

0

s

2

T

0 s

2

a).

The inequality constraints

a have the form

(11.67)

si

-1 sj

@

8 s

n+l

> -

a

ij

for a l l ( i , j ) € E . D i s a l w a y s n o n e m p t y a s it c o n t a i n s s d e f i n e d

by si:= e f o r I E N a n d s,+~:=

maxis

ij

I (i,j) E E )

(e d e n o t e s

t h e n e u t r a l e l e m e n t o f R'). If

1 is

a f i n i t e e i g e n v a l u e and y E ( R t I n a f i n i t e e i g e n v e c t o r

is a dual feasible solution with then s = ( Y , X ) ~ cTos =

1=

A(A)

.

Let p denote an elementary circuit in G

A

with

x

=

(1/1 ( p )) O w ( p )

k h e n x E Qy d e f i n e d by

{

=

x. lj

1/1 (PI

if (irj) E P ,

0

otherwise

,

is a primal feasible solution with x

T

o a = (l/l(p)) nw(p).

-

D u e t o t h e w e a k d u a l i t y t h e o r e m x T O a < cT u s f o r a l l f e a s i b l e N

pairs

N

thus A = w = A(AI =

(x;s);

1.

If no finite eigenvalue exists then the existence of a circuit in GA i m p l i e s

y 2

(11.34) s h o w s

=

M

N

h(A).

2

Further the strong duality theorem

A(A).

In any case A = w can be determined using the methods proposed in this chapter for the solution of linear algebraic programs

.

Linear Akebraic Optimization

252

(cf.

11.28 and 11.46).

As

(11.64) i s a maximization problem

t h e s e methods which a r e f o r m u l a t e d f o r m i n i m i z a t i o n p r o b l e m s h a v e t o be m o d i f i e d i n t h e u s u a l manner ( e . g . r e p l a c e a by ij -1 w a i j f o r 1 1 . 4 6 , e t c . ) . The o p t i m a l v a l u e A i s u s e d i n c h a p t e r 9 e i t h e r t o compute t h e f i n i t e e i g e n v e c t o r s ( i f

1

exists) or to

v e r i f y t h a t t h e f i n i t e e i g e n v a l u e p r o b l e m h a s no s o l u t i o n .

For t h e s p e c i a l c a s e o f t h e a d d i t i v e g r o u p of r e a l numbers t h i s a p p r o a c h is p r o p o s e d by D A N T Z I G , B L A T T N E R a n d R A O 1 1 9 6 7 1 and i s d i s c u s s e d i n CUNINGHAME-GREEN [ 1 9 7 9 ] .

12.

Algebraic Flow Problems

In this chapter we discuss the minimization of linear algebraic functions over the set of feasible flows in a network (algebraic flow problem). Although we assume that the reader is familiar with basic concepts from flow theory we begin with an introduction of the combinatorial structure which is completely the same as in the classical case. Then we develop the primal dual solution method which is a generalization of the min-cost max-flow algorithm in F O R D and F U L K E R S O N 1 1 9 6 2 1 . We discuss three methods for the minimization of linear algebraic functions over the set of feasible flows in a transportation network (algebraic transportation problem). These methods are generalizations of the Hungarian method, the shortest augmenting path method and the stepping stone method.

The underlying d-monoid H is weakly cancellative and an extended semimodule over the positive cone R + of a subring of the linearly ordered field of real numbers (cf. chapter 6). Only in the discussion of the Hungarian method for algebraic transportation this assumption can be weakened. Then H is a d-monoid which is a linearly ordered semimodule over R+.

We begin with an introduction of the combinatorial structure. G = (N,E)

denotes a simple, connected digraph with two special

vertices u and terminal vertex

called 80urce and sink. G contains no arc with

T U

or initial vertex T. W.1.0.g.

G is a n t i 8 y m m e t r i c I i.e.

253

we assume that

Linear Algebraic Optimization

254

B denotes the incidence matrix of vertices and arcs of G , i.e. 1

Bih:=

if h

{ -:

=

if h =

, (j,i) ,

(i,j)

otherwise

E f o r all i E N and f o r all h E E. Then x E R + is called a G if 0

5 B x

f l o w in

= -B x and

Bix = O Then g:= B U x is called the fzow v a l u e of x.

for all iEN\{o,r}.

Let c E ( R + \ {O})E be a given c a p a c i t y v e c t o r . Then a flow x is called f e a s i b t e if x

Cut if o E (i,j) E ( I

S

5 c.

A

partition (S,T) o f N is called a

and T E T . Then the sum of all capacities c . with 11

x J)

n E is called the c a p a c i t y Of t h e c u t

capacity is denoted by c(S,T).

(S,T). The

It is easy to establish the

relation (12.2)

9 =

1

(SXT)n~

xij -

I: (TXS) n E

< c(S,T)

x i’

-

f o r all feasible flows and f o r all cuts in G. A well-known

result of flow theory (cf. FORD and FULKERSON [I96211 is the

max-flow m i n - c u t f

theorem which states that the maximum v a l u e

of a feasible flow is equal to the m i n i m u m c a p a c i t y of a

cut in G.

A

feasible flow of maximum value is called a m a x i -

mum f l o w ; similarly a minimum c u t is defined. In ( 1 1 . 4 9 ) we introduced algebraic flows in regular matroids. A

generalization of the max-flow min-cut theorem for algebraic

flows in regular matroids is developed in HAMACHER

([1980a],

A

[1980c]).

A detailed study of such flows i s given in HAMACHER

1980bl.

There are many procedures for the determination of a flow with m a x i m u m f l o w value. B e s i d e s t h e f l o w a u g m e n t i n g p a t h m e t h o d i n F O R D and FULKERSON [I9621 such methods have been investigated by D I N I C [1970], E D M O N D S a n d K A R P [ 1 9 7 2 1 , Z A D E H [ 1 9 7 2 1 , K A R Z A N O V ([1974],

[19751), EERKASSKIJ 119771, MALHOTRA, PRADMODB KUMAR

a n d M A H E S H W A R I [ 1 9 7 8 ] , G A L I L 1 1 9 7 8 1 , S H I L O A C H 119781 and G A L I L a n d N A A M A D [1979]. E V E N [ 1 9 8 0 ] h a s w r i t t e n a n e x p o s i t i o n o f t h e m e t h o d o f K A R Z A N O V w h i c h h a s a c o m p l e x i t y b o u n d o f O ( NI

3

1;

n u m e r i c a l i n v e s t i g a t i o n s a r e d i s c u s s e d by H A M A C H E R 119791

With respect to a feasible flow x we introduce the increm n t a l d i g r a p h AG = ( N , E U E ) w i t h city vector ACER;'~

i : ={(j,i)I

(i,j) E E ]

and w i t h c a p a -

defined by

A C i s w e l l - d e f i n e d a s R is a s u b r i n g o f IR.

A

f e a s i b l e f l o w Ax

in AG i s c a l l e d a n i n c r e m e n t a z f l o w . T h e n x e A x r d e f i n e d by

for a l l ( i r j ) E E , i s a f e a s i b l e f l o w i n G. F o r c o n v e n i e n c e w e assume that an incremental flow has the property Ax

ij

for all ( i r j ) E E

> O

u i.

*

AX

ji

= O

C o n v e r s e l y t o (12.41, i f

is a feasible

Linear AIgebmic Optimization

256 N

flow i n G with g i n AG s u c h t h a t

2

g t h e n t h e r e e x i s t s a n i n c r e m e n t a l f l o w Ax Ax = x .

X *

t a l f l o w i n Ax t h e n g

+

I f Ag i s t h e v a l u e o f t h e i n c r e m e n -

Ag =

y.

T h u s x i s a maximum f l o w i f a n d

o n l y i f t h e v a l u e o f a maximum i n c r e m e n t a l f l o w i s 0 . L e t p d e n o t e a p a t h i n G from

t o T o r a c i r c u i t i n G.

0

Then

p ( 6 ) with

(12.5)

for a l l of

(i,j) Ep

,

otherwise

,

P . . ( 6 ) := 11

( i , j ) E E and 0

1. 6

E R is a flow i n G.

The c a p a c i t y 6 ( p )

p i s d e f i n e d by

5

Then p ( 6 ) i s a f e a s i b l e f l o w i f 0

6

5

6 ( p ) . Such a f e a s i b l e

f l o w i s c a l l e d a p a t h f l o w o r a c i r c u i t flow.

The f l o w v a l u e

o f a p a t h f l o w i s 6 a n d t h e f l o w v a l u e o f a c i r c u i t f l o w is 0. E a c h f l o w x h a s a r e p r e s e n t a t i o n a s t h e sum o f c e r t a i n p a t h flows and c i r c u i t f l o w s

(cf.

r

S

Pp(6 1 +

x =

(12.6)

BERGE [ 1 9 7 3 ] ) , i . e .

P=l

z

pp(Gp)

p=r+1

w h e r e t h e f i r s t s u m c o n t a i n s a l l p a t h f l o w s a n d t h e s e c o n d sum c o n t a i n s all c i r c u i t f l o w s .

S i m i l a r l y e a c h i n c r e m e n t a l f l o w Ax

i n AG i s t h e sum o f i n c r e m e n t a l p a t h flot)s a n d i n c r e m e n t a l

c i r c u i t flows r

S

(12.7)

The s e t P

cl

of a l l f e a s i b l e flows i n G with c e r t a i n fixed flow

257

Algebraic Flow Roblems

value g E R (0 5 g

5

f) can be characterized by linear con-

straints. Let b € R Y be defined by b all i L { c r I ? } .

0

=

-b

?

= g and bi = 0 for

Then

(12.8) For cost coefficients a

ij

( i , j ) E E we consider the

EH,

alge-

b r a i c f l o w problem N

(12.9)

z:=

inf XEP

xTma. 9

H is a weakly cancellative d-monoid which is an extended semi-

module over R+.

A

solution x o f

(12.9)

is called a minimum C o s t

f l o w . W . 1 . o . g . we assume that g = f, i.e. we solve a max-flow min-cost problem. The classical case is obtained from ( 1 2 . 9 ) f o r the special vectorspace

( J R , + 1 5over ) the real field. A

combinatorial generalization of the classical case is discussed in B U R K A R D and H A M A C H E R [ 1 9 7 9 ] .

They develop finite procedures

for the determination of a maximum matroid flow (cf. theorem 11.53) with minimum cost in IR. It is well-known that B i s a totally unimodular matrix (e.g. LAWLER [ 1 9 7 6 1 ) . Thus ( 1 1 . 4 3 ) plex method ( 1 1 . 4 6 )

is satisfied and the primal sim-

can be applied for a solution of the

algebraic flow problem ( 1 2 . 9 ) .

If the semimodule considered

is not extended then the primal simplex method may terminate

in a locally optimal solution. Later on in this chapter we will discuss the application of the prfmal simplex method to the special case o f algebraic transportation problems (cf. 12.33);

for classical transportation problems this method is

known to be the most efficient solution procedure.

258

Linear Akebmic Optimization

In the following we develop a primal dual method for the solution of the algebraic flow problem which generalizes the min-cost max-flow algorithm in FORD and FULKERSON 119621. T h i s method has previously been considered in BURKARD 119761,

BURKARD, HAMACHER and ZIMMERMANN, U. [1977b]).

([1976I, [1977a1 and

Based on Involved combinatorial arguments BURKARD,

HAMACHER and ZIMMERMANN, U .

[1976] proved the validity of

this method. Now we can provide a much simpler proof based on the duality theory in chapter 11. h'e

assume that a

ij

E H + for

all ( i , j ) E E . This assumption allows an obvious initial solution and guarantees that the objective function is bounded from below by e. Self-negative cost coefficients ( a , .< e) 11

will play a role in the solution of (12.9) only if N

(12.10)

a(z)

=

xo .

(12.10) holds if and only if there exists a maximum flow (with value f) in the partial graph G' = (N,E') with

This can easily be checked using one o f the usual maximum flow algorithms. w

If x ( z ) = h o then it suffices to solve the algebraic flow problem in the graph G'. This is a problem in the module

over R

GA 0

and can be solved similarly to the classical min-cost max-flow problem. For example, the necessary modification in the min-cost max-flow method in FORD and FULKERSON 119621 mainly consists in replacing usual additions by *-compositions and usual compari-

.

sons by 5 - c o m p a r i s o n s in G AO

Akebraic Flow Pmblem

Otherwise A(;)

> A.

259

and without changing the objective

< e

function value of any maximum flow we can replace a i j

by ai,:=

e.

In order to apply the duality theory of chapter 1 1 to the algebraic flow problem we assign dual variables u,v and t to the constraints Bx to ( 1 1 . 6 )

2

and ( 1 1 . 1 7 )

b, -Bx

2

2

-b and -x

-c. According

(u,v,t) is strongly dual feasib1.e if

for all (i,j) E E and i f

for all i E N . The complementarity condition ( 1 1 . 1 4 . 1 )

has the

form (12.13)

( c - x ) T ~ t= e = x T o a

with reduced cost coefficients (12.14)

As

ui

-

* v, * aij

=

ai,

a

defined by

* u, * vi * tij .

in the discussion of the modification ( 1 1 . 2 8 ' ) and ( 1 1 . 2 7 ' )

of the general primal dual method we consider only the case that all dual variables are elements of A

u

U (el for a certain

index p E A (cf. chapter 6 for the decomposition of H). Then a

in ( 1 2 . 1 3 ) and ( 1 2 . 1 4 ) is replaced by a (11). ij i, For flow problems it suffices to consider only dual variables

v v

j'

j E N with va = e. A pair (x;v) of a feasible flow x and

E (a,, u {ellN with va

if

- ( z ) is

= e and p c h

called p-compatibte

260

f o r all

Linear Algebmic Optimization

(i,j) E E .

If (x;v) i s U-compatible then w e can define

=

a strongly dual feasible solution (u,v,t) by u

1

(12.16)

f o r all

if vj >aij( p ) * v i and v

E

j

e and

=:E

*a

ij

(i,j)E E. All dual variables are elements of H

P

(11) * v i

u

,

{el

and (x;u,v,t) satisfies the complementary condition ( 1 2 . 1 3 ) with reduced cost coefficients defined by (12.17)

ui*v

j

* aij

= a

i j

(11) * u j * v i

* tij

for all ( i , j ) E E . (12.13) i s a sufficient condition for (11.18.1). Therefore theorem ( 1 1 . 1 8 ) applied to

shows that x is an optimal solution of ( 1 2 . 9 ) if x i s a feasible solution with flow value g .

(12.18) P r o p o s i t i o n Let (x;v) be v-compatible and let g denote the flow value of x. Then x is a minimum cost flow of value g and ( 11

Proof:

qnvT = x

T 0

a(ll)

*

cTtYt.

Due to our remarks it i s obvious that x is a minimum

cost flow. From theorem (11.18) we get (' - b )T+ n v = x T o a The definition of b in ( 1 2 . 8 )

*

( - b )T- n v

*

cT n t .

and va = e lead to ( 1 ) .

Algebraic Flow h b l e m s

261

The method proceeds In the following manner. The initial compatible pair is (x;v) with x

=

0 and v

=

.e

(?J =

Ao).

Then we

try to increase the value of x without violation of compatibility. If this is impossible then we try to increase the dual variables v without violation of compatibility. In this way the current flow i s of minimum cost throughout the performance of the method. We will show that the procedure leads to a flow x of maximum value in a finite number of steps. Then due to proposition ( 1 2 . 1 8 )

x is an optimal solution of ( 1 2 . 9 )

with

g = f. At first we try to change x using an incremental flow Ax such that ( x , x o Ax) is p-compatible, too. An arc (i,j E E is called

a d m i s s i b l e if (12.19)

an arc (i,j ) E

E

is called admissible if ( j ,I) E E is admissible.

Let 6 G = ( N , 6 E ) denote the partial graph of admissible arcs. Then feasible flow 6x in

AG

containing all

is called the a d m i s s i b i l i t y g r a p h . A

6G

6G I s

called an a d m i s s i b l e i n c r e m e n t a t flow.

Clearly the admissibility graph depends on the current flow x and the current variables v.

(12.20) P r o p o s i t i o n Let ( x ; ~ )be p-compatible, let g denote’the flow value of x and let 6 x be an admissible incremental flow of value 69. Then (1)

(x. 6x;v)

(21

(X

is p-compatible,

6xlT oa(p) = xT oa(p)

*

( 6 9 0vT).

262

Linear Algebraic Optimization

Proof:

( 1 ) Since 6x is an incremental flow x o 6x is a feasible

flow of value g + 69. Admissibility of 6x shows that x is changed only in admissible arcs. Thus (x o 6x;v) is v-compatible. ( 2 ) Proposition (12.18)

leads to

These equations in the module G

over R imply (2).

v

m

Proposition ( 1 2 . 2 0 ) shows that augmentation of x with an admissible incremental flow 6x does not violate compatibility. 6x can be determined by a maximum flow procedure. In particular, this is a purely combinatorial step with complexity bound O(lNI

3

)

if we use the method of KARZANOV [ 1 9 7 4 1 .

Secondly we discuss the case that the value of a maximum admissible incremental flow is 0. Let (S,T) be a minimum cut of 6G. Then (12.21)

Ac

ij

for all (i,j) E (S

x

= 0

T) fl 6E. (12.21) follows from the max-flow

min-cut theorem. If Acij = 0 for all (i,j) E (S

x

T)

n

(E U

i)

then

( S I T )is a minimum cut in AG of value 0 o r , equivalently, (S,T)

is a minimum cut in G of value f. Then x is a maximum flow in G and the procedure terminates. Otherwise we know that each

flow x in G of flow value y > g has either N

(12.22)

x

rs

for Some (r,s) E F : =

> x

rs

{ ( i , j )I

x

ij

C

c

ij

n

(S x T)

or

263

Algebraic Flow h b l e m s M

x

(12.23)

f o r some

< x

rs

rs

{ ( i , j )I

( r , s )E B:=

o <

x

1 n (T x s). s i n c e

< ars(v) *vr

v for a l l

( r , s ) E F and

for a l l

( r , s ) E B we may d e f i n e a v

i j

* a

s

rs

:= a

rs

( r , s ) E F U B by

(11)

*vr

if ( r , s ) E F

,

(v)

*vr

if

(r,s)E B

.

(12.24)

v We r e m a r k t h a t a

=:a

rs

*a

-

= a

rs

rs

for

rs

( r , s ) E F and a

rs

=

t r s f o r ( r , s ) E B.

T h e r e f o r e we may r e v i s e t h e d u a l v a r i a b l e s i n t h e f o l l o w i n g ' w a y .

(12.25)

Dual r e v i s i o n procedure

'Step 1

D e t e r m i n e t h e c o n n e c t e d component S p a r t i a l graph of Ac

i j

if

Step 2

T E S then return

F:=

(exit 1).

{(i,j)I x

o < x

{(i,j)l

Compute a

a:= m i n { a

v:=

of t h e

which c o n t a i n s a l l a r c s w i t h

i j

1 n

i f FUB = @ then return

Step 3

5 N)

> 0;

T:= N \ S ; B:=

6G

( U E S

for

rs rs

I

i j

< c

il

1 n

(SXT);

(TXS);

(exit 2).

(r,s) E F U B from ( 1 2 . 2 4 ) ;

( r , s )E F U B } ;

)i(a). if j ES,

Step 4

R e d e f i n e a l l d u a l v a r i a b l e s by a n d go t o s t e p 1 .

v

j

:=

vJ * a

i f j E T,

2 64

Linear Algebm'c Optimization

Due to our previous remarks it is clear that all steps are well-defined. In particular P = (12.25).

*

If F U B

0 in step

0

in the first call of

2 then

IS1 increases after a

revision of the dual variables in step 4 at least by one. Thus the dual revision procedure terminates after at most IN1

-

1 iterations either at exit 1

( 6 G contains an incremen-

tal flow of nonzero value) or at exit 2

(x is a maximum flow

in G ) . Each iteration contains at most O ( l N I tions, O(lNI 2 )

*-compositions and O ( l N I 2 )

2

usual opera-

5 - comparisons.

(12.26) P r o p o s i t i o n Let (x;v) be u-compatible and let g denote the value of x

< f). If the revision procedure leads to revised dual vari-

(g

ables denoted by (x;;)

(1)

with revised index

i then

is ;-compatible.

Proof: It suffices to prove ( 1 ) for one iteration in the dual revision procedure. The definition ( 1 2 . 2 4 ) U-compatibility implies h ( Q )

-

N

u 5 )i(z). Otherwise

a =

a

rs

2

P in step 3 .

of a

rs

If p =

together with

i then

( u ) > e for some (r,s) E F. Thus

.

Since a , . = t for (i,j) E B we find B = 0. Therefore rs 11 ij * N each flow x in G with flow value q > q has x > x for some ij ij (i,j) E F. Thus It remains to prove the inequalities 5 A(;).

a = a

(12.15)

u

for (x;;).

(T x S ) ]

n

It suffices to consider arcs (i,j) E [ ( S x T )

E. If (i,j) E F U B then a

5

a

ij

implies the

corresponding inequality. If (i,j) E (S x TI X F then x

> a.

and

j -

G 3.

5 a

ij

11

*vi.

*ii .

ij

=

c

ij

If (i,j) E ( T x S ) X B then x . = 0 and ij w

265

Algebraic Flow h b l e m s

Proposition ( 1 2 . 2 6 ) shows that a dual revision step does not violate compatibility provided that g

C

f. In the last step

it may happen that before actually finding a minimum cut LJ is raised beyond A(:).

Therefore if a minimum cut is detected

then an optimal solution of the possible dual programs (cf. chapter 1 1 ) has to be derived from the values of the dual variables before entering the dual revision procedure for the last time. Optimality of x is guaranteed in any case. By alternately solving a max-flow problem in the admissibility graph and revising dual variables we construct a minimum cost flow x E P f . We summarize these steps in the following algorithm.

( 1 2 . 2 7 ) Primal dual m e t h o d f o r a l g e b r a i c f l o w s x

Step 2

Determine a maximum flow 6x of value 6g in 6 G ;

R

0; v

e; z:= e; p : = h o .

Step 1

x:= x o 6x; z : = z Step 3

*

-

(69 o v T ) ; v:= v.

Revise dual variables by means of ( 1 2 . 2 5 ) ;

-

if ( 1 2 . 2 5 ) terminates at exit 2 then v:= v stop; otherwise go to step 2 .

At termination x is an optimal solution of ( 1 2 . 9 ) for g = f and v leads to an optimal solution (u,v,t) of the possible dual programs (cf. chapter 1 1 ) .

z is the optimal value; the

recursion for z in step 2 follows from proposition ( 1 2 . 2 0 ) . We remark that explicit knowledge of the ordinal decomposition (cf. chapter 6) of the semimodule is not necessary. The recursion for the index p is deleted and ars(p) in step 4

Lineor Algebraic Optimization

266

of the dual revision procedure is determined by

a

(v):= rs

1 ,"

if v T * a r s > a if v

*a T

In step 5 the dual variables v.(u) for i E

rs S

if a * v i >

This follows from proposition

rs

= v

'

.

can be found by a

,

(6.17). Admissibility can be

found similarly. It remains to show that the primal dual method ( 1 2 . 2 7 ) is finite.

( 1 2 . 2 8 ) Theorem Let H be a weakly cancellative d-monoid which is an extended semimodule over R+. Then the primal dual method

(12.27) is

finite. Proof: The performance of step 1, step 2 and step 3 is finite. Therefore it suffices to prove that step 3 in ( 1 2 . 2 7 ) is carried out only'a finite number of times. At termination of step 2 the admissibility graph with respect to (x;v) does not contain any path p from a to

T

of capacity 8(p) > 0. If the current

flow x is not a maximum flow then at termination of step 3 the admissibility graph with respect to (x;v) does contain such a path p .

Then v

-

< v

.

With respect to x the arcs of p are

partitioned in F U B defined by

267

Algebraic Flow Problems

The number of partitions is finite as well as the number of different paths from U to T in

G.

Thus it suffices to prove

that p did not appear in a previous iteration of the primal dual method with the same partition of its arcs. Admissibility and

U

=

;*

(12.29)

i

where

e lead to

(v)

(i,j) E B

s=v

denotes the composition of all weights a

ij

(i) with

((i,j) E F).

Now assume that p previously appeared in an admissibility graph N

N

with respect to (x;v), with capacity z(p) > 0 and with the same partition of its arcs. Similarly to ( 1 2 . 2 9 ) N

v

(12.30)

*T=Y N

with respect to weights a

> e implies

Now Thus

we get

7

ij

( P I . Further

> e and from

T 5

1. v T

<

7

and

T N

5

G.

-

we conclude Y = y .

- - - *i. v

(12.31)

* B = v

N

u-compatibility shows

for all (r,s) E B . Thus lead to N

v

T

=

-

v

11

=

';.

and

z.

= N

contrary to v

A(T) 5 y.

T

<

Then

iT *i 2 iT

> e and ( 1 2 . 3 1 )

Cancellation in the module G- implies

v .

L1

rn

Proposition ( 1 2 . 2 8 )

shows that the primal dual method is valid;

therefore this method provides a constructive proof of the existence of an optimal solution of the algebraic flow problem. Further a strong duality theorem is implied.

Linear Algebraic Optimization

168

(Network f l o w d u a l i t y theorem)

( 1 2 . 3 2 ) Theorem

Let H be a weakly cancellative d-monoid which is an extended E

semimodule over R+. For cost coefficient vector a E H + an optimal feasible pair (x;u,v,t) for the algebraic flow problem exists. (x;u,v,t) is complementary and satisfies (1)

fovr

(2)

(-c)

T

=

xToa

*

*

f

Ut(P)

cTat, 0VT

= xT a a

for

In discrete semimodules ( R = 2 2 ) we get 6g the primal dual method ( 1 2 . 2 7 ) .

2 1

u

N

=

X(z).

in step 2 of

Therefore at most O(f) itera-

tions are performed until termination occurs. Each iteration 3

consists in step 2 and step 3 and needs O ( l N 1 ) usual operations, O(lNI

2

)

*-compositions and O ( l N I

3

)

5 - comparisons (maximum flow

with method of K A R Z A N O V 1 1 9 7 4 1 ) . We remark that a solution of an equation a * y : =

B for a 5 B ,

a , B E H is counted as one *-composition.

In the next part of this chapter we discuss three methods for the solution of an algebraic flow problem which generalizes the classical (balanced) transportation problem. We begin with a description of the underlying graph which shows the special combinatorial structure of this problem. The subgraph of G = ( N , E ) induced by N \ { u , r }

is bipartite.

Let N = { u , ? } U I U J with I:= {l,2, Then

. . . ,m}

and J:= { m + l ,

...,m + n ) .

Akebraic Flow h b l e m s

269

Capacities on arcs (0,i) and ( j , ~ )are denoted by ci and c

j'

, iEI

j E J ; there are no capacity constraints on all other

arcs (cij =

m).

W.1.0.g.

we assume that the capacities are

b a l a n c e d . i.e.

Thus the value of a maximum flow i s f = rici = 1 c

j j'

Cost coeffi-

cients on arcs (u,i) and (j,?) have value e. The other cost coefficients are elements of A. Such a graph is called a t r a n s p o r -

t a t i o n n e t w o r k (Figure 6 ) .

J

I

T

U

Figure 6

Transportation network with assigned capacities, cost coefficients

The corresponding algebraic flow problem is called a l g e b r a i c

t r a n s p o r t a t i o n p r o b l e m . The linear description of the set of feasible solutions can be simplified in this case. Let C denote the incidence matrix of vertices and arcs for the partial subgraph (I u J , I

x

J), i . e .

2 70

Linear Algebmk Optimization

if h = ( k , j ) or h = ( i , k ) ,

1

'kh:=

otherwise.

Now a flow x in G is uniquely defined by its values on arcs in I X J ; therefore it suffices to describe the set of all corresponding y = x

IX

The set P

J'

T

of all y corresponding to a

maximum flow is

The a l g e b r a i c t r a n s p o r t a t i o n p r o b l e m is N

(12.33)

z:= ruin YEPT

y

T

na

for a cost coefficient vector a E H

mn

.

If a 6 HYn then let (cf.

proposition 4 . 7 )

Then

a

defined by : . . : =

a *a

11

ij

satisfies

a ij

H + for

( i ,j ) E I x J and

for all 7 , y E P

T

provided that H is a d-rnonoid which is a linear-

ly ordered semimodule over R+. y

T

na

*

foa = y

(12.34)

T

follows f r o m

na

mn for a l l y E P T . Therefore we can assume a E H + in the following w.l.0.g. Due to the description of P

T

w and s to the constraints Cy

it suffices to assign dual variables

2 c IUJ

is strongly dual feasible if (12.35)

wi*wj

< -

a

i j

*s. * s . 1 1

and -Cy

2

-cIuJ. Then (w,s)

27 1

Algebraic Flow Roblems

for all (i,j) E I

x J

and if

for all k € 1 U J . Complementarity is satisfied if yToa = e where ,EHyn (12.36)

is defined by

-

wi

* wj * ai, : =

for all (i,j ) E I

x

ai,

*

si

* s,

J.

If H is a weakly cancellative d-monoid which is an extended semimodule over R + then we can apply the primal dual method (12.27).

The constructed sequence of p-compatible pairs (x;v)

defines a sequence of pairs (y;w,s) by y:= x if k E J wk:'

I X J

and by

,

otherwise,

(12.37)

if k E I 'k:'

,

otherwise.

= e for all (i,j) E I x J (c = m) ij ij and therefore (w,s) is strongly dual feasible with respect to

p-compatibility implies t

(12.33).

Further (y;w,s) is complementary. Theorems ( 1 1 . 1 8 )

and ( 1 1 . 2 5 ) lead to the following strong duality theorem.

(12.38) Theorem

(Transportation duality theorem I )

Let H be a weakly cancellative d-monoid which is an extended semimodule over R+. For a cost coefficient vector a E H y n there exists a feasible optimal pair (y;w,s) for the algebraic transportation problem which is complementary and satisfies

Linear Algebraic Optimization

272

cTowJ = yToa J

*

T (-cI) o s I ( u )

cTow = y J J

*

cTos I

1

T

.

N

oa

for u = X ( z )

The application of the primal dual method to the algebraic transportation problem is called the H u n g a r i a n m e t h o d .

In the

following we will prove that this method is valid even if we only assume that H is a d-monoid which is a linearly ordered semimodule over R+. We remark that all steps of the method are well-defined in such d-monoids. Nevertheless a proof of its validity in this case but for the general algebraic flow problem is not known. A more detailed analysis is necessary to develop new arguments on which w e can base a proof of its validity for algebraic transportation problems. B U R K A R D [1978a1 develops the Hungarian method based on the concept of 'admissible transformations'. We will derive a description of our method which shows that both methods proceed in almost the same manner. Many arguments are drawn from B U R K A R D [1978a].

Let H be a d-monoid which is a linearly ordered semimodule over R+.

Arguments with respect to u-compatibility hold in this case,

too. Thus the primal dual method constructs a sequence of U compatible pairs (x;v). On the other hand finiteness and optimality are proved using certain cancellation arguments. These do not hold in d-monoids, in general. The values x .

a1

x jT

,

j E J are never decreased in

compatibility shows that arcs ( o , i

' i E I and

he procedure. Therefore

,

i E I and

( j , T ) ,

j E J re-

main admissible until in a certain iteration such an arc is

s a t u r a t e d , i.e.

u-

273

Akebraic Flow Problems

We conclude that xoi

' i'

x

'c

*

(12.39)

jT

*

j

v i = v

"j

U

= e ,

= "T

for all i E I, j E J is satisfied. Further u-compatibility implies (12.40)

*

v, 5 aij(u)

vi

for all i E I, j E J which shows that (12.41)

xij > 0

*

(i,j) admissible

for all i E I, j E J . Let (S,T) be a cut determined in step 1 and step 2 of the dual revision procedure ( 1 2 . 2 5 ) .

then

S x

Let

T = L U K . Further Q = (T x S ) f l E satisfies Q:=

( T n I)

x

(Sfl J)

(cf. Figure 7 ) .

Figure 7

The cut (SIT) in a transportation network

Linear Algebraic Optimization

274

Due to the construction of (S,T) and ( 1 2 . 4 1 ) x (12.42)

This

oi

= c

i

x I. T

= cj

x

= o

i j

we find

for all (0.i) E K

,

for all ( j , ~ E) K

,

for all ( i ,j ) E L U Q.

leads to a simple finiteness proof.

(12.43)

Proposition

Let H be a d-monoid which is a linearly ordered semimodule over R+. Then the Hungarian method terminates after finitely many

steps with a maximum flow x. Proof: Due to ( 1 2 . 4 2 )

the value g of the current flow x satis-

f ies (12.44)

g =

1 c. TflI

z

+

c

snJ

j

in step 3 of the Hungarian method. There are only finitely many flow values of such a form.

As

the flow value is strictly in-

creased in an iteration of the Hungarian method after finitely many iterations a maximum flow x is constructed and termination occurs.

For a discussion of optimality of the final flow x a combina-

torial property of maximum flows in transportation networks is o f particular interest. A maximum flow

X

saturates all arcs

(U,i), i E I and all arcs ( j , ~ )j, E J. Therefore ( 1 2 . 4 4 ) g = 1 K (12.44)

ij

.

Then ( 1 2 . 2 )

r L x ij

=

applied to f-g

+

leads to

I:

Q

-

x

ij

'

implies

A4pebraic Flow Problem

275

Another important combinatorial property is

C (i,j)E

EI

o

n (T x

c xij3

#.

SI =

Therefore in the dual revision procedure ( 1 2 . 2 5 )

the determina-

tion of a can be simplified. We get (12.45)

a = min{aijI

Due to ( 1 2 . 4 0 )

v,

*aij:=

aij(v) * v i , ( i , j ) E L I .

reduced cost coefficients

a ij

may be defined as

solutions of (12.46)

vj

* aij:=

a

ij ( u ) * v i

for all (i,j) E I X J . Nevertheless these reduced cost coefficients are not uniquely determined by ( 1 2 . 4 6 ) . then we define

-

a

ij

a ij

=

e a s usual. Otherwise v

j

If v . = a 7

< aij

* vi

and

In general d-monoids,reduced cost coeffi-

cients are chosen as proposed in BURKARD [ 1 9 7 8 a ] ; a

( v ) *vi

is uniquely determined if H is weakly cancellative (cf.

proposition 4 . 1 9 . 3 ) .

-

(y)

i. j

initially

a. Then after determination of a from a = min{a.,

(12.47)

13

I

(i,j) E

LJ

in the dual revision procedure the reduced cost coefficients are redefined by if (i,j) E Q (12.48)

I

if (i,j) E L and a

* E : = a- ij

'

otherwise. In particular

-

E:=

e if a = aij. It is easy to see that ( 1 2 . 4 6 )

is always satisfied. Even ( 1 2 . 4 8 ) all reduced cost coefficients. determined.

If we add

dual method ( 1 2 . 2 7 )

a

=

does not uniquely determine

In any case p = A ( a ) is uniquely

a in the initial step 1 of the primal

then the following modification of the

276

Linear A(gebroic Optimization

dual revision procedure is well-defined.

(12.49)

Modified dual r e v i s i o n procedure

Step 1

Determine the connected component S ( u E S C_ N ) of the partial graph of AG which contains all arcs with A c , . > 0; if 11

Step 2

T:=

N \ S ., L:=

if L Step 3

Step 4

E S then return (exit 1).

( S n I ) x ( T n J ) ; Q:=

(TflI) x (SflJ);

then return (exit 2 ) .

=

a:= min{a. . / 11

p:=

T

(i,j) E L I ;

A(al.

Redefine all dual variables by

j v .:= 1

Vj(U)

1 v 3.

i a

if j E s , if j E T ,

all reduced cost coefficients by

(12.40)

and go to

step 3 .

(12.49)

specifies ( 1 2 . 2 5 )

for algebraic transportation problems

in such a way that we can give a proof of the optimality of the final U-compatible pair constructed by the following Hungarian method.

(12.50)

H u n a a r i a n method w i t h d u a l v a r i a b l e s

Step 1

x

Step 2

Determine a maximum flow 6x of value 6 9 in 6G; x:=

0; v

x

0

3

6x;

e; z : = e ;

z:= z

*

u

(6g

= A

0;

-

a:= a -

v T ) ; v:= v .

277

Akebraic Flow Roblems

Step 3

Revise dual variables and reduced cost coefficients by means of ( 1 2 . 4 9 ) ; if ( 1 2 . 4 9 )

terminates at exit 2 then v:=

stop;

otherwise go to step 2 . The final p-compatible pair (x;v) is shown to be optimal in the proof of the following strong duality theorem.

( 1 2 . 5 1 ) Theorem

( T r a n s p o r t a t i o n d u a l i t y t h e o r e m 11)

Let H be a d-monoid which is a linearly ordered semimodule mn over R+. For a cost coefficient vector a E H + there exists a feasible optimal pair (y;w,s) for the algebraic transportation (y;w,s) i s complementary and satisfies

problem.

cTow J

J

=

*

yT0a

cTos I I '

Further z defined recursively in step 2 of the Hungarian method (12.50)

is the optimal value of the algebraic tr.ansportation

problem. Proof: We introduce an auxiliary variable B in the Hungarian method. Initially B : = (12.52)

B

B:=

*

e. Then add

[ (f - g)

0

a1

in step 4 of the modified dual revision procedure ( 1 2 . 4 9 ) . g denotes the flow value of the current flow x. We remark that A(B)

= p.

(12.53)

We will prove that ~

*

T

oy a = y T n a

is satisfied for all Y E P Hungarian method ( 1 2 . 5 0 ) .

T

throughout the performance of the In the initial step ( 1 2 . 5 3 )

i s ob-

278

Linear AIgebmic Optimization

-

viously satisfied since a = a. It suffices to consider an iteration in the modified dual revision procedure. Let ( 1 2 . 5 3 )

-

be satisfied before step 3 and denote the revised variables N

-

N

by v , a , p and

7.

yT o

(12.54)

i

Let y E P T . = yT

Q

o

iQ *

Then

5

A

T y L o aL

*

yH

(2)

T

0

5

(yT

0

a) leads to

-

aH

Due to ( 1 2 . 4 4 ) we get

Together with (12.55)

Then

7

=

(12.54) and ( 1 2 . 4 8 ) this leads to

yT0a = y B *[(f-g)

“f-g)

Ll:

o a ]

proves

0

a1

.

(12.53) with respect to the rede-

fined variables. Let (y;w,s) denote the final pair derived from the final compatible pair (x;v) of the modified primal dual -T

method. Then y E P T and y

-

-T y o a = ~ ( 1 2.56)

for all Y E P

< T’

Hence

O a = e. Therefore

-T * oy a =

ii

i * y T o a = yTna

y

is optimal.

The equation ( 1 ) is derived from the equations w

j

*a

i j - aij

*

‘i

by collecting all these equations after composition with

i.

The recursion for the optimal value is derived f r o m ( 1 2 . 5 6 )

-

shows B = y- T O a (another recursion!).

which

NOW let (a , g u ) be the

u

constructed sequence of variables a in step 3 of (12.49) and corresponding current flow values 9 ; then 5 satisfies (cf. 12.52)

Akebraic Flow Problem

*

B =

[(f-gp) o aU 1

219

.

p=1,2, ...fr Let 69p : = g p - g p - l for p = 2.3

,...,r

and let v :

,

p = 1,2

,...,r

denote the corresponding sequence of values of the dual variable v

. Then

v1 = e and T

vp = a for p = 2,3

,...,r.

*a2

* ... * a P-1

Thus

*...,

B =

~=2,3,

69

r

1!

mvy = z .

If we are not interested in the optimal dual variables then we may delete dual variables from (12.50). In this case it is necessary to identify admissible arcs without explicit use of dual variables. Due to the remarks on admissibility before (12.39) we may consider all arcs ( O , i ) , i E I and (j

f

~

)

jEJ

as admissible throughout the performance of the Hungarian method without changing the generated sequence of compatible pairs. Now ( i f j )E I

x

J is admissible (cf. 12.19 and 12.46) if and only

if (12.57)

vj

* aij

vj

=

-

Then ( j , i ) is admissible, too. If H is weakly cancellative then this is equivalent to (12.58)

8

*iij

=

a ij

=

e as well as to

B

where B is recursively defined by (12.52). BURKARD [1978a] uses (12.58) for a definition of admissibility. In general d-monoids H it is possible that these definitions lead to differing admissibility graphs. Let (1,j ) € I

x

J be B-admissible

if (12.58)

280

Lineor Algebmic Optimization

is satisfied. Then (j,i) is called 5 - a d m i s s i b Z e , too. Let 6G(B)

denote the partial subgraph of AG containing all arcs

from ( E U E ) \ ( ( I

J ) u ( J x 1)) and all 5-admissible arcs. This

x

leads to the following variant of the Hungarian method (12.50).

(12.59)

Hungarian method w i t h o u t dual variables

Step 1

x

Step 2

Determine a maximum flow 6x of value 69 in 6 G ( B ) ;

=

0; g : = 0;

x:= x Step 3

m

B:=

6x; g:= g

-

e ; a:= a ; f:= Tici.

+ 69.

Revise reduced cost coefficients and 5 by means of (12.60);

if ( 1 2 . 6 0 ) terminates at exit 2 then stop; otherwise g o to step 2 .

(12.60) R e v i s i o n o f r e d u c e d c o s t c o e f f i c i e n t s Step 1

Determine the connected component

S

(0

E S C_ N ) of the

partial graph of 6 G ( B ) which contains all arcs with Ac if Step 2

ij

> 0;

T E S

T:= N \ S ;

then return (exit 1 ) . L:=

(S n I) x (T fI J ) ; Q:=

if L = Ql then return (exit 2 ) . Step 3

a:= min{aij

I

(i,j) E L);

(T n I ) x ( S n J ) ;

28 1

Algebraic Flow Problems

Step 4

Redefine all reduced cost coefficients by ij

if (i,j) E Q

*a

if (i,j) E L and u a

-

I

*

E:=

aij

I

otherwise

ij

and go to step 1 .

Obviously all steps in the Hungarian method are well-defined. The reduced cost coefficients (i,j) E I

x J

a ij

satisfy

a ij -> # e for

all

throughout the performance of the Hungarian method.

Further (12.61)

x

ij

*

> 0

for all (i,j) E I

x J

(i,j) is B-admissible

i s always satisfied. This follows from the

transformation of the reduced cost coefficients in step 4 of (12.60). X 4

ij

= 0

-

a

ij

is increased only on arcs (i,j)E Q ; therefore

for ( i , j ) E Q and dom B

(cf. 4 . 1 8 . 2 )

5

dom(6

*

(f-g) O a )

lead to ( 1 2 . 6 1 ) .

The cardinality of

S

is increased

at least by one after such a transformation. Thus ( 1 2 . 6 0 ) minates after at most (n + m

-

1)

transformations. The same argu-

ment as used in the proof of proposition ( 1 2 . 4 3 )

x.

Then due to ( 1 2 . 6 1 )

-T D a - = e for y

y:=

IXJ'

can be proved in a similar manner as before. Thus -T y ma = 0

5 O*yToa

for all y E PT since

a E HYn.

shows that

is finite and leads to a maximum

the Hungarian method ( 1 2 . 5 9 ) flow

ter-

=

y

Hence

Toa is optimal.

(12.53)

282

Linear Algebraic Optimization

(12.59)

is developed in B U R K A R D 11978a1. T h e particular case

of algebraic assignment p r o b l e m s

(ck = 1 , k € I and R = Z ) is

solved in the same manner in B U R K A R D , HAHN and Z I M M E R M A N N , U. [19771.

It should be noted that ( 1 2 . 5 9 )

and ( 1 2 . 5 0 )

lead to

the same sequence of flows x if H is weakly cancellative. I n the form ( 1 2 . 5 9 )

the Hungarian method can be interpreted as a

sequence of (admissible) transformations o f the reduced cost

a

coefficients

-

+

a

such that y

for all Y E P

T'

T

o a = y

*

-

yToa

The composition o f a finite number of (admissible)

transformations is again an (admissible) transformation. The transformations are constructed in a systematic manner such that a final admissible transformation a

+

mn a E H + is generated

satisfying y for all Y E P

T

T

*

oa = 0

yToa

and f3 =

8

*

-T for some y € a . A s we have shown before such

are optimal.

In discrete semimodules ( R = Z Z ) we get similar complexity bounds as for the algebraic network flow problem. In particular for algebraic assignment problems an optimal solution is found in O(n

4

1 usual operations, O(n

4

)

*-compositions and O(n

4

)

5

-

comparisons. In the classical case the Hungarian method [ ( 1 2 . 5 0 )

a s well as

( 1 2 . 5 9 1 1 reduce to the Hungarian method for transportation

Algebraic Flow Problem

283

problems (cf. MURTY 119761) which is developed for assignment problems in KUHN [19551. For bottleneck objectives it reduces to the second (threshold) algorithm of GARFINKEL and RAO [1971]. In this case the algorithm remains valid even in the case of finite capacities (c C -1 ij

on arcs (i,j) E I

XJ.

This follows

from its threshold character (cf. BURKARD [1978a], EDMONDS and FULKERSON [1970]). We will discuss two further methods for a solution of algebraic transportation problems. Both methods are well-defined in d-monoids H which are linearly ordered semimodules over R+; on the other hand a Proof of validity and finiteness is only known in the case that H is a weakly cancellative d-monoid which i s an extended semimodule over R+. NOW we develop an algorithm proposed in BURKARD and ZIMMERMANN,U. [1980]. The method proceeds similar to the Hungarian method by

alternately flow augmentations and dual revision steps. Different from the Hungarian method flow augmentation is performed along certain shortest paths. The method consists of m stages (k = l,Z,

...,m).

In the k-th stage the flow x satisfies

={c i

x

01

for all i E I and 0

5

0

xk

C

i f i C k i f i > k

ck. In order to augment such a flow x

we consider paths in the r e d u c e d i n c r e m e n t a l g r a p h AG

k

with

set of arcs {(u,k)) U I x J

u

( ( j , i ) l (i,j)EIxJ,

xij

>ol u

{ ( j , r ) I XjT c C j L

If xk < ck then AGk contains at least one elementary path p

from u to T. Then 6(p) > 0 (capacity of p), p(6(p))

is an

Linear Algebraic Optimization

284

incremental path flow, and we may augment x to

xo

p ( 6 (p))

.

At the k-th stage the current flow x and the current dual variables v

j E N will be a l m o s t p - c o m p a t i b l e ,

j

i.e. all

conditions (cf. 12.15) are satisfied with the possible exception of v At

> v . which is necessary while x

a -

1

ui

< ci for i

5 k.

the end of the k-stage x o k = ck. If we augment x by p ( 6 ( p ) )

then almost u-compatibility is violated, in general. Due to the choice of p it will be possible to revise the dual variables i n such a manner that the new current x and v are almost p-compatible, again.

For a determination of p we assign weights a to the arcs of AGk.

Since x and v are almost v-compatible the reduced cost

coefficients

i,

can be defined by

1j

for all ( i , j ) E I

x

J. Let zij:=

a ij

for (i,j) E I

N

x

J and a

.= e

ij'

otherwise. We remind that by definition x.

11

for all (i,j) E I X J

> O

-

a

-b

(cf. 12.46).

ij

= e

Let w(p) denote the weight of N

a path p in AGk with respect to weights a .

lj

Let N':=

...,k}

{If2,

U J U

IT).

(cf. chapter 8 ) .

Shortest paths from u to i E N '

can be determined using the algebraic version of ( 8 . 4 ) with (8.10) since

yijE H +

together

for all arcs (i,j) of AGk.

The following shortest path method is a suitable version of (8.4)

applied to AGk. During its performance a label [ni; p(i)]

is attached to vertices i E N ' . r i

is the weight of the current

Algebraic Flow Roblems

285

d e t e c t e d s h o r t e s t p a t h from a t o i and p ( i ) i s t h e p r e d e c e s s o r o f i on t h i s p a t h .

(12.61)

Shortest paths i n A G ~

Step 1

L a b e l v e r t e x k w i t h [ e ; o ] and l a b e l a l l v e r t i c e s j E J

[ zk j ; k ] ;

with

Step 2

I':= {1,2,

Determine h E J ' II

h

= minIn

If

x

hr

;

J':= J.

with j

I

j EJ')

and r e d e f i n e J':=

Step 3

...,k - l }

J'

{h}.

< c h t h e n go t o s t e p 6 ;

l e t S ( h ) : = { i E 1'1 x . > 01; ih i f S(h) =

$ ,t

t h e n go t o s t e p 2 ;

o t h e r w i s e l a b e l a l l v e r t i c e s i E S ( h ) w i t h [ n h ; h ] and redefine I':= I ' \ S ( h ) .

Step 4

For a l l

( i , j ) E S ( h ) x J' do

if

then l a b e l vertex j with

Step 5

Go t o step 2.

Step 6

Label

'I

N

[ni*a i j;I].

w i t h [ n h ; h ] and d e f i n e

ni:=

=h

f o r a l l v e r t i c e s i E I U J which s a t i s f y ai >

kh

o r which

are not labeled.

A t t e r m i n a t i o n t h e s i n k T i s l a b e l e d and n

is t h e w e i g h t o f a

N

s h o r t e s t p a t h p from a t o

'I

which c a n e a s i l y be f o u n d by back-

tracing using the predecessor labels.

I f t h e weight of a

286

Linear Algebraic Optimization

shortest path from is equal to n after O(lNI O(lNI

2

)

2-

2

)

i'

to i E N '

0

Otherwise

is not greater than =

TIi

TI

usual operations, O(lN1

2

then it

algorithm terminates )

*-compositions and

comparisons.

(12.62)

Augmenting p a t h method

Step 1

k:= 1 ; x

Step 2

. The

TI

0; v

3

e; U:=

Determine weights n .

I'

Ao.

iEN\{o)

and the shortest path

N

p with capacity -

using the shortest path method ( 1 2 . 6 1 ) .

N

~(61, u:=

* n T 1.

Step 3

x:=

Step 4

Redefine the dual variables by

step 5

I f xok < c

x o

k

A(v

T

go to step 2 ;

if k = m stop; k:= k+l and go to step 2 .

All steps in ( 1 2 . 6 2 )

are well-defined in a d-monoid H provided

that (x;v) is almost U-compatible throughout the performance of (12.62).

The necessary determination of the reduced cost coeffi-

cients is not explicitly mentioned.

As

in the Hungarian method

it is possible to derive a recursion for the reduced cost coefficients; then the dual variables can be deleted from the algorithm with the exception of v T . The determination of

TI

i

in

step 2 is also possible using cuts and dual revisions as in the

Algebraic Flow Problems

287

Hungarian method; but the shortest path method leads to a better complexity bound: O ( l N 1 2 ) instead of O(lNI

3

)

with

respect to all types of operations considered (cf. 1 2 . 2 5 ) . This is due to the fact that we assign nonnegative weights M

to the arcs of AGk which admits the application of a aij fast shortest path procedure. It should be noted that in the

case

T

= e the subsequent determination of shortest paths

of zero-weight can be replaced by the determination of a maximum incremental flow in AGk using only arcs with weight 'c; ij=e For a proof of finiteness and validity of the augmenting path method we assume that H is a weakly cancellative d-monoid which is an extended semimodule over R+.

A

proof for general

d-monoids which are linearly ordered semimodules is not known.

(12.63) P r o p o s i t i o n The augmenting path method ( 1 2 . 6 2 )

generates a sequence of

almost v-compatible (x;v). Further x is of minimum cost among N

all flows

with xoi = x

Proof: In step 1

oi

for all i E I.

x a 0 and v

e are Ao-compatible. Thus it

suffices to prove that an iteration does not violate almost u-compatibility. If at the end of the k-th stage (x;v) is almost u-compatible then the same holds at the beginning of the k+lth stage since xdk = ck. Thus it suffices to discuss. an iteration at the k-th stage. Let (x;v) denote the current pair in step 2 and let (x'Iv') denote the redefined variables. We assume that (x;v) is almost p-compatible and we will prove that (x';v') is almost p'-compatible

(11'

=

A(vr * a r ) ) .

288

Linear Algebraic Optimization

At first we consider the validity of the inequalities in (12.15).

Obviously, these inequalities are satisfied on arcs

(U,i), 1

5

i < k and ( j , ~ ) , j E J in the same way as in the

Hungarian method

-

v. * a , . = a

(12.64)

11

3

ij

due to our assumption. If Otherwise

71,

1

> n

Let (i,j) E I

(cf. 1 2 . 3 9 ) .

i

x J.

Then

(PI *vi

TI,

3

implies n

5

71.

>

TI

then v ! < a . . ( P ' ) *v; 3

i'

-

13

Therefore rri

.

is the value

-

> n Hence v . * T I of a shortest path in AGk and TI * a i ij - j' 7 < v + a . . * a = a ( 1 1 ) * v . * a i . u 5 P ' implies v . < a . . ( p ' ) * v i . - j 11 i ij 3 13 Now let x!

ij

> 0. Then either x

ij

> 0 or (i,j) €,;

path from a to T. If x . . > 0 then v . = a 7

13

-

a , . = e. Therefore ?ri =

( p ) * v i implies

which leads to v ! = a . . ( u ' ) * v i . We

TI

j

11

ij

the shortest

3

13

r e m a r k that N

x T o a = e.

(12.65)

Otherwise ( i , j ) € c . Then = a , . ( P ) *vi* 1 3

TI^.

u

5

TI,

3

=

a 11 , . * n .1

which implies

P' leads to v! = a 3

Secondly we remark v' = e and v'E(H a u'

ij

(11')

*v;

v ,

1

* T ,

3

.

u {ellN. Now we will show

N

u ' 5 x(z).

(12.65)

shows that x is a minimum cost flow with N

respect to weights a. Let

be a flow in G with

7ai

=

x' for ai

N

all i E I. Then x = x D A X where Ax is an incremental flow in hGk.

Due to ( 1 2 . 7 )

we know that Ax has a representation as

sum of incremental path flows and circuit flows r Ax =

x

S

pP(bp) +

p=r+l

p=l

and weights with respect to

x

2

satisfy

pp(dp)

Algebraic Flow Problems

289

If

i s o f minimum c o s t among a l l s u c h f l o w s t h e n

too.

Since w(p)

N

N

5

w(p 1 f o r a l l p = 1 , 2 ,

-

P

..., r

we f i n d t h a t x '

i s o f minimum c o s t w i t h r e s p e c t t o a , t o o .

G

Now l e t N

t o a. i

E

b e a maximum f l o w i n G o f minimum c o s t w i t h r e s p e c t

N

x contains a p a r t i a l flow

x

5

with

= x'

ai

oi

for a l l

I . Then

By t h e o r e m ( 1 2 . 3 8 ) t h e a l g e b r a i c t r a n s p o r t a t i o n p r o b l e m h a s an optimal solution yEPT. cT o v J

Therefore

*

J

GT 0 ; 5

5 yToa

yTo; yT 0;

*

c;ov,

(12.64) leads t o T

*

cI o v I =

N

2

*

c

T I D V I

-

implies

(x')Toz<

;*

T

-

cI U V I

lead t o

Since

u'

= h(v

T

* r T ) w e conclude t h a t x '

and v '

a r e almost

u'-

compatible. F i n a l l y w e c o n s i d e r a network G ' c a p a c i t i e s ci:= (?,TI)

x

oi

which d i f f e r s from G i n t h e

f o r i E I and c o n t a i n s an a d d i t i o n a l a r c

t o a n a u x i l i a r y s i n k T' w i t h c a p a c i t y

cTT,s=

xici

a T ~ .-, e . Let x T T , %= c

TT'

( x ; v ) i s u-compatible

and t h e r e f o r e due t o p r o p o s i t i o n

and v

I=

and

v T . Then t h e e x t e n d e d p a i r (12.18)

Lineor Algebraic Optimizatioii

290

of minimum cost. This implies that x in

G

is of minimum

cost.

Proposition ( 1 2 . 6 3 ) shows that all steps in the augmenting path method are well-defined. Further, if the method is finite, then at the end of the k-th stage a maximum flow x of minimum cost is determined.

(12.66)

Theorem

Let H be a weakly cance--ative d-monoid w ic

is an extended

semimodule over R + . Then (1 1

the augmenting path method

(2)

the final pair (x;v) is p-compatible,

(3)

(y;w,s) defined b y T

cJ0wJ = y T (-cl)

T

*

oa

0SI(U)

*

(12.37)

(12.62)

is finite,

satisfies

cTos I I ’

T

CJ O W J

=

yT.a

for u

N

=

A(z)

.

Proof: At first we assume finiteness. Then the final pair is iJ-compatible. Hence (y;w,s) is complementary and satisfies the claimed equations (cf. theorem 1 2 . 3 8 ) . Secondly we prove finiteness in a similar manner as in the proof of ( 1 2 . 2 8 ) .

It is sufficient to prove that the number of

performances of step 4 is finite for fixed parameter k. Assume that we find a shortest augmenting path of weight e in step 2 . Then in step 4 the dual variables are not changed. Subsequent iterations of this type are equivalent to the determination of

a maximum incremental flow in the partial graph o f b G k con-

Algebraic Flow h b l e m s N

29 1

. = e . The number o f s u b s e q u e n t i t e r a -

t a i n i n g only a r c s with a

13

t i o n s i s f i n i t e i f we c h o o s e a u g m e n t i n g p a t h s s u b j e c t t o t h e f i n i t e n e s s r u l e o f P O N S T E I N 1 1 9 7 2 1 o r i f we d i r e c t l y a p p l y a maximum f l o w a l g o r i t h m ( e . g . and X ( a T )

2 u

since

ij

E Hp

-

-

where

i

index

F.

= Vh(P)

u

{ e l . Let

Then a

path determined i n s t e p 2. Vh

KARZANOV [ 1 9 7 4 ] ) . O t h e r w i s e

* TIh

(h,T)

cz,

TI

> e

the shortest

= a h and

> vh

denotes the redefined dual variables with redefined N

The a r c s o f p a r e p a r t i t i o n e d i n t o

Suppose t h a t

p"

y

E and

5 i.

a p p e a r e d i n a p r e v i o u s i t e r a t i o n w i t h t h e same N

p a r t i t i o n of i t s arcs.

L e t v denote t h e corresponding redefined N

dual v a r i a b l e s with index

u

a f t e r t h e previous determination of

N

p.

Then

-

v

(12.67)

N

h

> v > v h . h -

-

N

F u r t h e r we r e m i n d t h a t v k = v k s i n c e Then

5

T . ~ j~

e a t t h e k-th

and

(12.68)

for all

(i,j)

€y

U {(s,r)I (r,s)

€ z } which

f o r c o n s t a n t s a , B d e f i n e d by

a:=

*

(i,j)

'

leads to

stage.

29 2

Linear Algebraic Optimization

(12.68) implies h(a.. 11

h(u)

( 7 ) )5 7

for all (j,i) €

5 ' ; and X ( B ) 5 ' ; . If '; < =;,=v

Therefore

then N

; h

5.

d. ,

c v

k -

h

contrary to (12.67). Otherwise cancellation of B ( i )

ih*B ( i ) -

in

N

= Vh* B(U)

N

leads to vh = v

h

contrary to (12.67).

Again we may derive complexity bounds in the discrete case (R = Z )

from the observation that 6C;)

5 1 in step

2.

Then

any iteration mainly consists in a determination of a shortest path and the redefinition of flow values and dual variables. These steps need O((m+n) positions, and O((m+n)

2

)

2

)

usual operations, O((m+n)

2

)

*-corn-

5 - comparisons. There are a t most

f = 1 c such iterations. In the particular case of assignment i i problems this leads to a complexity bound of O(n operations, O(n 3 )

*-compositions and O(n 3 )

3

)

usual

5 - comparisons.

These bounds are better than corresponding bounds for the Hungarian method

(by a factor o f O(n+m)).

We remark that in the classical case the augmentation method reduces to a method proposed by TOMIZAWA [1972];

computational

experience is reported in DORHOUT [1973] in the case of assignment problems. For bottleneck objectives computational experience is discussed in DERIGS and ZIMMERMANN, U.

([1978b],[1979]).

All methods that we previously described in this chapter can be interpreted as primal dual methods. Nevertheless, we shortly

Algebraic Flow Problem

293

mentioned the possible application of the primal simplex method (cf. 11.46) to the algebraic flow problem. In the following we discuss this method in the case of algebraic transportation problems.

We assume that H is a weakly cancellative d-monoid which is an extended semimodule over R+. W.1.o.g. it is well-known

let a E H y n . Since

(cf. LAWLER 119761) that the constraint ma-

trix describing the polyhedron P

T

in the algebraic transporta-

tion problem ( 1 2 . 3 3 ) is totally unimodular all steps of the primal simplex method are well-defined. In particular, for transportation problems it is easy to find an initial feasib'le basic solution. Only slight modifications are necessary (if any) in order to find such a solution for algebraic transportation problems. Then the primal simplex method generates a sequence of feasible basic solutions. We assume that cycling does not occur

-

the same assumption is usually made in the classi-

cal transportation problem. Thus we may forgo an explicit discussion of finiteness. Due to the analysis of the simplex method for linear algebraic programs we know that the final feasible basic solution x is i-dual feasible with

I:=

A(x

T

o a ) . Theorem

(11.41) implies that x is optimal. In the following we discuss

the primal simplex method in more detail and derive a particular form of the optimality criterion (11.42). mn Let x E R + be a basic feasible solution of PT (cf. 1 2 . 3 3 ) and let B denote the set of all arcs (i,j) for basic variables x Let v : =

X(x

T

ij'

ma). Then a u-complementary solution (cf. 11.38)

294

sE

Linear AIgebmic Optimization

(G,,

m+n

(12.70

can be determined solving the system of equations s .

1

=

a

ij

(p)

* si

for all (i,j) E B . We remark that

G

P

is the group containing H

P

(cf. discussion of extended semimodules in chapter 6 ) . The n e u t r a l element of G

u

is identified with e. It is well-known

that B is the arcset of a connected partial graph T = (I u J , B ) of the bipartite graph ( I U J , I

X J ) .

T contains no cycle and is

called a s p a n n i n g t r e e .

I

F i g u r e 8.

If a , . 11

=

Spanning tree T in (I U J . 1

X J )

e for all (i,j) E B then x is an optimal feasible s o l u -

tion. Otherwise a solution of ( 1 2 . 7 0 )

can be determined in the

following way. Let (r,t) E B with e < a rt ( P ) E G U . Let s

s := a and solve ( 1 2 . 7 0 ) t rt

along the tree using ( 1 2 . 7 0 )

sively. In general, this leads to a solution s E G

:=

e,

recur-

m+n which conP

tains some negative components. Then let 6:=

[min{skl

which exists in the group

k E I U J11-l G

.

?J

of ( 1 2 . 7 0 )

by

We define a solution v E ( H U{e))+ P

m+n

Algebraic Flow Problem

295

vk:= sk * 6 for all k E I UJ. Then v contains some components of value e ; v is the m i n i m a l p o s i t i v e S o Z u t i o n of ( 1 2 . 7 0 ) IBl = m + n

-

in

G

P

.

Since

1 such a solution can be determined with O ( m + n

*-compositions and O(m + n

-

1)

-

1)

5-comparisons. We remark that

(12.71)

for all (1,j) E B with a . E H \{el.

u

lj

(12.72)

Theorem

Let H be a weakly cancellative d-monoid which is an extended semimodule over R+. Let a E HY". with basis B and P : =

A(x

solution of ( 1 2 . 7 0 ) .

If

T

Let x E P

T

be a basic solution

ma). Let v denote the minimal positive

vj 5 ai,(v) * v i

(1)

5 v then x is an optiij ma1 basic feasible solution of the algebraic transportation for all (i,j) d B with v

j

> e and h (a

problem ( 1 2 . 3 3 ) . Proof: We define dual variables w

k

E G

P '

k E I U J by

Then ( 1 ) implies that w is u-dual feasible and v-complementary (cf. 1 1 . 3 8 and 1 1 . 3 9 ) .

Theorem ( 1 1 . 4 1 )

shows that x is an

optimal basic solution of the algebraic transportation problem (12.33)

and w is an optimal solution of the corresponding dual

problems. rn

Linear Algebraic Optimization

796

An optimality criterion similar to (12.72) is discussed in ZIMMERMANN,

[1979d].

U.

Here the involved combinatorial argu-

ments in that paper are not necessary since we can apply theorem (11.41) based on the duality principles from Z I M M E R M A N N , U. [1980a]. The main difference lies in the fact that instead of m+n a 'minimal' solution v E H + of the system of equations v. = a 3

ij

* v

i

for all ( i,j ) E B with I(v.1

I

5

a minimal positive Solution Of

LI

(12.70) is considered. The advantage of using (1) and not (cf. 1 1 . 4 2 ) w.

+ w . c a. 3 -

1

11

(IJ)

as optimality criterion lies in the fact that due to the choice of a minimal solution several v. will have value 0. Thus we 3

expect that ( 1 ) may lead to computational savings.

If the current basic solution x does not satisfy (1) then we may use a 'most negative reduced cost coefficient rule' similar to the classical case. Let

and let a ,

be defined by

11

(12.73)

v

j

=:

a ij * a ij ( l J ) * v i

for (i,j) E K . Then choose x (12.74)

a

rs

=

max(a.

lj

I

rs

with (r,s) E K and with

(i,j) E K )

as new basic variable in the new basis.

Algebraic Flow Roblem

297

It is easy to derive a generalization of the primal simplex method applied to transportation problems. An efficient implementation using the tree-structure of the basic solution is described in MURTY [ 1 9 7 6 1 . We give only an outline of the method showing the differences to the classical case.

(12.75)

Primal t r a n s p o r t a t i o n a l g o r i t h m

Step 1

Apply any of the methods for the determination of an initial feasible basic solution

with basis

g; let

-T u:= X(x na).

Step 2

Determine a minimal positive solution v E (I3

mtn

lJ

U {el)+

of the equations ( 1 2 . 7 0 ) ; if v

X(ai,) Step 3

ai,(p) * v i for (I,,)

C

j -

@ s with

5 u then stop (G is optimal).

v

,

> e and

Choose ( r ls ) E K with ars :=

max{ai,l

(if,) E K , vj:= ai, *aij(u) e v i l

and determine the new basis by introducing x

rs

into

the basis; redefine

i,

and

u and qo to step

2.

We remark that with these modifications of the general primal simplex method ( 1 1 . 4 6 ) it is possible to avoid an explicit use of the associated negative part of the groups variables in ( 1 2 . 7 5 )

G

lJ

.

All

are elements of the given semimodu1.e H.

The choice of the nonbasic variable in step 3 which is introduced into the basis leads in the classical case to the

298

Linear Algebmic Optimization

classical stepping stone method; for bottleneck objectives the method is equivalent to those of BARSOV 119641, SWARCZ (119663, 119711) and HAMMER

(119691, 119711) in the sense that

it generates the same sequence of basic solutions provided that the starting solutions are the same. In all these methods the entering nonbasic variable is determined investigating certain cycle sets defined with respect to the bottleneck values in the current basic solution. The above proposed rule using dual variables looks more promising as well from a theoretical as from a computational point of view.

Finally we give some remarks on particular algebraic transportation problems which have been considered in more detail. We forgo an explicit discussion of the classical problem and refer for this purpose to standard textbooks. Algebraic assignment problems were solved first in BURKARD, HAHN and ZIMMERMANN, U.

119771 using a generalized Hungarian

method. The augmenting path method was directly developed for algebraic transportation problems by BURKARD and ZIMMERMANN, U. 119801; that paper contains a short discussion of the specific

case of algebraic assignment problems, too. The primal simplex method for algebraic transportation problems is given in ZIMMERMANN, U.

[1979dl without explicit discussion of algebraic

assignments. FRIEZE [1979] develops a method for the algebraic assignment problem similar to the classical method of DINIC and KRONROD 119691. This method is of order O(n 3 ) similar to the augmenting path method. The solution of algebraic transportation problems is used for

Algebraic Flow Problems

299

deriving bounds in a branch and bound method for the solution of certain algebraic scheduling problems in BURKARD [ 1979c1.

Particular objective functions differing from the classical case which have found considerable interest in the literature are bottleneck objectives and lexicographical objectives (cf. 11.58 and 11.59). For lexicographical objectives a FORTRAN program is developed from the algebraic approach in ZIMMERMANN, U. 119761. Although the program is structured in order to show the impact of the algebraic approach it is easy to derive an efficient version for lexicographic objectives, in particular. Computational experience is discussed, too. Bottleneck assignment and bottleneck (or time) transportation problems have been considered by many authors. The following list will hardly be complete but hopefully provides an overview which seems to be necessary since even today similar approaches are separately published from time .to time. Such problems were posed and solved by BARSOV 119641 (1959 in russian). He proposed a primal solution method. Independently GROSS [1959] stimulated by FULKERSON, GLICKSBERG and GROSS [19531 investigated the same objective function for assignment problems. In the following years GRABOWSKI ([1964]., [1976]), SWARCZ "19661,

[19711) 'and

HAMMER ([1969], [1971]) developed solution methods. In particular HAMMER ([1969],

[1971]), SWARCZ [19711, SRINIVASAN and THOMPSON

[1976], ZIMMERMANN, U.

([1978a],

[1979d]) consider primal methods

equivalent to BARSOV'II method. They only differ in the method for the determination of the pivot element. GARFINKEL 119711 and

3 00

Linear Algebraic Optimization

G A R F I N K E L and R A O [1971] discuss primal dual methods based

on the threshold algorithm in E D M O N D S and F U L K E R S O N [1970]. The augmenting path method is finally developed within the scope of an algebraic approach by B U R K A R D and Z I M M E R M A N N , U. [19801. Further discussions can be found in S L O M I N S K I

(119761,

i19771, [19781). Computational experience is discussed by P A P E and S C H o N [1970], B U R K A R D [1975a], S R I N I V A S A N and T H O M P S O N Z I M M E R M A N N , U.

([1978b],

[19761, D E R I G S and

[19791), F I N K E and S M I T H 119791, and

D E R I G S [1979d]. F O R T R A N programs of the augmenting path method

are contained in D E R I G S and Z I M M E R M A M N , U.

([1978b],

[19791).

Computational efficiency has shown to be highly dependent on the choice of initial heuristics; by now, augmenting path methods and primal methods have been developed to nearly the same computational efficiency. Both seem to be faster than versions of the Hungarian method (also called primal dual or threshold method).

13.

Algebraic

Independent Set Problems

In this chapter we discuss the solution of linear algebraic optimization problems where the set of feasible solutions corresponds to special independence systems (cf. chapter 1 ) . In particular, we consider matroid problems and 2-matroid intersection problems; similar results for matching problems can be obtained. We assume that the reader is familiar with solution methods for the corresponding classical problems (cf. LAWLER [ 1 9 7 6 ] ) . Due to the combinatorial structure of these problems it suffices to consider discrete semimodules ( R = 72 1 ;

in fact, the

appearing primal variables are elements of {O,l}.

We begin with a discussion of linear algebraic optimization problems in rather a general combinatorial structure. In chapter 1 we introduce independence systems F which are subsets of the set P(N) of all subsets of N:= assume that F is norrnaz, i.e.

{1,2,

...,n}.

W.1.o.g.

we

{i) E F for all i E N . By defini-

tion (1.23) F contains all subsets J of each of its elements IEF.

In the following we identify an element I of F with its

i n c i d e n c e v e c t o r x E {0,1)n defined by (13.1)

x

j

= 1

CI*

jEI

for all j E N . Vice versa I = I ( x ) is the s u p p o r t of x defined by I = {j E N 1 xj = 1). A

linear description of an independence system F can be derived

in the following way. Let H denote the set of all flats (Or closed sets) with respect to F and let r: P ( N )

30 1

+

Z + denote the

301

Linear Algebraic Optimization

rank function of F. A denotes the matrix the rows of which are the incidence vectors of the closed sets C E H ; b denotes a vector with components b

:=

r(C), C E n . Then the set P of

all incidence vectors of independent sets is (13.2)

P = {x E Z:

I

Ax

5 b}.

We remark that such a simple description contains many redundant constraints; for irredundant linear descriptions of particular independence systems we refer to G R ~ T S C H E L 119771.

Let H be a linearly ordered, commutative monoid and let a E H

n

.

We state the following linear algebraic optimization problems: (13.3)

minfxToal ~ E P I ,

(13.4)

max{xToal

(13.5)

min{xT

0

a1 x E P ~ I ,

(13.6)

max{xT

0

a1 x E P ~ I ,

with Pk:=

~ E P I I

I x E P I E x . = k) for some k E N . 7

(13.3) and (13.5) can

trivially be transformed into a problem of the form (13.4) and (13.6) in the dual ordered monoid. Since P and Pk are finite sets all these problems have optimal solutions for which the optimal value is attained. Theoretically a solution can be determined by enumeration; in fact, we are only interested in more promising solution methods.

An element in P of particular interest i s the lexicographically

maximum Vector ;(PI. defined by

A related partial ordering

on 22

n

+

is

303

Algebraic Independent Set Problems

(13.7)

X'y

f o r x,y E

k 1 j=l

VkEN:

k

z:.

( 1 3 . 8 ) Proposition I f x E P i s a maximum o f P w i t h r e s p e c t t o t h e p a r t i a l o r d e r i n g (13.7)

t h e n x = x(P).

Proof:

I t s u f f i c e s t o show t h a t x

x,yEP.

>

Let x

f o r some k E N .

y.

= yi

Then x i

<, y i m p l i e s

for a l l 1

5

x -$ y

for all

i < k b u t x k > Yk

Therefore

k

k j=1

which shows t h a t x

k

y.

We i n t r o d u c e two v e c t o r s x

+

a n d xk d e r i v e d f r o m x E ZZ

i f a

>

j -

e

,

(13.9) otherwise

(13. 1 0 )

:=

{z.l

if j

5

I

j (k),

otherwise

f o r a l l j E N a n d f o r a l l k E 22+

w i t h j ( k ) : = max{jENI

j Z

'

x. zk}. i=1 The r e l a t i o n s h i p o f t h e s e v e c t o r s a n d t h e p a r t i a l o r d e r i n g (13.7) i e described i n t h e following proposition.

Linear Algebraic Optimization

3 04

(13.11) P r o p o s i t i o n Let ii be a linearly ordered, commutative monoid. Hence H is a

linearly ordered semimodule over

LI =

+ xx.

and

3

e

(1)

Proof:

(2)

V

xy

=

j'

Z + .

Let x,y E

with

P,:

Then

.

5 (x+lT0a

( 1 ) is an immediate implication of (13.9).

y 5 x implies

k(x'):= rnin{jl

yj

*

x'.

-

Exv = v . j

Assume y

*

X I : =

xv and let

Then 6:= x ' - y k > 0. Define x" by k

XI).

j

6

i f j = k ,

x " :=

if j = k+l, otherwise

Then y 5 x " and due to 6 O a

k+l

-<

.

6 O a k we get

-

( x " ) ~ O< ~ ( x ' ) ~ O ~

If y = x " then (2) is proved. Otherwise we repeat the procedure for x " . Since k(x") > k(x') after a finite number of steps we find y = x " . (31

If

r

5

P

5

y then xr j

- xj

= 0 for all j with a

j

< e. There-

f o r e xr < xs for all j E N implies the claimed inequality. If 1 - j

;

S for all j with a . 2 e. Further x 2 xj xj I for all j E N implies x r O a < xs O a f o r all j with a < e. j j - j j j This proves the claimed inequality.

LI

5 s 5

(4)

r then xr =

I

From (2) we get y T o a 5 ( x " ) ~O a . From ( 3 ) we know

Algebraic Independent Set Pmblems

305

( ~ ~ ) 5~ (o ~ a+ ) ~ ofor a all k E Z + . F o r k = u we find the claimed inequality.

The importance of proposition

( 1 3 . 1 1 ) for the solution of

the linear algebraic optimization problems (13.4) and ( 1 3 . 6 ) can be seen from the following theorem which is directly implied by ( 1 3 . 1 1 ) and (13.8).

(13.12) T h e o r e m Let H be a linearly ordered, commutative monoid. Hence H is a linearly ordered semimodule over Z + .Let

x

denote the lexico-

graphically maximum solution in P and let k E N with 1

< -

max(Ex.1 1

xEPI.

5 k 5

~f

...

(1)

a 1 -> a 2 ->

(21

P has a maximum with respec

> -

an ' to the partial o r ering

(13.7)

;+lToa

= max{xTnal

~ E P I ,

x -k T o a = maxfxT m a 1 XEP,}.

Theorem ( 1 3 . 1 2 ) gives sufficient conditions which guarantee that a solution of (13.4) and (13.6) can easily be derived from the lexicographically maximum solution

x

in P. It is well-known

(cf. LAWLER [ 1 9 7 6 ] ) that the following algorithm determines the lexicographic maximum x(P); ej vector o f 23

n

.

, jE

N denotes the j-th unit

306

Linear Algebraic Optimization

(13.13)

Greedy algorithm

Step 1

x

Step 2

If x + e . E P 1

Step 3

If j = n then stop; j:=

0; j:= 1 .

j + 1

then

x:= x + e . 7

.

and go to step 2.

If we assume that an efficient procedure is known for checking x i e . E P in step 2 then the greedy algorithm is an efficient 3

procedure for the determination of x(P).

I t is easy to modify

the greedy algorithm such that the final x is equal to

Condition ( 1 3 . 1 2 . 1 )

x+

-k or x

.

can obviously be achieved by rearranging

the components of vectors in Z n .Let Il denote the set of all permutations n: N

-B

N. For n E Il we define a mapping

n:

Zn+Zn

by

which permutes the components of x E 22 inverse permutation of n). to achieve (13.12.1)

If

TI

accordingly (n-'

is the

is the necessary rearrangement

then fi[Pl:= {i(x)I

x E P 1 is the corres-

ponding new independence system. We remark that +[PI does not necessarily have a maximum with respect to the partial ordering

(13.7) if P satisfies (13.12.2).

If ;[PI

satisfies ( 1 3 . 1 2 . 2 )

for all n E TI then the lexicographically maximum solution ;(;[PI) leads to an optimal solution if

307

Algebraic Independent Set Problems

Such independent systems are matroids as the following theorem shows which can be found in ZIMMERMANN, U.

119771. Mainly

the same result is given in GALE [1968]; the difference lies in the fact that GALE explicitly uses assigned real weights.

(13.14) T h e o r e m Let T(P) denote the independence system corresponding to P. Then the following statements are equivalent: (1)

T(P) is a matroid,

(2)

for all n E I I there exists a maximum of ;[PI

with

respect to the partial order (13.7). Proof: (1)

*

(2).

If T(P) is a matroid then T(n[P])

is a matroid

since the definition of a matroid does not depend on a permutation of the coordinates in P. Thus it suffices to show that P contains a maximum with respect to the partial ordering (13.7). Let

=

;(PI

Suppose y

&

be the lexicographical maximum of P and let y E P.

x.

Then there exists a smallest k E N such that k

k

I: Y j >

I:

j=l Let J : = {jI y j =. 1, j

5

k);

j=l

xj

.

-

I:= {jl x

=

j

1, j < k}. Then

IJl > 111 and due to (1.24) there exists an independent set IU

{ u ] for some

x'

%

( 2 ) =+

x

p E J L I . The incidence vector x' of I satisfies

contrary to the choice of

-

X.

(1). Assume that T(P) is not a matroid. Then there exist

two independent sets 1,J with 111

C

IJI such that for all

j E J X I the s e t I U {j} is dependent. We choose n E l l such that

n[I] < n[J\I]

< n[N\ (IUJ)]

.

3 08

Linear Algebraic Optimization

Let u denote the incidence vector of J. If ;(PI

contains a

maximum n(y) with respect to the partial ordering (13.7) then proposition (13.8) shows that the lexicographical maximum

-

- -

x = X ( n ( P ) ) satisfies

= n(y).

yi =

Therefore i(u)

0

for all i E J \ I .

that ;[PI

Then y

i

1 for all i E I and

=

i(y) which implies

does not contain a maximum with respect to the parti-

cular ordering (13.7).

Due to the importance of the lexicographical maximum x ( P ) we describe a second procedure for its determination. Let

5 be a partial ordering on Zn. Then we define

5'

x

(13.15)

-

y

O(y)

5

on 22

n

by

G(x)

for a € ll with o ( i ) = n - i + 1 for all i E N .

In particular we

may consider the partial orderings i'and 4 ' . We remark that x 5' y implies x -s'

y.

(13.16)

M o d i f i e d greedy a l a o r i t h m

Step 1

x

Step 2

If x + e

Step 3

If j = 1 then stop; j:=

0;

j

-

j:= n.

j

EP

then

xi= x + e

j'

1 and go to step 2 .

The final vector x in this algorithm is denoted by x ' ( P ) . Since the application of this algorithm to P is equivalent to the application of the greedy algorithm to ;[PI

.

we get x ' ( P ) =;(:[PI)

(13.15) shows that x' Is the minimum of P with respect to 4'

.

Algebraic Independent Set Problem

1

L e t r:= max{Ix. 1

p*:=

309

x E PI and d e f i n e Iy€{0,11"1

3 xEP,:

y

5

1

- X I .

T h e n P* c o r r e s p o n d s t o a c e r t a i n i n d e p e n d e n c e s y s t e m T ( P * ) .

In p a r t i c u l a r , i f T ( P ) i s a m a t r o i d t h e n T ( P * ) i s t h e d u a l matroid

( c f . c h a p t e r 1 ) . T h e r e f o r e w e c a l l P* t h e d u a l of P.

L e t s:= m a x { I y i l

Y E P * ) . Then r + s = n.

(13.17) Proposition I f P h a s a maximum w i t h r e s p e c t

-

then x(P) = 1 Proof:

S i n c e ;(PI

t i a l ordering x

x

i

- x'

t o t h e p a r t i a l ordering (13.7)

(P*).

is t h e maximum o f P w i t h r e s p e c t t o t h e pa.r-

(13.7),;EPr.

Thus 1 - , € P i .

Let xEPr.

Then

means by d e f i n i t i o n k

r

-

j=l xJ

for a l l kEN.

k

<

X

F u r t h e r Ex n

I:

>

-

j -k+ 1 x J f o r a l l k = O,l,.

. ., n - 1

i s t h e maximum o f P

r

i

j-1

j

-

j

= Ex

j

= r . Thus

n E x j j=k+l

i ( x ) . Therefore

which shows

w i t h respect t o

i'. Hence

x

i s the maxi-

mum o f P r w i t h r e s p e c t t o 4 ' . T h u s

1 - x 6' 1 - x for a l l x E P Let y € P*xPs. implies

and 1

-x

i s t h e minimum of P* w i t h r e s p e c t t o 4 ' . 9

Then t h e r e e x i s t s

P:.such

that y

5

which

4' y . T h e r e f o r e 1 - x is t h e minimum o f P* w i t h

-

r e s p e c t t o 6'. H e n c e 1 - x = x ' ( P * ) .

Linear Algebraic Optimization

310

Proposition ( 1 3 . 1 7 )

shows that the lexicographical maximum

x(P) can be derived a s the complement of x'(P*I determined by ( 1 3 . 1 6 )

which is

applied to P*. The application of the

modified greedy algorithm to P * is called the d u a l g r e e d y

a z g o r i t h m . This is quite a different method for the solution of the optimization problems ( 1 3 . 4 )

and ( 1 3 . 6 ) .

In particular,

checking x + e . E P in step 2 of the greedy algorithm is reI placed by checking x + e . E P * in the dual greedy algorithm. 3

Therefore the choice of the applied method will depend on the computational complexity of the respective checking procedures.

For a more detailed discussion on properties o f related partial orders we refer to ZIMMERMANN, U.

([19761,

[19771).

The greedy

algorithm has been treated before by many authors. KRUSKAL 1 1 9 5 6 1 applied it to the shortest spanning tree problem, RADO 1 1 9 5 7 1 realized that an extension to matroids is possible.

WELSH 1 1 9 6 8 1 and EDMONDS 1 1 9 7 1 1 discussed it in the context of matroids. The name 'greedy algorithm' is due to EDMONDS. Related problems are discussed in EULER 1 1 9 7 7 1 ;

a combinatorial

generalization is considered in FAIGLE 1 1 9 7 9 1 . For further literature, in particular for the analysis of computational complexity and for related heuristics we refer to the bibliographies of KASTNING 1 1 9 7 6 1 and BAUSMANN 1 1 9 7 8 1 . For examples of matroids we refer to LAWLER [ 1 9 7 6 1 and WELSH 1 1 9 7 6 1 .

The following allocation problem is Solved in ALLINEY, BARNABEI and PEZZOLI [ 1 9 8 0 1 for nonnegative weights in the group ( I R , + , z ) . Its solution can be determined using a special (simpler) form of the greedy algorithm ( 1 3 . 1 3 ) .

Algebraic Independent Set Problems

(Allocation Problem)

Example A

31 1

system with m units (denoted a s set T) is requested to serve

n users (denoted as set S). It i s assumed that each user can engage at most one unit. Each user i requires a given amount a

i

of resources and each unit j can satisfy at most a given required amount b unit j iff ai

5 b

with vertex sets

of resources. Thus user i can be assigned to j' S

Let G = (S U T , L ) denote the bipartite graph and T and edge set L where (i,j) E L iff

< b.. A matching I is a subset of L such that no two diffei I rent edges in I have a common endpoint. A set A 5 S is called

a

assignable if there exists a matching I such that

A =

{il

(i,j)EI).

The set of all assignable sets is the independence system of a matroid

(cf. EDMONDS and FULKERSON 1 1 9 6 5 1 or WELSH [ 1 9 7 6 ] ) .

In

particular, all maximal assignable sets have the same cardinality. We attach a weight c i E H + to each user i where (A,*,() a linearly ordered commutative monoid.

(13.12)

is

and ( 1 3 . 1 4 )

show that we can find an optimal solution of the problem max{xTUcI

x E PI by means of the greedy algorithm ( 1 3 . 1 3 )

notes the set of all incidence vectors of assignable sets).

(P de-

Due

to the special structure o f the matroid it is possible to simplify the independence test ( x + e E P ) in step 2 o f the greedy j algorithm. In fact, we will give a separate proof o f the validity o f the greedy algorithm for the independence system T(P). Together with theorem ( 1 3 . 1 4 )

we find that T(P) is a matroid.

= max{ckl k E S 1 . If Ti:= {j € T I ai 5 b 1 = 0 then there i j exists no unit which can be assigned to user i. Then G ' denotes

Let c

the subgraph generated by (S\{i))

U T. If we denote the optimal

Linear Algebraic Optimization

312

value of an assignable set of (11

z ( G ' )

If Ti

$

= z G)

0 then let b . 3

by

z(G)

then

-

min{b

=

G

k

I k E T i } . Let I denote a match( I J , E~ ) I}.

ing corresponding to an optimal assignable set {vl

Every such matching is called an "optimal matching". We claim (i,j) E I. Let V:= { u , p l

that we can assume w.1.o.g. If i , j 6 V then I U {(i,j)}

( I J , ~ )E

I}.

is an optimal matching, too. If

i E V , j e V then we replace (i,k) E I by

(i,j). If i @ V , j E V

then we replace (k,j) € 1 by (i,j); since c . is of maximum value the new matching is optimal, too. If i,j E V but then ,we replace (i,v), ( I Jj) , E I by (i,j) and ( p , v ) . that

( I J , ~ )E

Let G ' Then I

L since a

We remark

5 bv. Thus we can assume (i,j) E I.

< b.

u -

1

U (T'{j}).

denote the subgraph generated by (S'{i}) { (i,j) 1 is a matching in G '

.

Therefore z

(G)

On the other hand, if I' is an optimal matching in I' U { (i,j)) is a matching in G and thus Therefore in the case Ti (2) ( 1 ) and

z ( G ' )

(2)

*

ci =

$

(i,j) a I

z ( G ' )

*

c

< i -

5 G ' ,

z (G')

* ci.

then

z ( G ) .

@ we find

z ( G ) .

show the validity of the following variant of the

> c2 > greedy algorithm. W e assume c 1 -

0;

Step 1

I =

Step 2

If T.:= {b

T:= { 1 , 2 ,

i:= 1 ;

k

1 -

I:= ~ U { ( i , j ) l

;

j

=

Cn.

...,m}.

I a . < bk, k E T }

find j E T . such that b

T:= ~'{j}.

. - -->

=

0 then go to step

min Ti

;

3;

313

Algebraic Independent Set Problems

Step 3

If i = n then stop; i:= i

+

1 and go to step 2 .

At termination I is an oFtimal matching which describes the assignment of the users i of an optimal assignable set A* = {il

(i,j) E 11 to units j. It should be clear (cf. theo-

rem (13.12)) that {ii

( i , j ) € 1 1 is optimal among all assign-

able sets of the same cardinality at any stage of the algorithm and for arbitrary c E H i

..

(iE.5)

satisfying c 1

5 cn . In particular, we may consider (IR,+ , L ) and

min,l). Then for ciEIR, i

E S

2

c2

2

..

(1R U Em},

we find

for all assignable sets A with

IAl

=

lA*l;

i.e. the sum as

well as the minimum of all weights is maximized simultaneously.

In the remaining part of this chapter we discuss linear algebraic optimization problems for combinatorial structures which do not contain a maximum with respect to the partial ordering (13.7), in general. We assume that H is a weakly cancellative d-monoid. Hence w.1.o.g.

H is an extended semimodule over Z +

(cf. 5.16 and chapter 6). F r o m the discussion in chapter 1 1 we know that max- and min-problems are not equivalent in d monoids due to the asymmetry in the divisor rule (4.1). In

Linear Algebraic Optimization

314

chapter 1 1 we developed reduction methods for both types of algebraic optimization problems which reduce the original problem to certain equivalent problems in irreducible sub-d-monoids of H ; since H is weakly cancellative these sub-d-monoids are parts of groups. Let ( H A ; A E h ) denote the ordinal decomposition of H. Then

for an incidence vector x of an independent set ( A o : =

min A ) .

According to ( 1 1 . 2 1 ) and ( 1 1 . 5 4 ) the optimization problems (13.3)-

-

( 1 3 . 6 ) are reduced to optimization problems in H

(13.18) (13.19)

u1 u2

=

min{A(xTma)

I

~ E P ,I

=

max{A(xTna)

I

x~ P I

(13.20)

p 3 = minlA(x

(13.21)

p4

T

u with

,

a) I x E pkI,

= maxrA(xTna)l

~ E P ~ I .

Since o E P the reduction ( 1 3 . 1 8 ) of ( 1 3 . 3 ) is trivial:

u 1 - Ao.

The reduction ( 1 3 . 1 9 ) of ( 1 3 . 4 ) is also simple; since the corresponding independence system is normal the unit vectors e

j

,

j E N are elements of P. Therefore

uz

=

maxIA(aj)

I

j EN).

For the determination of u 3 we assume that it is possible to compute the value of the rank function r : P(N)

+

Z+

for

a given subset of PJ by means of an efficient (polynomial time) procedure. This is a reasonable assumption since we will consider only such independence systems for which the classical

Algebmic Independent Set Problems

315

combinatorial optimization problems can efficiently be solved; then an efficient procedure for the determination of an independent set of maximum cardinality in a given subset of N is known, too. In chapter 1 the closure operator u: P ( N )

-D

P(N)

is

defined by

A

subset I of N I s called c l o s e d if U(1) = I. Clearly, if the

rank function can efficiently be computed then we may efficiently construct a closed set J containing a given set I. J is called a m i n i m a l c l o s e d c o v e r of I. Every set I C_ N has at least one minimal closed cover J , but in general J is not unique. Let s : = k - r ( J )

2

0 (cf. 13.5 and 13.6).

Then we define the

b e e t r e m a i n d e r s e t AJ by i f s = O , (13.22)

AJ:=

{il,i2,...,is}

5

otherwise,

...

.

5 < a. The It i2 following method for the determination of p 3 is a refinement

where N L J = {il,i2,...,it}

and a

a

of the threshold method in EDMONDS and F U L K E R S O N 119701.

(13.23)

R e d u c t i o n m e t h o d f o r (13.5)

Step 1

u:=

Step 2

If r(K) 2 k then stop;

max{A(a.)I

I

j€A@};

K:= {j

€NI

A(a,)

determine a minimal closed cover J of K.

5

p}.

Linear Algebraic Optimization

316

In the performance of (13.23) we assume k method terminates in O(n) steps.

A

5 r(N). Then the

similar method for the deri-

vation of thresholds for Boolean optimization problems in linearly ordered commutative monoids is proposed in ZIMMERMANN, U. [ 1978cl.

(13.24)

Proposition

u in

The final parameter

(13.23) i s equal to

u 3 in

(13.20).

Proof: Denote the value of u before the last revision in step 3 by

i.The

corresponding sets are denoted by

r ( K ) < k. Since Pk

*

@ we conclude

.

u3 >

k , 5 and

Aj.

Then

Further r ( ? ) = r ( K )

shows that an independent set with k elements has at least s = k

- r(k)

elements in N L j . Thus "3 > maxE)i(aj)

I

jE

which implies u 3 2 u. Now r ( K )

A ~ I

5 k shows

p3

5 u.

W e remark that for bottleneck objectives (cf. 11.58)

the re-

duction method (13.21) leads to the optimal value of (13.5) since A(x

T

a) = xT m a in this case.

In the determination of u 4 we consider certain systems F j

, jEN

derived from the underlying independence system F. We define (13.25)

FJ:= {I E F I

I

u

{j} E F , j

d I)

for j E N.

Algebraic Independent Set Problems

317

Obviously F j is an independence system for each j E N . Its rank function is denoted by r j

a

1

> -

a2 >

.

W.1.o.g.

we assume

- - .-> a n

in the following method.

(13.26)

Reduction method f o r ( 1 3 . 6 1

Step 1

w:=

Step 2

If r ( N )

Step 3

w:=

1. W

w

+

2

k

-

1 then stop ( p = X(a,)).

1;

to step 2 .

go

Finiteness of this method is obvious.

(13.27)

Proposition

The final parameter w in ( 1 3 . 2 6 ) satisfies p 4 = h(aw) for p 4 in ( 1 3 . 2 1 ) . Proof: Let

V

denote the final parameter in ( 1 3 . 2 6 ) .

Then there

exists an independent set I E F of cardinality k such that j E I. Therefore h ( a w ) r

w-1

5

pa,

If v = 1 then equality holds. Otherwise

(N) < k-1 shows that there exists no I E F of cardinality k

such that j €.I. Thus p 4

5

A(aw).

Again, we remark that for bottleneck objectives the reduction method ( 1 3 . 2 7 )

leads to an optimal value of ( 1 3 . 6 ) .

After determination of the corresponding index optimization problems ( 1 3 . 3 )

-

(13.6)

u the algebraic

are reduced according to

L i n w Algebraic Optimization

31R

(11.23)

and ( 1 1 . 5 5 ) . The sets of feasible solutions P and P k

are replaced by P : = {xMI x E P M

and x

(Pk)M:= CxMl x E P k and x with M =

M(p)

= {j E N 1

X(a.1 7

5 u).

= 0 for all j E M ) ,

j

= 0

j

for all j E M }

I f F denotes the correspond-

ing independence system then ( 1 3 . 2 8 )

shows that the reduced sets

of

feasible solutions correspond to the restriction F

M,

i.e.

M

of F to

(13.28)

which is again a normal independence system. Theorems ( 1 1 . 2 5 ) and ( 1 1 . 5 6 )

show that ( 1 3 . 3 )

-

(13.6)

are equivalent to the

following reduced l i n e a r a l g e b r a i c o p t i m i z a t i o n probZems T

(13.29)

min{x

(13.30)

max{x

(13.31)

minIx

(13.32)

max{xT

T

o a ( ~ ) ~XIE P # I

,

oa(u)MI XEP,}

,

T 0

a(v),l

xE

a(ujMI x E (pklM)

with respect to M = M ( p ) and p = p i , i = 1 , 2 , 3 , 4 defined by (13.18)

-

(13.32)

are problems in the extended module G

p = pi

, i

(13.21).

Since H i s weakly cancellative ( 1 3 . 2 9 ) 1!

over ZZ

-

for

= 1,2,3,4.

Next we consider two particular independence systems. Let F denote the intersection F

n

F 2 o f two matroids F 1 and F

2

where F 1 and F 2 are the corresponding independence systems. An element I E F is called an i n t e r s e c t i o n . Then ( 1 3 . 3 )

-

(13.6)

Algebraic Independent Set Problem

319

are called a l g e b r a i c m a t r o i d i n t e r s e c t i o n p r o b l e m s . In particular, ( 1 3 . 5 )

and ( 1 3 . 6 )

are called a l g e b r a i c m a t r o i d k - i n t e r -

s e c t i o n p r o b l e m s . Let (V,N) denote a graph. Then I

N is

called a m a t c h i n g if no two different edges in I have a common endpoint. We remark that in a graph (i,j) and ( j , i ) denote the same edge. The set of all matchings is an independence system (13.3)

F. Then

-

(13.6)

In particular, ( 1 3 . 5 )

are called a t g e b r a i c matching p r o b t e m s .

and ( 1 3 . 6 )

are called a t g e b r a i c k-match-

ing probtems.

The classical matroid intersection problem is well-solved; efficient solution methods are developed in LAWLER ( [ 1 9 7 3 ] ' , [19761), [1979].

IRI and TOMIZAWA [ 1 9 7 6 1 , FUJISHIGE 1 1 9 7 7 1 and EDMONDS An augmenting path method and the primal dual method

are described in the textbook o f LAWLER [ 1 9 7 6 ] ;

FUJISHIGE [ 1 9 7 7 ]

considers a primal method. The classical matching problem is well-solved, too; an efficient primal dual method f o r its solution is given in EDMONDS [ 1 9 6 5 1 and is described in the textbook of LAWLER 1 1 9 7 6 1 . A primal method is considered in CUNNINGHAM and MARSH 1 1 9 7 6 1 . The necessary modification of the primal dual method for the classical versions of ( 1 3 . 5 ) (13.6)

is discussed in WHITE [ 1 9 6 7 ] .

and

For the classical matroid

intersection problem such a modification is not necessary since in the shortest augmenting path method optimal intersections of cardinality 0 , 1 , 2 , . . .

are generated subsequently (cf.

LAWLER [ 1 9 7 6 ] ) .

In particular, efficient procedures for the determination of a

Linem Algebraic Optimization

320

matching

( m a t r o i d i n t e r s e c t i o n ) o f maximum c a r d i n a l i t y i n a N a r e d e s c r i b e d i n LAWLER

g i v e n s u b s e t N'

[1976].

Thus t h e

e v a l u a t i o n o f t h e rank f u n c t i o n i n step 2 of t h e r e d u c t i o n method

is possible i n polynomial t i m e f o r both pro-

(13.23)

blems. Let F be t h e

(cf.

13.25)

w h e r e N'

set of a l l m a t c h i n g s i n t h e g r a p h

is t h e s e t of a l l m a t c h i n g s i n t h e g r a p h

is t h e set of

common w i t h j sections.

( V , Nj )

a l l edges which have no e n d p o i n t i n

(jEN). L e t F = F 1 n F 2 be t h e set of a l l inter-

Then F j

= Fi

nF;.

a l l i n d e p e n d e n t s e t s of of

(V,N). T h e n F J

F u r t h e r F'

1

( a n d F:)

a certain matroid

t h e o r i g i n a l m a t r o i d t o N'

{ j),

cf.

is t h e set of

(called contraction

WELSH

[1976]).

Thus the

e v a l u a t i o n of t h e r a n k f u n c t i o n r v , V E N i n s t e p 2 o f t h e r e d u c t i o n method

(13.26)

is p o s s i b l e i n polynomial t i m e

for both

problems. The r e s t r i c t i o n FM ( c f .

1 3 . 2 8 ) l e a d s t o t h e set o f a l l match-

i n g s i n t h e g r a p h (V,M) and t o t h e i n t e r s e c t i o n o f t h e t w o restricted matroids blems

(13.29)

-

( F 1 l M f l( F 2 ) M . T h e r e f o r e t h e r e d u c e d p r o -

( 1 3 . 3 2 ) are a l g e b r a i c matching

section) problems provided . t h a t matching

-

( 1 3 . 6 ) are algebraic

(matroid i n t e r s e c t i o n ) problems.

It remains t o

modules.

(13.3)

(matroid inter-

solve t h e reduced problems i n t h e respective

A l l methods f o r t h e s o l u t i o n o f

the classical matroid

i n t e r s e c t i o n problem are v a l i d and f i n i t e i n modules.

A re-

f o r m u l a t i o n of t h e s e m e t h o d s i n g r o u p s c o n s i s t s o n l y i n rep l a c i n g t h e u s u a l a d d i t i o n of r e a l numbers by t h e i n t e r n a l composition i n t h e u n d e r l y i n g group and i n r e p l a c i n g t h e u s u a l l i n e a r o r d e r i n g o f t h e r e a l numbers by t h e l i n e a r o r d e r ing i n t h e underlying group.

Optimality o f t h e augmenting p a t h

32 1

Algebraichiependent Set Problems

method is proved by KROGDAHL (cf. LAWLER 119761) using only combinatorial arguments and the group structure (i.e. mainly cancellation arguments) of the real additive group. Thus his proof remains valid in modules. Optimality of the primal method in FUJISHIGE [I9771 is based on similar arguments which remain valid in modules, too. Optimality of the primal dual method is based on the classical duality principles. From chapter 1 1 we know how to apply similar arguments in modules. In the following we develop these arguments explicitly; for a detailed description of the primal dual method we refer to LAWLER 119761. Let Ax

5 b and

xx

5

be the constraint systems (cf. 13.2) with

respect to the restricted matroids. Then (13.33)

In modules it suffices to consider max-problems since a minproblem can be transformed into a max-problem replacing the cost coefficients by inverse cost coefficients. We assign dual variables ui and vk to the closed sets 1 and k of the restricted matroids. Then (u,v) is dual feasible if A

(13.34)

T

*

ou

NT A ov

N

where a = a(v),

and where

2

N

a, ui

2

e l vk

2

e

u denotes the respective index of

the reduction. The set of all dual feasible (u,v) is denoted by DM. Then the algebraic dual of (13.30) (13.35)

min{b

T

ou

*

gTovl

is

(u,v) E DH).

In the usual manner we find weak duality, i.e. x T O z 5 bT O u

NT b Ov.

Linem Algebraic Optimization

322

The corresponding complementarity conditions are

(13.37)

*

ui > e

(Ax)i = b i

(13.38) N

for all j E M and all closed sets i and k. If a j E M then x

0, u

E

e, v

=

< e for all

1 -

e is an optimal pair of primal

and dual feasible solutions. Otherwise the initial solution x s 0 , v s e , ui = e for all i

*

M and

u : = maxIZ. I j E M ) M

3

is primal and dual feasible. Further all complementarity conditions with the possible exception of (13.39)

are satisfied. We remark that bM is the maximum cardinality Of an independent set in one of the restricted matroids. Such a pair (x;u,v) is called compatible, similarly to the primal dual method for network flows. Collecting all equations ( 1 3 . 3 6 ) after external composition with x (

j

we find

13.40)

for compatible pairs. The primal dual method proceeds in stages. At each stage either the primal solution is augmented or the values of the dual variables are revised. At each stage no more than 21MI dual variables are permitted to be non-zero. Throughout the performance of the method the current pair (x;u,vl is compatible. This is achieved by an alternate solution of

323

Algebraic Independent Set Pmblems

max{x

(13.41)

T

O

-

a1 x E P M

(x;u,v) compatible)

I

for fixed dual feasible solution (u,v) and of minjb

(13.42)

T

* FT 0 vI

0u

(u,v) E DM

, (x;u,v) compatible}

f o r fixed primal feasible solution x. From

that ( 1 3 . 4 1 )

i s equivalent to

(13.43)

max{ 1

xEPM

I

we conclude

(x;u,v) compatible)

1

jEM

is equivalent to

and that ( 1 3 . 4 2 )

(u,v) E D M

min{uMI

(13.44)

I

x

(13.40)

,

(x;u,v) compatible).

An alternate solution of these problems is constructed in the same way as described in LAWLER [ 1 9 7 6 1 for the classical case. In particular, the method is of polynomial time in the number of usual operations, *-compositions and (-comparisons.

The

final compatible pair (x;u,v) is complementary. Thus it satisfies xT

0:

= bT

u

* FT 0 v;

then weak duality shows that

(x;u,v) is an optimal pair. Therefore this method provides a constructive p r o o f of a duality theorem for algebraic matroid intersection problems. N

M

5 c and Cx 5 c be the constraint systems (cf.

Let Cx

13.2)

with respect to the two matroids considered. Then (13.45) A

closed set I of one of the matroids with I C_ M is closed,

with respect to the restricted matroid, too. We remark that N

c, = b

j

u

(c, = b

j

)

f o r all j E M .

We assign dual variables u i , v k to the closed sets i and k of the matroids. Then (u,v) is dual feasible with respect to

Linear Algebraic Oprimization

37-4

maxfx

(13.4')

T

oa(p2)1 X E P )

if

*

cTou

(13.46) The set of

-T

c

2

o v

a l l dual feasible

a l g e b r a i c dual o f

(13.4)

min{cTou

2

a ( u 2 ) , ui

2

e, v k

e

( u , v ) i s d e n o t e d by D 2 .

. Then t h e

is

*

-T

c

( u , v ~E D ~ I .

0v1

Algebraic dual programs with r e s p e c t t o (13.3')

minix

T

oa(ul)

I

XEPI

a r e c o n s i d e r e d i n c h a p t e r 11. with respect t o

(13.3') e

An o b j e c t i v e

if

za(ul)

X(ui)

(u,v) i s strongly dual feasible

*

C

5 u1 ,

~

h(vk)

*

-T

c

O

o ~v

,

5 u l , e 5 ui

function is defined according t o

,

e z vk '

(11.19).

( 1 3 . 4 7 ) Theorem ( M a t r o i d i n t e r s e c t i o n d u a l i t y theorem) L e t H be a weakly c a n c e l l a t i v e d-monoid.

Hence H i s an e x t e n -

d e d s e m i m o d u l e o v e r Z + .T h e r e e x i s t o p t i m a l f e a s i b l e p a i r s ( x ; u , v ) f o r t h e a l g e b r a i c matroid i n t e r s e c t i o n problems and

(13.4')

which a r e complementary a n d s a t i s f y

(11

e = x T o a * c

(2)

( - c l T 0 u ( u1 1

(31

cTou

Proof: -

(13.3')

T

*

-T

o u * c - T

(-c)

(for 13.3')

o v

ov(v,) = x

T

ma

TT0v = xT0a

( 1 ) . The m i n - p r o b l e m

( f o r 13.3'1, (for 1 3 . 4 ' ) .

( 1 3 . 2 9 ) i s t r a n s f o r m e d i n t o a max-

problem o f t h e form ( 1 3 . 3 0 ) r e p l a c i n g a

j

(v,)

by i t s i n v e r s e i n

325

Algebmic Independent Set Problems

for a l l j E M = M(pl).

t h e group G

- - -

(x;u,v) denote the

Let

p1 f i n a l p a i r generated i n the application of the primal dual Then f o r j E M :

m e t h o d t o t h i s max-problem. a.(pl) 7 when A a n d

-1

5

(

~

~

*0 . i;i T o v )j ?

a r e s u b m a t r i c e s o f C and

( c o l u m n s j E M and rows

o f c l o s e d s e t s 1 5 M a n d row a ( M ) w i t h t h e c l o s u r e f u n c t i o n of t h e respective matroid). L e t if ] E M , x j:=

otherwise then x E P .

Further l e t if i $ M u := i

I

if i = U 1 ( M ) ,

e

otherwise

where a l is t h e c l o s u r e f u n c t i o n o f t h e m a t r o i d F 1 and l e t i f k 5 M v

k

I

i f k = a2(M),

:=

otherwise where a 2 is t h e c l o s u r e f u n c t i o n o f t h e m a t r o i d F 2 .

e for a l l j E M .

*

aj

T (C n u

*

ZTov)

j

N o w A ( a . ) > u1 f o r j 6 M shows

I

e

Therefore

5

Then

5

a(pl)

*

cTou

*

-T

c

~v

(u,v) is strongly dual f e a s i b l e ,

m e n t a r y and ( I ) and

(2)

. ( x ; u , v ) is c o m p l e -

are satisfied.

(31 Let (x;u,v) denote the f i n a l p a i r generated i n the applic a t i o n o f t h e p r i m a l d u a l method t o

(13.30).

Since N = M ( p 2 )

Linear Algebraic Optimization

326

we find

Therefore (u,v) is dual feasible, (x;u,v) is complementary and (3) is satisfied.

A valuable property o f compatible pairs leads to the solution

of algebraic matroid k-intersection problems. Again we consider a solution in the respective group. Then

1

(PkIM = ( x E Z f

Ax

N

5

b, Ax

5

z,

x

= kl.

x

jEM

j

We assign a further dual variable X to E x . = k . Then (u,v,x) is 1

dual feasible if (13.48)

A

T

nu

*

-T A

o v

*

[A]

zZ,

ui

2 e , vk 2 e.

The dual variable X is unrestricted in sign in the respective group. The set o f all dual feasible (u,v) is denoted by (DkIM. Then the algebraic dual o f (13.32) is

Again we find weak duality xTO;5b

T

Ou

*

-T b Ov

*

(koX)

for all primal and dual feasible pairs

(x;u,v,X). We apply the

primal dual method to (13.301,but now for v

v

v

A sequence of compatible pairs ( x ;u ,v

EX'!

I

= v

for v =

o,I,...

)

u:= u 4 and M:= M(u4). is generated with

.

We modify the dual revision procedure slightly in order to admit negative uM (cf. LAWLER [1976], p. 347: let 6 = min{6 6v.6wl).

The stop-condition uM = 0 is replaced by Ex

j

= k.

U

,

327

Algebmic Independent Set Problems

T h e n i t may h a p p e n t h a t uM b e c o m e s n e g a t i v e d u r i n g t h e p e r A l l other compatibili-

formance o f t h e p r i m a l d u a l method. t y c o n d i t i o n s remain v a l i d .

{:

u' := i

and l e t A:=

Hence

u

M'

(x;u',v,A)

Now r e p l a c e u by u '

Then

i

$

MI

i

=

M

s a t i s f i e s (13.pB) and

(x;u',v,h)

i s o p t i m a l f o r (13.32)

primal feasible, i.e.

= k.

i f xx

w e f i n d a s t r o n g d u a l i t y theorem.

W e a s s i g n d u a l v a r i a b l e s ui

and

j

5

k.

(13.49) i f x i s

S i m i l a r l y t o theorem

(13.47)

From (13.45) w e g e t

a n d vk t o t h e c l o s e d s e t s o f t h e

m a t r o i d s a n d two f u r t h e r d u a l v a r i a b l e s A t o Ex

d e f i n e d by

t o xx

I n t h e c a s e o f a max-problem w i t h

> k and A+

j -

= M\N(p4)

$

@

w e have t o a d j o i n t h e i n e q u a l i t y

< o

t - X

j -

M

with assigned dual variable Y. PI:= k {X€Pk[ 1 ; (u,v,A+,A-,Y)'is (13.6')

Then x,

f 01.

dual feasible with respect t o max{x

T

aa(p4)

I

XEP;I

if

ui where

Y

2

e , Vk

2

e l A-

2

e,

denotes t h e vector with j-th

A+

:e ,

Y

2

e

I

c o m p o n e n t y f o r j €;

and

Linear Algebraic Optimizarwn

328

j-th component e otherwise. Without this additional dual variable y which does not appear in the objective function it is necessary that at least one of the other dual variables has the index A(max(a.1 7

j

€:I)

> p4.

This leads to a duali-

ty gap which can be avoided by the introduction of y:= max{a j

€GI.

I

j

The set o f all dual feasible solutions according to

(13.50)

is denoted by D;.

f: Di

H can be defined with respect to (13.6') similarly as

+

in ( 1 1 . 1 9 ) .

Let a:= k O A -

6:= c T O u

Then a dual objective function

and

*

-T c Ov

*

(kOA+)

.

Then f is defined by

if a where

5 B or m

A(a) = A ( E ) = Ao.

Otherwise let f(u,v,A+,A-,y):=

-

denotes a possibly adjolnt maximum o f H. The dual of

(13.6') is

Dual programs with respect to

are defined as in chapter 1 1 .

(u,v) is strongly dual feasible

if

e

5 A- , e 5

A+

,

e

5 u i , e 5 vk

.

An objective function is defined according to ( 1 1 . 1 9 ) .

329

Algebmic Independent Set Pmblerns

(13.51) Theorem

(Matroid k-intersection duality theorem)

Let H be a weakly cancellative d-monoid. Hence H is an extended semimodule over Z+

.

There exist optimal feasible pairs

(x;u,v,A+,A-) and (x;u,v,A+,A-,Y) for the algebraic matroid k-intersection problems ( 1 3 . 5 ' ) and ( 1 3 . 6 ' ) which are complementary and satisfy T

*

-T

c

Ov

T (-c) Ou(v3)

*

(-c)~OV(U)=xToa

(1)

koA- = x

(2)

ko6(U ) 3

(3)

xToa

(41

xToa = cT0u

with 6 ( p 3 ) : = A

*

Oa

*

*

cTou

*

(koA-) =cTOu

*

(A+)

* -1

-T

c

*

Ov

and

*

(for 1 3 . 5 ' )

(kOAy)

N

3

-T

c

*

Ov

(for 1 3 . 6 ' 1 ,

(koA+)

(for 1 3 . 6 ' )

ko€(p4)

€(u4)

:=

A+

(for 1 3 . 5 ' 1 ,

*

(A_)

-1

.

Proof: The proof of ( 1 ) and ( 2 ) is quite similar to the proof of ( 1 ) and ( 2 ) in theorem ( 1 3 . 4 7 ) . The value of the variable in the final pair generated by the primal dual method is assigned to A +

or A -

in an obvious manner.

The proof of ( 3 ) and ( 4 ) follows in the same manner as the proof of ( 3 ) in theorem ( 1 3 . 4 7 ) if we choose y = max{a

j

I

jEN)

;

the variable y does not appear in the objective function.

If we are not interested in the determination of the solutions of the corresponding duals then we propose to use the augmenting path method instead of the primal dual method for a solution of the respective max-problem in a group. This method generates subsequently primal feasible solutions x

I

v=1,2,

...

330

Linear Algebraic Optimization

of cardinality V which are optimal among all intersections of the same cardinality. It should be noted that such solutions are not necessarily optimal among all intersections of the same cardinality with respect to the original (not reduced) problem; optimality with respect to the original problem is implied only for those v which satisfy

where u denotes the index used in the reduction considered. The solution of algebraic matroid intersection (k-intersection) problems is previously discussed in ZIMMERMANN, U. [1978b],

([1976],

[1978c], [1979a]) and DERIGS and ZIMMERMANN, U. [1978a].

A solution of the reduced problems in t h e case of matching problems can be determined in the same manner. Again it suffices

to consider max-problems. For a detailed description of the primal dual method we refer to LAWLER [1976]. We may w.l.0.g. assume that the respective group G is divisible (cf. proposition 3.2). Hence G is a module over Q. The primal dual method remains valid and finite in such modules. Again a reformulation of the classical primal dual method for such modules consists only in replacing usual additions and usual comparisons in the group of real numbers by the internal compositions and comparisons in the group considered. A l l arguments used for a proof of the validity and finiteness in the classical case carry over to the case of such modules. Optimality follows from similar arguments as in the case of matroid intersections.

Algebraic Independent Set Roblems

33 1

Let A denote the incidence matrix of vertices and edges in the underlying graph (V,N). Let S k be any subset of V of odd cardinality 2 s k + l . Then

is satisfied by the incidence vector x of any matching. We

represent all these constraints in matrix form by Sx (13.52)

n P = { x E Z , I Ax

5

8.

Then

5 1, Sx 5 9 1 .

For the reduced problem with M = M ( p ) we find (13.53)

P

M

=

5 1, SMx 5

Ex E 22: I AMx

We assign dual variables u

i

s].

to the vertices i E V and dual

variables vk to the odd sets S k . Then (u,v) is dual feasible if (13.54)

A

T nu M

*

2

s;ov

N

a

N

where a : = a(llIj for j E M = M(p). The set of all dual feasible j

solutions is denoted by DM. In the usual manner we find weak duality, i.e.

-

xToY<

*

lT.U

sTov

for primal feasible x and dual feasible (u,v). The corresponding complementarity conditions are (13.55) (13.56)

u

i

for all (i,j) E N ,

> e

I .

E x

j i j

=1,

for all vertices V and for all odd sets T k Z N .

Let (13.58)

V(x):= l i E V l

r jx ij

< 11,

Linem Algebraic Optimization

332

i.e. V(x) is the set of all vertices which are not endpoint N

of some edge in the matching corresponding to x. If a for all (i,j) E M then x

*

0, u

e, v

< -

e

is an optimal pair

e

9

ij

of primal and dual feasible solutions. Otherwise let 6:=

(maxIY;.. I

(I/z)

(i,j) E M I )

11

which is well-defined in the module considered. The initial solution x

0, v

E

2

6 is primal and dual feasible,

e and u

satisfies (13.55) and (13.57) and all dual variables vi for i E V ( x ) have an identical value

rl.

Such a pair is called

c o n p a t i b z e , similarly to previously discussed primal dual methods. From (13.55) we conclude xToy

*

(1

-

AMx)

T

O u = lT0u

*

sTOv

for a compatible pair (x;u,v). Since all variables u . with ( 1 -AMxIi

(13.59)

+

0 have identical value rl we get

x

T

oa

*

( I V I- 2 zMxij)

0 0

= 1

T

ou

*

sTov.

Similar to the primal dual method for matroid intersections the primal dual method for matchings alternately proceeds by revisions of the current matching and the current dual solution without violation of compatibility. This is achieved by an alternate solution o f (

13.60)

T maxfx

O

a1 x E P M

, (x;u,v) compatible}

for fixed dual solution (u,v) and of (13.61)

T rninI1 n u

s

T

OvI

(u,v) E D M ,

(x;u,v) compatible)

for fixed primal solution x. From (13.59) w e conclude that

333

Algebmic Independent Set Problems

(13.60) is equivalent to (13.62)

maxI1 x

M ij

I

xEPM

,

( x ; u r v ) compatible)

and that (13.61) is equivalent to (13.63)

(U,V) E D M

min{nI

;

(x;u,v) compatible)

where TI is the common value of all variables u

i with i E V ( x )

(cf. 13.58). Finiteness and validity of the method can be shown in the same manner as described in LAWLER [19761 for the classical case. In particular, the method is of polynomial time in the number of usual operations, *-compositions and

<-comparisons.

The final compatible pair is complementary.

Thus it is optimal. Therefore the method provides a constructive proof of a duality theorem for algebraic matching problems.

We consider the algebraic matching problems (

13. 3" )

min{xT

(

13.4 " )

max{xT

0

a(ul) I x E P I , a(uZ) I x E PI.

The dual of (13.4") is (13.64)

min{lTnu

*

sTovl A T n u

*

sTnv

2 a(u2), u i z e , vk?e}.

Strong dual feasibility with respect to (13.3") is defined by

"k

and an objective function I s defined according to

11.19).

Linear Algebraic Optimization

334

(13.66) T h e o r e m ( M a t c h i n g d u a l i t y t h e o r e m ) Let H be a weakly cancellative d-monoid. Hence w.1.0.g. an extended semimodule over Q,.

H is

There exist optimal feasible

pairs (x;u,v) for the algebraic matching problems (13.3") and (13.4") which are complementary and satisfy (1) (2)

(3)

e = xToa

*

lTou

( - 1 ) T 0u(pl)

1

T

ou

*

*

*

(for 13.3"),

sTov

(-s)Tov(pl) = x T o a

(for 13.3"), (for 13.4").

sTov = xToa

Proof: Similar to the proof of theorem (13.47).

WHITE 119671 develops a parametric approach for k-matching problems (the classical case of (3.6)) which remains valid f o r modules over Q. DERIGS [1978a1 proposes a modification of the primal dual method which remains valid and finite in such modules. Compatibility is slightly modified, too. Then similar results as for the corresponding algebraic k-intersection problems (cf. theorem 13.51) can be derived. Algebraic matching problems have previously been discussed by DERIGS (11978a1, [1978b]).

In particular, solution methods for the perfect

matching problem

(cf. 11.37) are proposed in DERIGS ([1978b],

11979a1, [1979b]). A short joint discussion of algebraic problems is given in D E R I G S and ZIMMERMANN, U.

[1978a].

Matroid intersection problems for particular matroids are discussed in further papers. ZIMMERMANN, U.

119761 gives an ex-

tension of the Hungarian method for assignment problems using the concept of admissible transformations (cf. 12.59). If the

Algebraic Independent Set Pmblems

335

set of feasible solutions consists in the intersections of two matroids then a method based on admissible transformations is valid if and only if the two matroids are partition matroids (cf. theorem 12.15 and 12.16 in ZIMMERMANN, U.

119761). A fur-

ther extension to three partition matroids is considered in BURKARD and FROHLICH [19801 within a branch and bound scheme. Algebraic matroid intersection problems in the particular case that one of the matroids is a partition matroid are solved by DETERING 119781. Algebraic branching problems are solved by HAAG 119781 with generalizations of the various known methods. In particular, the derivation of EDMONDS' method

(cf. EDMONDS

119671) in KARP 119711 (cf. also C H U and LIU 119651) remains

valid in arbitrary d-monoids.

This Page Intentionally Left Blank

CONCLUSIONS

Although the material in this book covers several classes of optimization problems it should be emphasized that there are many areas of optimization problems which have not been analyzed in algebraic terms, by now. At first, we mention nonlinear optimization. For example, a formulation of a function which is quadratic over an ordered algebraic structure can easily be given. Some remarks on quadratic assignment problems can be found in BURKARD [1975b]. In particular, the duality theory in chapter eleven has consequences for such problems similarly to the classical case. cornbinatorial optimization problems with such objectives have linear relaxations which can be treated using methods described in this book. Piecewise linear algebraic functions and quotients of such functions are discussed in CUNINGHAME-GREEN and MEIJER [19781, and CUNINGHAME-GREEN ([1978],

[1979a]).

Secondly, the greater part of the results in part two of this book applies only to problems over linearly ordered algebraic structures. With the exception of path problems it seems that linear and combinatorial Optimization problems become much more difficult if linearity of the order relation is not assumed. For example, one of the simpler combinatorial problems, t h e . d e termination of a maximum spanning tree in a weighted graph, is efficiently solved by means of the greedy method provided that the weights are linearly ordered. The algebraic structure of the underlying semigroup is quite general. But if linearity is not assumed then no efficient solution method is known for

337

Conclusions

338

this problem. ZIMMERMANN,

U.

A

related bottleneck problem is proposed in [1980].

Thirdly, hard combinatorial optimization problems are usually solved using simpler relaxations within a branch and bound method. Again the methods developed in this book may be applied for a solution of algebraic versions of such relaxations. For example S C H O L Z [1978] and BURKARD [1979c] discuss certain algebraic scheduling problems which have algebraic transportation problems as relaxation, and BURKARD and FR8HLICH [19801 consider 3-dimensional assignment problems. Since the algebraic nature of the objective function is only of interest for deriving bounds, i.e.

for solving the relaxed problems,

the enumeration scheme remains unchanged. In this way a branch and bound method for the algebraic case of the original combinatorial optimization problem is derived.

BIBLIOGRAPHY

The following bibliography contains all papers and books referenced in this monograph. Some related papers, books and bibliographies are added. Journal titles are abbreviated according to Mathematical Reviews.

AHO, A.V.; HOPCROFT, J.E.; ULLMAN, J.D.: The D e s i g n and

A n a l y s i s of Computer A l g o r i t h m s , Addison-Wesley, Amsterdam (1974). ALLINEY,

S.;

BARNABEI, M.; PEZZOLI, L.: W-optimal assign-

able sets and allocation problems, Boll. U. Mat. I t a l .

A 151

17 ( 1 9 8 0 )

ARLAZAROV, v.L.;

131

-

136.

DINIC, E.A.; KRONROD, M.A.; F A R A D ~ E V , I.A.:

On economical construction of the transitive closure of an oriented graph, S o v i e t Math. DokZ. 1 1

(1970)

1209 - 1 2 1 0 .

BACKHOUSE, R.C.;

CARRE, B.A.: Regular algebra applied to

I n s t . Math. A p p Z . 15 ( 1 9 7 5 )

path-finding problems, J . 161

- 186.

BARSOV, A.: BELLMAN, R.: (1958)

what i s l i n e a r p r o g r a m m i n g , Heath, Boston (1964). On a routing problem, Q u a r t . A p p l . Math. 88

-

16

90.

BELLMAN, R.; KALABA, R.: On k-th best policies, SIAM J . A p p t .

Math. 8

(1960)

582

- 588. 339

Bibliography

3411

BENZAKEN, C.: Structures alggbraiques des cheminements: pseudotreillis, gerbiers de carre nul; in Network and S w i t c h i n g

T h e o r y (Biorci, G . ; 40

-

Ed.), Academic Press, New York ( 1 9 6 8 )

47.

BERGE, C.: Graphs and H y p e r g r a p h s , North Holland, Amsterdam (1973).

BERGE, c.;

GHOUILA-HOURI,

Programme,

A.:

Spiele,

Transportnetze,

Teubner, Leipzig ( 1 9 6 9 ) . BIRKHOFF, G.: L a t t i c e T h e o r y , American Mathematical Society, Providence ( 1 9 6 7 ) . BLAND, R.G.: New finite pivoting rules for the simplex method,

Math. O p e r . R e s . 2 ( 1 9 7 7 )

103

-

107.

BLONIARZ, P.: A shortest path algorithm with expected time O ( n L log n

log*n), P r o c e e d i n g s of t h e 1 2 t h ACM Symposium

o n t h e T h e o r y o f C o m p u t i n g , L o s Angeles ( 1 9 8 0 ) . BLONIARZ, P.; FISCHER, M.J.;

MEYER, A.R.:

A note on the average

time to compute transitive closures, in A u t o m a t a L a n g u a g e s

and P r o g r a m m i n g ,

(Michaelson; Milner; Eds.),

Edinburgh

University Press (1976). BLUM, E.; OETTLI, W.: M a t h e m a t i s c h e o p t i m i e r u n g : G r u n d l a g e n

u n d V e r f a h r e n , Springer, Berlin (1975). BLYTH, T.S.: Matrices over ordered algebraic structures, J.

London Math. S O C . 39 ( 1 9 6 4 )

427

-

432.

BLYTA, T.s.: Module T h e o r y , An A p p r o a c h t o L i n e a r A l g e b r a , Clarendon Press, Oxford BLYTH, T.S.; JANOWITZ, M.F.:

(1977).

R e s i d u a t i o n T h e o r y , Pergamon Press,

Oxford (1972). B O R ~ V K A ,0.: 0 jistkrn problkmu minimglnim. P r a c e MoravskL

PGirodovZdeckk S p o l e E n o s t i 3 translated to German).

(1926)

37

-

58

(partially

34 1

Bibliography

BRUCKER, P.: VerbBnde stetiger Funktionen und kettenwertige Homomorphismen, Math. Ann.

187 (1970)

77

- 84.

BRUCKER, P.: R-Netzwerke und Matrixalgorithmen, C o m p u t i n g 1 0 (1972)

271

-

283.

BRUCKER, P.: T h e o r y of M a t r i x A l g o r i t h m s , Mathematical systems in economics 13, Hain, Meisenheim am Glan (1974). BURKARD, R.E.:

Kombinatorische Optimierung in Halbgruppen,

in O p t i m i z a t i o n a n d O p t i m a l C o n t r o l (Bul'irsch, R.; Oettli, Stoer, J.; Eds.),

Lecture notes in mathematics, Springer,

Berlin, 477 (1975a) BURKARD, R.E.:

w.;

-

1

17.

Numerische Erfahrungen mit Summen- und Bottle-

neck-Zuordnungsproblemen, in N u m e r i s c h e M e t h o d e n b e i gra-

p h e n t h e o r e t i s c h e n u n d k o m b i n a t o r i s c h e n P r o b l e m e n (coilatz, L. ; Meinardus, G.; Werner, H.; Eds.), 9

-

Birkhsuser, Basel, (197513)

25.

BURKARD, R.E.:

Flusse in Netzwerken rnit allgemeinen Kosten, in

Graphen, Algorithmen, D a t e n s t r u k t u r e n (Noltemeier, H.; Ed.), Hauser, Miinchen (1976) BURKARD, R.E.:

123

-

134.

A general Hungarian method for the algebraic

transportation problem, D i s c r e t e Math. 22 (1978a) 219

-

232.

BURKARD, R.E.:

An application of algebraic transportation

problems to scheduling problems, in P r o c e e d i n g s o f t h e

P o l i s h - D a n i s h M a t h e m a t i c a l P r o g r a m m i n g S e m i n a r (Krarup, J.; Walukiewicz, S.; Eds.)

,

Instytut Badan Systemowych, Polska

Akademia Nauk, Warsaw (1978b) BURKARD, R.E.:

124

-

138.

Uber eine Anwendung algebraischer Transportpro-

bleme bei Reihenfolgeproblernen, in N u m e r i s c h e M e t h o d e n bei g r a p h e n t h e o r e t i s c h e n u n d k o m b i n a t o r i s c h e n P r o b lemen (Collatz, L.; Meinardus, G.; Wetterling, W.; Eds.), Birkhluser, Basel, 2 (1979a)

22

-

36.

342

Bibliography

BURKARD, R.E.:

On the use of algebraic objective functions in

combinatorial programming, Report 79-3, Mathematisches Institut der Universitat z u Kdln (1979b). BURKARD, R.E.:

Remarks on some scheduling problems with alge-

braic objective functions, O p e r a t i o n s R e s e a r c h V e r f a h r e n 32 ( 1 9 7 9 ~ ) 6 3 -77. BURKARD, R.E.; FROHLICH, K.: Some remarks on 3-dimensional assignment problems

(extended abstract), O p e r a t i o n s

R e s e a r c k V e r f a h r e n 36 (19801

31

-

35.

BURKARD, R.E.; AAHN, W.; ZIMMERMANN, U.: An algebraic approach to assignment problems, Math. P r o g r a m m i n g 12 (1977)

318

-

327.

BURKARD, R.E.;

HAMACHER, H.:

Minimal cost flows in regular

matroids, to appear in Math. P r o g r a m m i n g Stud.

(1981).

BURKARD, R.E.; HAMACHER, H.; ZIMMERMANN, U.: T h e algebraic network f l o w problem, Report 76-7, Mathematisches Institut der UniversitZt z u Kdln, Kdln (1976). BURKARD, R.E.; HAMACHER, H.; ZIMMERMANN, U.: Einige Bemerkungen zum alqebraischen FluOproblem, O p e r a t i o n s R e s e a r c h Ver-

f a h r e n 23 (1977a) BURKARD, R.E.;

252 - 2 5 4 .

HAMACHPR, H.; ZIMMERMANN, U.:

FluBprobleme mit

allqemeinen ~ o s t e n ,in N u m e r i s c h e M e t h o d e n b e i g r a p h e n -

t k e o r e t i s c k e n P r o b l e m e n (Collatz, L.; Heinardus, G.; Wetterlinq, W.; E d s . ) , BURKARD, R.E.;

Birkhauser, Basel 3 ( 1 9 7 7 b )

9- 22.

ZIMMERMANN, U.: T h e solution of algebraic assign-

ment and transportation problems, in O p t i m i z a t i o n a n d

O p e r a t i o n s R e s e a r c h (Henn, R.; Korte, B.; Oettli, W.; Eds.), Lecture notes in economics and mathematical systems, Springer, Berlin, 157 (1978)

55 -65.

Bibliography

BURKARD, R.E.;

ZIMMERMANN, U.:

343

Weakly admissible transformations

for solving algebraic assignment and transportation problems,

Math. Programming S t u d .

12 (1980)

1

- 18.

CARRE, B.A.: An algebra f o r network routing problems, J. I n s t .

Math. A p p l . C A R R ~ ,B.A.:

7 (1971)

273

-

294.

Graphs and N e t w o r k s , Oxford University Press (1979).

EERKASSKIJ, B.V.:

An algorithm for the construction of the

maximal flow in a network with a labor expenditure of

o( IV I

JIE()actions,

M a t h e m a t i c a l m e t h o d s for t h e s o l u t i o n

of e c o n o m i c p r o b l e m s ( S u p p l . t o Ekonom. i M a t . M e t o d y ) 7 (1977)

CHAEIJAN, L.G.:

117

-

126.

A polynomial algorithm in linear programming,

S o v i e t Math. D o k l . 20 ( 1 9 7 9 )

191

-

194.

CHRISTOFIDES, N.: Graph T h e o r y , An A l g o r i t h m i c A p p r o a c h , Academic Press, New York (1975). CHU, Y.J.;

LIU, T.H.:

On the shortest arborescence of a direc-

ted graph, Sci. S i n i c a 14 (1965) CLIFFORD, A.H.:

Naturally totally ordered commutative semi-

groups, Arner. J. Math. CLIFFORD, A.H.:

76 ( 1 9 5 4 )

631 - 6 4 6 .

Totally ordered commutative semigroups,

Bull. Amer. Math. SOC. 6 4 ( 1 9 5 8 ) CLIFFORD, A.H.:

1396 - 1 4 0 0 .

305 - 3 0 6 .

Completion of semi-continuous ordered commuta-

tive semigroups, Duke Math. J .

26 ( 1 9 5 9 )

CONRAD, P.: Ordered semigroups, Nagoya Math. J . 51

-

44

-

59.

16 ( 1 9 6 0 )

64.

CONRAD, P.: L a t t i c e O r d e r e d G r o u p s , Preprints and Lecture Notes in Mathematics, Tulane University, New Orleans (1970).

3 44

Bibliogmphy

CONRAD, P.; HARVEY, J.; HOLLAND, C.: The Hahn embedding theorem for lattice-ordered groups, T r a r . s . Arner. Math. S o c . (1963)

CONWAY, J.H.

143 :

-

108

169.

. ? e g u l a r A l g e b r a a n d F i n i t e M a c h i n e s , Chapman and

F?all, London (1971). CRUON, R.;

HEEVE,

P.: Quelques rgsultats relatifs d une struc-

ture algkbraique et d son application au probldme central de l'ordonnancement, Revue - f r a n g a i s e d e r e c h e r c h e o p l r a t i o -

n e z l e 34 ( 1 9 6 5 )

3

-

29.

CUNNINGHAM, W.H.; MARSH, A.B.:

A primal algorithm for optimum

matching, Technical report 262, Department of Mathematical Sciences, John Hopkins University, Baltimore CUNINGHAME-GREEN, R.A.:

(1976).

Process Synchronisation in a steelwork-

a problem of feasibility, in P r o c e e d i n g s of

t h e 2nd I n t e r -

n a t i o n a l C o n f e r e n c e on O p e r a t i o n a l R e s e a r c h , English University Press ( 1 9 6 0 ) CUNINGHAME-GREEN, R.A.:

-

323

328.

Describing industrial processes with

inference and approximating their steady-state behaviour,

O p e r a t i o n a l R e s . Q u a r t . 1 3 (1962) CUNINGHAME-GREEN, R.A.:

-

100.

Projections in minimax algebra. Math.

Programming 10 ( 1 9 7 6 ) CUNINGHAME-GREEN, R.A.:

95

111

-

123.

An algebra for the absolute centre of

a graph, Report, Department of Mathematical Statistics, University o f Birmingham (1978). CUNINGHAME-GREEN, R.A.:

Minimax A Z g e b r a , Lecture notes in

economics and mathematical systems 166, Springer, Berlin (1979).

CUNINGHAME-GREEN, R.A.:

An algorithms for the absolute centre

of a graph, Report, Department o f Mathematical Statistics, University of Birmingham

(1979a).

345

Bibliography

CUNINGHAME-GRFEN, R.A.; BORAWITZ, W.C.:

Some algebraic ideas

relevant to operations research I , the structures J,K,L,M, Report, Department of Mathematical Statistics, University

of Birmingham (1978a). CUNINGHAME-GREEN, R.A.; BORAWITZ, W.C.:

Some algebraic ideas

relevant to operations research 11, non-commuting z-transforms, Report, Department of Mathematical Statistics, University of Birmingham (1978b). CUNINGHAME-GREEN, R.A.; MEIJER, P.F.J.:

An algebra for piece-

wise-linear minimax problems, Report, Department of Mathematical Statistics, University of Birmingham (1978).

DANTZIG, G.B.:

On the shortest route through a network,

Management S c i . 6 (1960) DANTZIG, G.B.:

187

-

190.

All shortest routes in a graph, in T h e o r y of

Graphs (Rosenstiehl, P.; Ed.),Gordon 91

(1967)

-

and Breach, New York

92.

DANTZIG, G.B.; BLATTNER, W . O . ;

RAO, M.R.: Finding a cycle in a

graph with minimum cost to time ratio with application to a ship routing problem, in T h e o r y of Graph8

(Rosenstiehl,

P.; Ed.), Gordon and Breach, New York (1967a) DANTZIG, G.R.; BLATTNER, W.O.;

RAO, M.R.:

77 - 8 3 .

All shortest routes

from a fixed origin in a graph, in T h e o r y

Of

Graphs

(Rosenstiehl, P.; Ed.), Gordon and Breach, New York (1967b) 85

-

90.

DEO, N.; PANG, C.: Shortest path algorithms: taxonomy and annotation, Report CS-80-057, Washington State University

.

(1980)

DERIGS, U.: Algebraische Matchingprobleme, Doctoral thesis, Mathematisches Institut der Universitat z u Koln, KBln (1978a).

Bibliogmphy

346

DERIGS, U . :

On solving symmetric assignment and perfect mat-

ching problems with algebraic objectives, in O p t i m i z a t i o n

and O p e r a t i o n s R e s e a r c h (Henn, R.; Korte, B.; Oettli, W.; Eds.), Lecture notes in economics and mathematical systems, Springer, Berlin, 157 (1978b)

79

-

86.

DERIGS, U.: Duality and the algebraic matching problem, Opera-

t i o n s R e s e a r c h V e r f a h r e n 2 8 ( 1 9 7 8 ~ ) 253 - 2 6 4 . DERIGS, U . :

A generalized Hungarian method for solving minimum

weighted perfect matching problems with algebraic objective, D i s c r e t e A p p l . Math.

1 (1979a)

167

-

180.

DERIGS, U.: A shortest augmenting path method for solving minimal perfect matching problems, Report 79-3, Mathematisches Institut der Universitat zu Koln (1979b). DERIGS, U.: Duality and admissible transformations in combinatorial optimization, 2 . 251

-

O p e r a t i o n s R e s . S e r . A 23 ( 1 9 7 9 ~ )

267.

DERIGS, U.: On three basic methods for solving bottleneck transportation problems, Report, Industrieseminar der Universitdt z u K61n (1979d). DERIGS, U.: On two methods for solving the bottleneck matching problem, in O p t i m i z a t i o n T e c h n i q u e s K.; Walukiewicz,

S . ;

Eds.),

(Iracki, K.; Malanowski,

Lecture notes in control and

information sciences, Springer, Berlin, 23 (1980a) DERIGS, U.; ZIMMERMANN,

U.:

176 -184.

Duality principles in algebraic

linear programming, Report 78-6, Mathematisches Institut der Universitat zu Kciln, Kdln (1978a). DERIGS, U.; ZIMMERMANN,

U.:

An augmenting path method for solving

linear bottleneck assignment problems, C o m p u t i n g 19 (1978b) 285

-

295.

DERIGS, U . ;

ZIMMERMANN,

U.:

An augmenting path method for solving

linear bottleneck transportation problems, C o m p u t i n g 22 (1979)

1

-

15.

341

Bibliography

PAIR, C.: ProbtBrnes d e Cherninernent d a n s l e s

DERNIAME, J.c.;

C r a p h e s , Dunod, Paris (1971). DETERING, R.:

Ein Ldsungsverfahren fiir eine spezielle Klasse

von algebraischen Matroid-Intersektionen Problemen mit Anwendung auf das algebraische verallgemeinerte Zuordnungsproblem, Diploma thesis, Mathematisches Institut der Universitst z u K6ln (1978). DIJKSTRA, E.W.: A note on two problems in connexion with graphs, Numerische Mathernatik 1 DINIC, E.A.:

(1959)

269 - 2 7 1 .

Algorithm for solution of a problem of maximum

flow in a network with power estimation, S o v i e t Math.

DokZ.

(1970)

11

DINIC, E.A.;

1277

KRONROD, M.A.:

-

1280. An algorithm for the solution of

the assignment problem, S o v i e t Math. D o k l . 1324 DOMSCHKE,

-

10 ( 1 9 6 9 )

1326.

w. : K i i r z e s t e Wege i n C r a p h e n : A l g o r i t h m e n , V e r f a h -

r e n s v e r g t e i c h e , Mathematical systems in economics 2 , Hain, Meisenheim am Glan (1972). DORHOUT, B.:

Het Lineaire Toewijzungsproblem, Vergelijken von

Algorithmen, Report BU 21/73, Stichting Mathematisch Centrum, Amsterdam (1973). D R ~ G A N ,I. : Un algorithme pour la resolution de certains problsmes parametriques, avec un seul paramstre contenu dans la fonction Bconomique, R e v . R o m a i n e Math. P u r e s A p p l . 11

(1966)

447 - 4 5 1 .

EDMONDS, J.: Maximum matching and a polyhedron with 0 - 1 vertices, J . 125

-

130.

R e s . N a t . Bur. S t a n d a r d s S e c t . B 69 (1965)

Bibliogmphy

348

EDMONDS, J.:

Optimum branchings, in M a t h e m a t i c s of t h e D e c i s i o n

Sciences

(Dantzig, G.B.; Veinott, A.F.;

Mathematical Society, Providence, 1

Eds.), American

(1968)

-

346

361.

EDMONDS, J.: Submodular functions, matroids and certain polyhedra, in C o m b i n a t o r i a l 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 (Guy, R.; Hanam, H.; Schonheim, J . ; Eds.), Breach, New York ( 1 9 7 0 )

-

Gordon and

87.

J.: Matroids and the greedy a gorithm, Math.

EDMONDS 1

1971)

-

Programming

127 - 136.

J.: Matroid intersection, Ann

EDMONDS 39

69

D i s c r e t e Math. 4 ( 1 9 7 9 )

51.

EDMONDS, J.; FULKERSON, D.R.: Transversals and matroid partition, J.

P e s . f l a t . Bur. S t a n d a r d s S e c t .

B 69 (1965)

EDMONDS, J.; FULKERSON, D.R.: Bottleneck extrema, J .

t o r i a l Theory 8 (1970) EDMONDS, J.; KARP, R.M.:

229

-

147

19 ( 1 9 7 2 )

153.

Combina-

306.

Theoretical improvement in algorith-

mic efficiency for network flow problems, J. A s s o c .

Pack.

-

248

-

Cornput.

264.

EDMONDS, J.: PULLEYBLANK, W.: Faces of 1-matching polyhedra, in

Hypergraph Seminar

(Berge, C . ;

Ray-Chaudhuri, D.; Eds.),

Lecture notes in mathematics, Springer, Berlin, 411 ( 1 9 7 4 ) 214

-

242.

EILENBERG, S.: A u t o m a t a ,

L a n g u a g e s and M a c h i n e s , Volume

A,

Academic Press, New York ( 1 9 7 4 ) . EULER, R.: Die Restimmung maximaler Elemente in monotonen Mengensystemen, Report 7 7 6 1 , Institut fur Okonometrie und Operations Research, Universitiit Bonn (1977). EULER, R.: On rank functions of general independence systems, Report 7893, Institut fur Skonometrie und Operations Research, Universitiit Bonn (1978a).

349

Eiblwgraphy

EULER, R.: Rank-axiomatic characterizations of independence systems, Report 78126, Institut fur bkonometrie und Operations Research, Universitlt Bonn

(1978b).

EVEN, SH.: The maximal flow algorithm of Dinic and Karzanov, to appear in T h e o r . Comput. S c i .

(1980).

FAIGLE, U.: The greedy algorithm f o r partially ordered sets,

D i s c r e t e Math. 28 (1979)

153

-

159.

FAUCITANO, G.; NICAUD, J.F.: Algorithme de Gauss pour l a quasiinversion de matrices, rapport de DEA sur la complexit6 des algorithmes (J.Berstel), Universitk de Paris VI (1975). FINKE, G.; AHRENS, J.H.: algorithm, INFOR 1 6

A variant of the primal transportation

(1978)

35 -46.

FINKE, G.; SMITH, P.A.: Primal equivalents to the threshold algorithm, O p e r a t i o n s R e s e a r c h V e r f a h r e n 31 (1979)

185

-

198.

FISCHER, M.J.

;

MEYER, A.R.: Boolean matrix multiplication and

transitive closure, C o n f e r e n c e R e c o r d I E E E 1 2 t h Annual

S y m p 0 6 i u m o n S w i t c h i n g and A u t o m a t a T h e o r y (1971) FLETCHER, J.G.:

A

129

-

more general algorithm for computing closed

semiring costs between vertices of a directed graph,

C o m m . ACM 23 (1980) FLOYD, R.W.:

Algorithm 9

350

,

-

351.

shortest path, C o m m . ACM 5 (1962)

345. FORD, L.R.; FULKERSON, D R.: F l o w s i n N e t w o r k s , Princeton University Press (1962). FORTET, R.: Application de l'alg6bre de Boole en recherche opkrationelle, Revue F r a n p a i s e R e c h e r c h e O p 6 r a t i o n e l Z e

4 (1960)

17 - 2 6 .

131.

Bibliogrnphy

350

Calculating k-th shortest paths, I N F O R 11 ( 1 9 7 3 )

FOX, B.: 66

-

70.

FRATTA, L.; MONTANARI, U.: A vertex elimination algorithm for enumerating all simple paths in a graph, Networks 5 ( 1 9 7 5 ) 151

-

177.

FREDMAN, M.L.: New hounds o n the complexity of the shortest path problem, SIAM J .

Cornput. 5 ( 1 9 7 6 )

83

-

89.

Bottleneck linear programming, OpePatiOnaZ R e s .

FRIEZE, A.M.:

Quart. 26 (1975) FRIEZE, A.M.:

871

-

874.

Algebraic linear programming, Report, Department

of Computer Science and Statistics, University of London (1978).

FRIEZE, A.M.:

An algorithm for algebraic assignment problems,

D i s c r e t e AppZ. Math.

1 (1979)

253

-

259.

FuCHS, L.: T e i l w e i s e g e o r d n e t e a z g e b r a i s c h e S t r u k t u r e n , Vandenhoeck and Ruprecht, Gdttingen

(1966).

FUJISHIGE, S.: A primal approach to the independent assignment problem, J .

Operations Res. SOC. Japan

FULKERSON, R.; GLICKSBERG, I.; GROSS, 0.:

A

20

(1977)

1

14.

production line

assignment problem, RAND Research Memorandum R M - 1 1 0 2 FURMAN, M.E.:

-

(1953).

Application of a method o f fast multiplication

of matrices in the problem o f finding the transitive closure o f a graph, S o v i e t Math. D o k t .

GABOVI?!,

E. J.

:

11 ( 1 9 7 0 )

1252.

An equivalency problem in discrete programming

over ordered semigroups, Semigroup F o r u m 8 ( 1 9 7 4 )

69 -73.

GABOVIi', E.J.: Constant problems of discrete optimization over permutation sets, K y b e r n e t i k a 5 ( 1 9 7 6 a ) (Russian).

128 - 1 3 4

Bibliography

GABOVIE, E.J.:

35 1

Fully ordered semigroups and their applications,

R u s s i a n Math. S u r v e y s

31

(1976b)

147

- 216.

GALE, D.: Optimal assignments in an ordered set. An application of matroid theory, J. C o r n b i n a t o r i a l T h e o r y 4

(1968)

-

176

180.

GALIL, 2.:

A new algorithm for the maximal flow problem, in

P r o c e e d i n g s of t h e 1 9 t h I E E E Symposium o n F o u n d a t i o n of Computer S c i e n c e , Ann Arbor University, Mich,igan ( 1 9 7 8 ) . GALIL, 2.; NAAMAD, A.: Network flow and generalized path compression, Report, Department of Computer Sciences, University of Tel Aviv ( 1 9 7 9 ) . GAREY, M.R.; 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 T h e o r y of N P - C o m p l e t e n e s s , Freeman, San Francisco ( 1 9 7 9 ) . GARFINKEL, R.: An improved algorithm for the bottleneck assignment problem, O p e r a t i o n s R e s . GARFINKEL, R.S.; RAO, M.R.:

19

(1971)

1747 - 1 7 5 1 .

The bottleneck transportation problem,

Naval R e s . L o g i s t . Q u a r t .

18

(1971)

465

-

472.

GARFINKEL, R.S.; RAO, M.: Bottleneck linear programming, Math.

Programming 11

(1976)

291

- 298.

GIFFLER, B.: Scheduling general production systems using schedule algebra, Naval R e s .

Logist. Quart.

10 ( 1 9 6 3 )

237

-

255.

GIFFLER, B.: Schedule algebra: a progress report, N a v a l R e s .

Logist. Quart.

15

(1968)

255

- 280.

GLOVER, F.; KARNEY, D.; KLINGMAN, D.: The augmented predecessor index method for locating stepping stone paths and assigning dual prices in distribution problems, T r a n s p o r t a t i o n 6

(1972)

171

- 179.

sci.

352

Bibliography

GOLDEN, B.L.; MAGNANTI, T.L.:

Deterministic network optimiza-

tion: a bibliography, N e t w o r k s 7 (1977) GONDRAN, M.:

-

183.

AlgPbre linbaire et cheminement dans un graphe,

!?AIR0 R e c h e r c h e OpOr. 9 (1975) GONDRAN, M.:

149

77

-

99.

Path algebra and algorithms, in C o m b i n a t o r i a l

P r o g r a m n i n g : M e t h o d s and A p p l i c a t i o n s (Roy, B. Reichel, Dordrecht (1975a)

137

-

;

Ed.),

148.

GONDRAN, M.: Alg6bre des chemins et algorithmes, Bull. D i r e c -

t i o n E t u d e s R e c k e r c h e s SOP. C Math. I n f o r m a t . 57

-

(1975b)

(French version of 1975a).

64

GONDRAN, M.:

L'algorithme glouton dans les algdbres de chemins,

B u z l . D i r e c t i o n E t u d e s R e c h e r c h e s SOr. C Math. Informat. ( 1 9 7 5 ~ ) 25 GONDRAN, M.:

-

32.

Les elements p-rbgulidres dans les semi-anneaux,

Note Electricie de France HI 1778/02

(1975d).

GONDRAN, M.: Eigenvectors and eigenvalues in hierarchical classification, in R e c e n t DeveZopments i n S t a t i s t i c s Ed.),

North Holland

(1977)

(Barra, J.R.;

775 -781.

GONDRAN, M.: Les elements p-reguliers dans les dioldes,

D i s c r e t e M a t h e m a t i c s 25 (1979) 33 GONDRAN, M.:

-

39.

The dio'id theory and its applications, private

communication (1980). GONDRAN, M.; MINOUX, M.: Valeurs propres et vecteurs propres dans les dioides et leur interpretation en thkorie des graphes, BUZZ. D i r e c t i o n E t u d e s R e c h e r c h e s S l r .

Informat.

(1977)

25

GONDRAN, M.; MINOUX, M.:

C Math.

- 41. L'independance lineaire dans les dioldes,

B u l l . D i r e c t i o n E t u d e s R e c h e r c h e s S l r . C . Math. I n f o r m a t . (1978)

67

-

90.

353

Bibliogmphy

GONDRAN, M.; MINOUX, M.: Eigenvalues and eigenvectors in semimodules and their interpretation in graph theory, in

S u r v e y of Mathematical Programming (Prkkopa, A.; Ed.), North Holland, Amsterdam, 2 (1979a)

333 - 3 4 8

(English

version of 1977). GONDRAN, M.; MINOUX, M.: Graphes e t A l g o r i t h m e s , EyrolleS, Paris (197923). GRABOWSKI, W.: Problem of transportation in minimum time,

Bull. Acad. P o l o n . S c i . Sir. S c i . Math. Astronom. P h y s . 12 (1964) 107 GRABOWSKI, W.:

-

108.

Lexicographic and time minimization in the

transportation problem, Z a s t o s . Mat.

15

(1976)

191

-

213.

GRXTZER, G.: General L a t t i c e T h e o r y , Birkhluser, Base1 (1978). GROSS, 0.: The bottleneck assignment problem, Report P-1630, RAND Corporation (1959).

Polyhedrische Charakterisierungen kombinatoris c h e r O p t i m i e r u n g s p r o b l e m e , Mathematical systems in econo-

G R ~ T S C H E L , M.:

mics 36, Hain, Meisenheim am Glan (1977). GUPTA, R.; ARORA, S.R.: Programming problems with maximin objective functions, Z. O p e r a t i o n s R e s . 22 (1978) 69 - 72.

HAAG, W.: Verfahren zur Bestimmung optimaler Branchings, Diploma thesis, Mathematisches Institut der Universitzt z u Kaln, Kdln (1978). HAHN, H.: uber die nichtarchimedischen GrbRensysteme, S i t z u n g s

b e r i c h t e d e r Akademie d e r W i s s e n s c h a f t e n Wien, I I a , 116 (1907) 6 0 1 -655. HAMACHER, H.: Numerical investigations on the maximal flow algorithm of Karzanov, Computing 22 (1979)

17

-

29.

Bibliography

354

HAMACHER, H.: Algebraic flows in regular matroids, D i s c r e t e

A p p l . Mat h. 2

(1980a)

27

- 38.

HAMACHER, H.: Algebraische FluRprobleme in regultiren Matroiden, Doctoral thesis, Mathematisches Institut, Universitat z u Koln, Koln (1980b). HAMACHER, H.: Maximal algebraic flows: algorithms and examples, in D i s c r e t e S t r u c t u r e s a n d A l g o r i t h m s Hauser, Munchen ( 1 9 8 0 ~ ) 153

-

(Pape, U.; Ed.);

166.

HAMMER, P.L.: Time minimizing transportation problems, N a v a l

R e s . L o g i s t . Q u a r t . 16 (1969)

345

-

367; 18 (1971)

487

-

490.

HAMMER, P.L.;

RUDEANU, S.: B o o l e a n Met hods i n O p e r a t i o n s R e s e a r c h ,

Springer, Berlin (1968). HANSEN, P.: An O ( m log D) algorithm f o r shortest paths, D i s c r e t e

A p p Z . Math. HARARY, F.; An

2 (1980)

NORMAN, R.z.;

151

-

153.

CARTWRIGHT, D.:

S t r u c t u r a l Models,

I n t r o d u c t i o n t o t h e T h e o r y of D i r e c t e d G r a p h s , Wiley,

New York (1965). HAUSMANN, D. (Ed.): I n t e g e r Programming and R e l a t e d A r e a s , a

C l a s s i f i e d B i b Z i o g r a p h y 1 9 7 6 - 1 9 7 8 , Lecture notes in economics and mathematical systems 1 6 0 , Springer, Berlin (1978).

HOFFMAN,

A.J.:

On abstract dual linear programs, Naval R e s .

Logist. Quart. HOFFMAN, A.J.:

10

(1963)

369

-

373.

On lattice polyhedra X I : Construction and examples,

Report RC6268, IBM Research Center, Yorktown Heights (1976). HOFFMAN, A.J.: On lattice polyhedra 111: blockers and antiblockers of lattice clutters. Math. (1978b)

197

-

207.

Programming S t u d . 8

Bibliography

HOFFMAN, A.J.;

SCHWARTZ, D.E.:

355

On lattice polyhedra, C o t t o q u i a

Mathematica Societatis Jdnos BoZyai, 1 8 ( 1 9 7 6 )

593 - 5 9 8 .

HOFFMAN, N.; PAVLEY, R.: A method for the solutions of the n-tn best path problem, J . A s s o c . Comput. Mach. 6 ( 1 9 5 9 )

-

506

514.

HOHN, F.E.;

SESHU, S.; AUFENKAMP, D.D.:

Trans. T.'R.B.

EC-6

(1957)

154

-

The theory of nets,

161.

HOLDER, 0.: Die Axiome der Quantitlt und die Lehre vom MaR,

B e r i c h t e iiber d i e V e r h a n d t u n g e n d e r K B n i g Z i c h S a c h s i s c h e n G e s e l t s c h a f t d e r W i s s e n s c h a f t e n z u Leipzig, M a t h e m a t i s c h Physikatische Ktasse 53 ( 1 9 0 1 ) HU, T.C.: 9

1

-

64.

The maximum capacity route problem, O p e r a t i O n S Res.

(1961)

898 -900.

IRI, M.; TOMIZAWA, N.: An algorithm for finding an optimal 'independent' assignment, J. O p e r a t i o n s Res. Soc. J a p a n 19

(1976)

32

-

57.

ISHII, H.: A new method finding the k-th best path in a graph,

J.

O p e r a t i o n s Res. S O C . J a p a n 2 1 ( 1 9 7 8 )

JANTOSCIAK, J.; PRENOWITZ, W.:

469 - 475.

J o i n Geometries, Springer,

New York, Berlin, ( 1 9 7 9 ) . JOHNSON, D.B.: Efficient algorithms for shortest paths in sparse networks, J . A s s o c . Comput. Mach. 2 4

KALABA, R.:

(1977)

1

-

13.

On some communication network problems, in Cornbina-

t o r i a t A n a l y s i s , American Mathematical Society, Providence ( 1 9 6 0 )

261

-

280.

356

Bibliography

KARP, R.M.:

A simple derivation of Edmonds’ algorithm for opti-

mum branchinqs, f l e t u o r k s 1 KARZANOV, A.V.:

(1972)

265 - 2 7 2 .

Determining the maximal flow in a network by

the method of preflows, S o v i e t Math. D o k l . 434

-

15

(1974)

437.

An efficient algorithm for the determination of

KARZANOV, A.V.:

a maximal flow, Ekonom.

i. Mat. Metody 1 1 ( 1 9 7 5 ) 721 - 7 2 9

(Russian). KASTNING, D. (Ed.): I n t e g e r Programming and R e l a t e d A r e a s ,

a

C l a s s i f i e d B i b l i o g r a p h y , Lecture notes in economics and mathematical systems 128, Springer, Berlin (1976). KAUFMANN, A.; MALGRANGE, Y.: Recherche des chemins et circuits hamiltoniens d ’ u n e graphe, Revue F r a n p a i s e R e c h e r c h e O p B r a -

tionelle KLEENE, S.C.:

(1963)

7

61

-

73.

Representation of events in nerve nets and finite

automata, in A u t o m a t a S t u d i e s Eds.), KLEIN, M.:

(McCarthy, J.; Shannon, C.;

Princeton University Press, Princeton (1956). A primal method for minimal cost flows with applica-

tions to the assignment and transportation problem,

Mar.agement S c i . 14 ( 1 9 6 7 )

205

-

220.

KLEIN-BARMEN, F.: uber gewisse Halbverbande und kommutative Semigruppen I , Math. 2.

4 8 (1942)

275 - 2 8 8 .

KLEIN-BARMEN, F.: Uber qewisse Halbverblnde und kommutative Semigruppen 11, Math. 2 . KOKORIN, A.I.;

KOPYTOV, V.M.:

48 (1943)

715

-

734.

F u l l y O r d e r e d G r o u p s , Wiley,

New York (1974). KORBUT, A.A.:

Extremal spaces, S o v i e t . Math. Pokl. 6 ( 1 9 6 5 )

1358 - 1 3 6 1 .

357

Bibliography

KRUSKAL, J.B.: On the shortest spanning subtree of a graph and the travelling salesman problem, P r o c . Amer. Math. SOC. 7

(1956)

48

-

50.

KUHN, H.W.: The Hungarian method for the assignment problem,

Navel Res. L o g i s t .

Q u a r t . 2 (1955)

83

-

97.

LAWLER, E.L.: Optimal cycles in a doubly weighted directed linear graph, in T h e o r y of Graphs

(Rosenstiehl, P.; Ed.),

Gordon and Breach, New York ( 1 9 6 7 )

209

-

213.

LAWLER, E.L.: Matroid intersection algorithms, Math. Programming 9 (1975)

31

-

56.

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 : N e t w o r k s and M a t r a i d s ,

LAWLPR, E.L.:

Holt, Rinehart and Winston, New York LEHMANN, D.J.:

Theor.

(1976).

Algebraic structures for transitive closure, in

Comput.

sci.

4 (1977)

59

-

76.

LUGOWSKI, H.: uber gewisse geordnete Halbmoduln mit negativen Elementen, P u b l . Math. LUNTS, A.G.:

D e b r e c e n 1 1 (1969)

23

- 41.

The application of Boolean matrix algebra to the

analysis and synthesis of relay contact networks, Dokl.

A k a d . Nauk S S S R 7 0 ( 1 9 5 0 )

421 - 4 2 3

(Russian).

LUNTS, A.G.: Algebraic methods of analysis and synthesis of relay-contact networks, I Z V . A k a d . 16 ( 1 9 5 2 )

MAHR, B . :

405

- 426

Nauk S S S R S e r . M a t .

(Russian).

Algebraische KomplexitZt des.allgemeinen Wegeproblems

in Graphen, Report 79-14, Institut fiir Software und Theoretische Informatik, Technische Universitat, Berlin, (1979). MAHR, B.:

A

birds eye view to path problems, Report 80-19,

Institut ffir Software und Theoretische Informatlk, Technische Universitat, Berlin,

(198Oa).

Bibliography

358

MAHR, B.: Semirings and transitive closure, in preparation (198Ob). MALHOTRA

PRADMODH KUMAR. M.; MAHESHWARI, S.W.:

V.M.:

O ( VI

3

)

algorithm for finding the maximal flow in net-

wo k s , rnformation P r o c e s s i n g Lett. 7 (1978) MARTELLI

An

277 - 27%.

A.: An application of regular algebra to the enumera-

tion of the cut sets of a graph, I n f o r m a t i o n P r o c e s s i n g 74 (19741 MARTELLI,

A.:

511 - 515.

A Gaussian elimination algorithm for the enumera-

tion of cut sets in a graph, J. A s s o c . C o m p u t . Mach.

58

(1976)

-

19

73.

McNAUGHTON, R.; YAMADA, H.: Regular expressions and state graphs for automata, I R E Trans. o n E l e c t r o n i c C o m p u t e r s

9 (1960) 39 - 4 7 . MLNIEKA, E.: On computing sets o f shortest paths in a graph,

Comm. A C M 17 (1974) MINIEKA, E.; SHIER, D.R.:

351

-

353.

A note on an algebra for the k-th

best routes in a network, J. I n s t . M a t h . Appl.

1 1 (1973)

145 - 149. MINOUX, M.: Structures algebraiques generalisees des problemes de cheminement dans les graphes: ThBorPmes, algorithmes et applications, R A I R O R e c h e r c h e Op&r. MINOUX, M.:

1 0 (1976)

33 - 6 2 .

Generalized path algebras, in S u r v e y Of M a t h e m a t i -

c a l Programming

(Prgkopa, E.; Ed.), North Holland,

Amsterdam, 2 (1979) MOISIL, G.C.:

359

-

364.

Assupra unor representari ale grafurilor ce in-

tervin I n probleme de economia transporturilor, Corn. Acad.

Rep. Pop. R o m i n e 1 0 (1960) 647 - 6 5 2 . MONGE, G.: DIblai et remblal, M & m o i r e s d e Z'Acadlmie Paris (1781).

de Sciences,

359

Bibliography

MULLER-MERBACH, H.: Operatione Research M e t h o d e n , Van Vahlen, Berlin ( 1 9 7 0 ) . MUNRO, I.: Efficient determination of the transitive closure of a directed graph, Information Processing Lett. 56

-

1

(1971)

58.

MURA, R.B.; RHEMTULLA, A.: Orderable G r o u p s , Lecture notes in pure and applied mathematics 2 7 , Dekker, New York ( 1 9 7 7 ) . MURCHLAND, J.D.:

A new method for finding all eiementary paths

in a complete directed graph, Report SE - T N T

-

22,

Transport

Network Theory Unit, London Graduate School of Economics (1965).

Linear a n d Combinatorial Programming, Wiley,

MURTY, K.G.:

New York ( 1 9 7 6 ) .

ORE, 0.: Theory of equivalence relations, Duke Math. J. 9 (1942)

PAGE, E.S.: 241

PAIR,

-

573

A

-

627.

note on assignment problems, Cornput. J. 6

(1963)

243.

Sur les algorithmes pour les problemes de cheminement

C.:

dans les graphes finis, in Theory of Graphs (Rosenstiehl, P.; Ed.),

Gordon and Breach, New York, ( 1 9 6 7 ) A new matrix calculus, SIAM J. A p p l .

PANDIT, S.N.N.: 632

-

271 9

-

300.

(1961)

639.

PANDIT, S.N.N.:

Minaddition and an algorithm to find the most

reliable paths in a network, IRE Transactions on Circuit

Theory CT-9

(1962)

190

-

191.

PAPE, U.: Eine Bibliographie z u "Kiirzeste Wegllngen und Wege in Graphen und Netzwerken", Elektronische Datenverarbeitung 6

(1969)

271

-

274.

3 60

Bibliography

PAPE, U.; SCHON, B.: Verfahren zur LBsung von Summen- und EngpaR-zuordnungsproblemen, Elektronische Datenverarbeitung

7 (1970)

PETEANU,

149

-

163.

v.: sur la compatibilitb du problPme central de l'ordon-

nancement, M a t h e m a t i c a ( C l u j ) 9 ( 1 9 6 7 a )

141

-

146.

PETEANU, V.: An algebra of the optimal path in networks, M a t h e m a -

tics ( C l u j ) PETEANU, V.:

9 (196733)

335

-

342.

Optimal paths in networks and generalizations,

part I , M a t h e m a t t c a f c l u j ) 1 1 PETEANU, V.:

(1968)

311

-

327.

Optimal paths in networks and generalizations,

part 11, M a t h e m a t i c a ( C l u j ) 12 ( 1 9 7 0 )

159

-

186.

PETEANU, V.: 0 generalizare a notiunii de centru a1 unui graf,

Rev. Anal. Numdr. T e o r i a A p r o x i m a t i e i ( C l u j ) 1 ( 1 9 7 2 ) 83

-

86.

PICHAT, E.: Algorithms for finding the maximal elements of a finite universal algebra, P r o c . I F I P C o n g r e s s E d i n b u r g h , Booklet A ( 1 9 6 8 ) PIERCE, A.R.:

9 6 -101.

Bibliography on algorithms for shortest path,

shortest spanning tree, and related circuit routing problems

(1956

-

1974), N e t w o r k s 5

(1975)

129 - 1 4 9 .

PONSTEIN, J.: On the maximal flow problem with real arc capacities, Math. P r o g r a m m i n g 3 ( 1 9 7 2 )

254

- 256.

PUDLAK, P.; TfiMA, J.: Every finite lattice can be embedded in the lattice of all equivalences over a finite set, (Preliminary communication), C o m m e n t a t i o n e s M a t h e m a t i c a e Uni-

v e r s i t a t i s C a r o t i n a e 18 ( 1 9 7 7 )

409 - 4 1 4 .

PUDLAK, P.; TbMA, J.: Every finite lattice can be embedded in the lattice of all equivalences over a finite set, to appear in A l g e b r a U n i v e r s a l i s ( 1 9 8 0 ) .

Bibliography

361

RADO, R.: Note on independence functions, P r o c . London Math. SOC.

7 (1957)

300-320.

ROBERT, P.; FERLAND, J.:

Gbneralisation de l'algorithme de

Warshall, Revue F r a n g a i s e d ' l n f o r m a t i q u e e t d e R e c h e r c h e

Op6rationeZZe 2 (1968)

71

-

85.

ROY, B.: Transitivith et connexitb, C . R . A c a d . S c i . Paris 249 (1959)

216

- 218.

ROY, B.: Chemins et circuits: enumeration et optimisation, in C o m b i n a t o r i a l Programming: Methods and A p p l i c a t i o n s (Roy, B.; Ed.) RUDEANU,

S.:

Reidel, Dordrecht ( 1 9 7 5 )

105

-

136.

On Boolean matrix equations, R e v . R o m a i n e Math.

P u r e e AppZ. 17 ( 1 9 7 2 )

1075

-

1090.

SALOMAA, A.: T h e o r y of A u t o m a t a , Pergamon Press, Oxford (1969). SATYANARAYANA, M.:

P o s i t i v e l y O r d e r e d S e m i g r o u p s , Lecture notes

in pure and applied mathematics 4 2 , Dekker, New York (1979). SCHNORR, C.P.: An algorithm for transitive closure with linear expected time, SIAM J. Comput. 7 ( 1 9 7 8 )

127

-

133.

SCHOLZ, K.: On scheduling multiprocessor systems with algebraic objectives, Computing 20 ( 1 9 7 8 ) SHIER, D.R.:

189

- 205.

A decomposition algorithm for optimality problems

in tree-structured networks, D i s c r e t e Math. 6 ( 1 9 7 3 ) 175

-

189.

SHIER, D.R.:

Iterative methods for determining the k-shortest

paths in a network, N e t w o r k s 6 ( 1 9 7 6 )

205

-

229.

SHILOACH, Y.: An O ( l V l ( l V l + l E l ) l o g 2 ( 1 V l + l E l ) maximum flow algorithm, Report, Department of Computer Science, Stanford University (1978).

362

Siblwgmphy

SHIMBEL, A.:

~ p p l i c a t i o n s of matrix algebra to communication

nets, B u l l e t i n o f M a t h e m a t i c a Z B i o p h y s i c s 13 ( 1 9 5 1 ) 165

-

178.

SHIMBEL, A.:

Structural parameters of communication networks,

B u Z l e t i n of M a t h e m a t i c a l B i o p h y s i c s 15 ( 1 9 5 3 ) SHIMBEL, A.:

501

-

507.

structure in communication nets, P r o c e e d i n g s of

t h e S y m p o s i u m o n I n f o r m a t i o n N e t w o r k s ( 1 9 5 4 1 , Polytechnic Institute of Brooklyn, New York ( 1 9 5 5 )

199-203.

Utilization of the operation of space dilatations

SHOR, N.Z.:

in the minimization of convex functions, C y b e r n e t i c s 6 (1970) SAOR, N.Z.:

7

-

15.

Cut-off method with space extension in convex pro-

gramming problems, C y b e r n e t i c s 13 ( 1 9 7 7 )

94

-

96.

SLOMINSKI, L.: Bottleneck assignment problem: an efficient algorithm, Report, System Research Institute, Polish Academy o f Sciences, Warschau (1976). SLOMINSKI, L.: An efficient approach to the bottleneck assignment problem, Polon.

Acad. 25 ( 1 9 7 7 )

S c i . S g r . Sci. Math. A s t r o n o m . Phys. BUZZ. 17-23.

SLOMINSKI, L.: Bottleneck discrete problems, in P r o c e e d i n g s of

the Polish-Danish Mathematical Programming Seminar (Krarup, J.; Walukiewicz, S.; Eds.), Instytut Badan Systemowych, Polska Akademia Nauk, Warsaw ( 1 9 7 8 )

107 - 1 2 3 .

SLOMINSKI, L.: Bottleneck assignment problems: an efficient algorithm, Arch. A u t o m a t . i T e l e m e c h . 24 ( 1 9 7 9 ) SRINIVASAN, V.; THOMPSON, G . L . :

469

-

482.

Accelerated algorithms for

labeling and relabeling of trees, with applications to distribution problems, J. Assoc. 712

-

726.

Comput. Mach. 19 ( 1 9 7 2 )

363

Bibliography

SRINIVASAN, V.; THOMPSON, G.L.:

Algorithms for minimizing total

cost, bottleneck time and bottleneck shipment in transportation problems, NaVaZ R e s . L o g i s t . Q u a r t . 2 3 ( 1 9 7 6 ) 567

-

595.

STENZEL, L.: Uber eine Klasse geordneter Halbgruppen mit negativen Elementen, W i s s e n s c h a f t t i c h e Z e i t s c h r i f t , Mathernatisch-

N a t u r w i s s e n s c h a f t t i c h e R e i h e , HochschuZe Potsdarn 13 ( 1 9 6 9 ) 411 - 4 1 5 .

SZWARC, W.: The time transportation problem, ZaStOS. Mat. 8 (1966)

SZWARC, W.:

231

-

242.

Some remarks on the time transportation problem,

Naval R e s . L o g i s t . Q u a r t . 18 ( 1 9 7 1 )

TARJAN, R.E.:

473

- 486.

Solving path problems on directed graphs, Report,

Computer Science Department, Stanford University, Stanford (1975).

TARJAN, R.E.:

Graph theory and Gaussian elimination, in S p a r s e

M a t r i x c o m p u t a t i o n s (Bunch, J.R.; Rose, D.J.; Eds.), Academic Press Inc., New York ( 1 9 7 6 )

3 -37.

TESCHKE, L.; HEIDEKRUGER, G.: Zweifach assoziative Rziume, Verbindungsraume, Konvexitatsraume und verallgemeinerte Konvexitatsraume, W i s s e n s c h a f t l i c h e Z e i t s c h r i f t d e r PH

" N . K . K r u p s k a j a " HaZze 14 ( 1 9 7 6 a )

21

-

24.

TESCHKE, L,; AEIDEKRUGER, G.: Trennung in verallgemeinerten Konvexitzitsraumen, 21.

Internationazee WissenschaftZiches

Koltoquiurn, TH Ilmenau (1976b)

73

- 76.

TESCHKE, L.; ZIMMERMANN, K.: Extremale Algebren und verallgemeinerte Konvexitatsraume, Ekonom. Mat. Obzor 74

15 ( 1 9 7 9 )

- 84.

TOMESCU, I.: Sur l e s methodes matricielles dans la thiiorie des reseaux, C . R . Acad. S c i . P a r i s 263 ( 1 9 6 6 )

826 -829.

364

Bibliography

TOMESCU, I.: Un algorithme pour la dbtermination des plus petites distances entre les sommets d'un rkseau, Revue

Frangaise d'lnforrnation e t de Recherche O p b r a t i o n e l l e 1 (1967)

133

-

139.

TOMESCU, I.: Sur l'algorithme matriciel de B Roy, Revue F r a n p a i s e

d ' I n f o r r n a t i o n e t d e R e c h e r c h e O p 6 r a t i o n e l Z e 2 (1968) TOMIZAWA, N.:

87 - 91.

On some techniques useful f o r the solution of

transportation network problems, N e t w o r k s 1 (1972)

179 -

194.

VOROBJEV, N.N.: 4 (1963)

The extremal matrix algebra, S o v i e t Math. D o k l .

1220- 1223.

VOROBJEV, N.N.:

Ekstremalnaja algebra polozitelnich matriz,

E Z e k t r o n i s c h e Informationsverarbeitung und K y b e r n e t i k 3 (1967)

39 - 7 1

VOROBJEV, N.N.:

(~ussian).

Ekstremalnaja algebra neotrizatelnich matriz,

E t e k t r o n i s c h e Informationsverarbeitung u n d K y b e r n e t i k 6 (1970)

WAGNER, R.A.:

303

A

-

312

(Russian).

shortest path algorithm for edge sparse graphs,

J . A s s o c . Comput. Mach. 23 (1976)

50

- 57.

A theorem o n Boolean matrices, SIAM J . A p p z . Math.

WARSHALL, S . : 9 (1962)

11

-

12.

Kruskal's theorem for matroids, P r o c e e d i n g s of

WELSH, D.J.A.:

t h e Cambridge P h i Z o s o p h i c a Z S o c i e t y 64 (1968) WELSH, D.J.A.: WHITE, L.J.:

3

- 4.

M a t r o i d T h e o r y , Academic Press, London (1976).

Parametric study o f matchings and coverings in

weighted graphs, Doctoral thesis, University of Michigan

(1967).

Bibliography

365

WHITNEY, H.: On the abstract properties of linear independence,

Amer. J . Math. 57 ( 1 9 3 5 )

509

- 535.

WONGSEELASHOTE, A.: An algebra for determining all path-values in a network with application to k-shortest-paths problems,

Networks 6 ( 1 9 7 6 )

307

-

334.

WONGSEELASHOTE, A.: Semirings and path spaces, D i s c r e t e Math. 26 ( 1 9 7 9 )

55

- 78.

WUSTEFELD, A.; ZIMMERMANN, U.: Nonlinear one-parametric linear programming and t-norm transportation problems, Naval R e s .

L o g i s t . Q u a r t . 27 ( 1 9 8 0 )

187

-

197.

YEN, J.Y.: Finding the k shortest loopless paths in a network;

Management S c i . 17 ( 1 9 7 1 ) YEN, J.Y.:

712

-

716.

Finding the lengths of all shortest paths in an

N-node non-negative distance complete network using 1/2 N 3 additions and N 3 comparisons, J . A s s o c . Comput. Mach. 19 (1972)

YEN, J.Y.:

4 2 3 - 424.

S h o r t e s t P a t h Network P r o b l e m s , Mathematical systems

in economics 18, Hain, Meisenhelm am Glan (1975). YOELI,

M.:

A note on generalization of Boolean matrix theory,

Amer. Math. Monthly 6 8 ( 1 9 6 1 )

552

- 551.

ZADEH, N.: Theoretical efficiency of the Edmonds-Karp algorithm for computing maximal flows, J . A s s o c . Comput. Mach. (1972)

184

-

19

192.

ZIMMERMANN, K.: Some properties of the systems of extremal linear inequalities, Ekonom. Mat. Obaor 9 ( 1 9 7 3 a ) (Russian).

212

-

227

Bibliogmphy

366

Z I M M E R M A N N , K.:

S o l u t i o n of s o m e o p t i m i z a t i o n p r o b l e m s in e x t r e -

m a 1 v e c t o r s p a c e , Ekonom. Mat.

Obzor 9 (1973b)

336

-

351

(Russian). ZIMMERMANN, K . :

S o l u t i o n o f a n o p t i m i z a t i o n p r o b l e m in e x t r e -

m a 1 v e c t o r s p a c e , Ekonom. Mat.

O b z o r 10 ( 1 9 7 4 a )

298 - 3 2 1

(Russian). Z I M M E R M A N N , K.:

Conjugate optimization problems and algorithms

in t h e e x t r e m a l v e c t o r s p a c e , Ekonom. Mat. (197413)

428

ZIMMERMANN, K . :

-

Obzor 10

439.

Einige Aufgaben auf dem extremalen Vektorraum,

U n t e r n e h m e n s f o r s c h u n g und a n g e w a n d t e S t a t i s t i k 5 5 ( 1 9 7 6 a ) T284

-

T286.

Z I M M E R M A N N , K.: E x t r e m a l n i A l g e b r a , V P z k u m n a p u b l i c a c e E k o n o m i c k o m a t e m a t i c k & L a b o r a t o f e p g i E k o n o m i c k h m fistava E S A V , 4 6 , Praha

(1976b).

ZIMMERMANN, K . :

A general separation theorem in extremal algebras,

Ekonom. Mat. O b z o r 13 ( 1 9 7 7 ) Z I M M E R M A N N , K.: t a t , Math. 10 ( 1 9 7 9 a )

Z I M M E R M A N N , K.:

Linearitat und Verallgemeinerungen der Konvexi-

Operationsforsch. S t a t i s t . Ser. Optimization 17 - 2 5 . A g e n e r a l i z a t i o n of c o n v e x f u n c t i o n s , Ekonom.

Mat. O b z o r 15 ( 1 9 7 9 b ) Z I M M E R M A N N , K.:

179 - 2 0 0 .

147

-

158.

Convexity in semimodules (extended abstract) ,

t o a p p e a r i n M e t h o d s of O p e r a t i o n s R e s e a r c h , A t h e n B u m et al.

(1981).

ZIMMERMANN, K . ;

J U H N K E , F.:

Lineare Gleichungs- und Ungleichungs-

systeme auf extremalen Algebren, K i s s e n s c h a f t l i c h e Z e i t s c h r i f t

d e r T e c h n i s c h e n H o c h s c h u t e O t t o v o n G u e r i c k e , Magdeburg 23

(1979)

103

-

106.

367

Bibliography

ZIMMERMANN, U.: Boole'sche Optimierungsprobleme mit separabler zielfunktion und matroidalen Restriktionen, Doctoral thesis, Mathematisches Institut der Universitat z u Kbln, Koln (1976). ZIMMERMANN, U.: Some partial orders related to Boolean optimization and the greedy algorithm, Ann. D i s c r e t e Math. 539

-

1 (1977)

550.

ZIMMERMANN, U.: A primal method for solving algebraic transportation problems applied to the bottlenepk transportation problem, in P r o c e e d i n g s of t h e P o l i s h - D a n i s h M a t h e -

m a t i c a l P r o g r a m m i n g S e m i n a r (Krarup, J.; Walukiewicz, S.; Eds.), Instytut Badan Systemowych, Polska Akademia Nauk, Warsaw (1978a) ZIMMERMANN, U.:

139

-

153.

Duality and the algebraic matroid intersection

problem, O p e r a t i o n s R e s e a r c h V e r f a h r e n 28 (1978b)

285

-

296. ZIMMERMANN, U.: Threshold methods for Boolean optimization problems with separable objectives, in O p t i m i z a t i o n T e c h -

n i q u e s (Stoer, J.; Ed.), Lecture notes in control and information sciences, Springer, Berlin, 7 ( 1 9 7 8 ~ ) 289 -298. ZIMMERMANN, U.: Matroid intersection problems with generalized objectives, in S u r v e y of M a t h e m a t i c a l P r o g r a m m i n g

(Prhkopa,

A.; Ed.), North Holland, Amsterdam, 2 (1979a)

- 392.

383

ZIMMERMANN, U.: A primal-dual method for algebraic linear programming, Report 79-16, Mathematisches Institut der Universitlt zu .Kbln (1979b). ZIMMERMANN,

u.:

On some extremal optimization problems, Ekonom.

Mat. O b z o r 15 ( 1 9 7 9 ~ ) 438 - 4 4 2 . ZIMMERMANN, U.: Duality principles and the algebraic transportation problem, in N u m e r i s c h e M e t h o d e n b e i g r a p h e n t h e o r e t i -

s c h e n P r o b l e m e n (Collatz, L.; Meinardus, Eds.), Birkhauser, Base1 2 (1979d)

G.;

234 -255.

Wetterling, W.;

368

Bibliography

ZIMMERMANN, U.:

Duality for algebraic linear programming,

L i n e a r A Z g e b r a a n d i t s A p p l i c a t i o n s 32 (1980a)

9 -31.

ZIMMERMANN, U.: Linear optimization for linear and bottleneck objectives with one nonlinear parameter, in O p t i m i z a t i o n

T e c h n i q u e s (Iracki, K.; Malanowski, K.; Walukiewicz, S.; Eds.),

Lecture notes in control and information sciences,

Springer, Berlin, 23 (1980b)

211-222.

ZIMMERMANN, U.: An algebraic approach to extremal optimization problems

(extended abstract), M e t h o d s of O p e r a t i o n s R e s e a r c h

37 ( 1 9 8 0 ~ ) 295 - 2 9 8 . ZIMMERMANN,

U.:

Some remarks o n algebraic path problems

(extended

abstract), to appear in M e t h o d s of O p e r a t i o n s R e s e a r c h , Athendum et al.

(1981).

AUTHOR I N D E X

AHO, A.V.: 1 1 1 I 124 ALLINEY, S.: 3 1 0 ARLAZAROV, V.L.: 124, 146

DERNIAME, J.C.: 124 DETERING, R.: 335 DIJKSTRA, E.W.: 117, 146 DINIC, E.A.: 25.5, 298 DORHOUT, 8.: 292

BACKHOUSE, R.C.: 124, 133, 139, 148, 152, 154 BARNABEI, M . : 3 1 0 BARSOV, A.: 298, 299 BLONIARZ, P.: 146, 147 BELLMAN, R.: 133, 167 BENZAKEN, C.: 124, 154 BERGE, C.: 1, 2 5 6 BIRKHOFF, G.: 25 BLAND, R.G.: 228, 2 3 9 BLATTNER, W.O.: 184, 186, 252 BLUM, E.: I l l , 212, 228, 239 BLYTH, T.S.: 96, 168 BORaVKA, 0.: 117, 146 BRUCKER, P.: 124, 139, 147, 163, 164 BURKARD, R.E.: 189, 214, 244, 257, 258, 272, 275, 279, 282, 283, 298, 299, 300, 335, 337, 3 3 8

EDHONDS, J.: 202, 221, 230, 255, 283, 300, 310, 311, 315, 319, 335 EILENBERG, S.: 152 EULER, R.: 3 1 0 EVEN, S.H.: 255 FAIGLE, U.: 3 1 0 FAUCITANO, G.: 137 FERLAND, J.: 124, 139, 148 FINKE, G. : 3 0 0 FISCHER, M.J.: 124, 146 FLETCHER, J.G.: 124, 1 4 0 FLOYD, R.W.: 139, 146 FORD, L.R.: 253, 254, 255, 258 FOX, 8.: 167 FRATTA, L.: 124, 154 FREDMAN, M.L.: 147 FRIEZE, A.M.: 212, 221, 225, 226, 248, 298 FROHLICH, K.: 335, ,338. FUCHS, L.: 30, 48, 69 FUJISHIGE, S.: 319, 321 FULKERSON, D.R.: 202, 221, 25.3, 254, 255, 258, 283, 299, 300, 311, 315 FURMAN, M.E.: 124, 146

CARRE, B.A.: 120, 124, 126, 130, 132, 133, 137, 139, 148, 152, 154, 155, 167 CARTWRIGHT, D.: 163 EERKASSKIJ, B.v.: 255 CHU, Y.J.: 335 CLIFFORD, A.H.: 111, 51, 54, 67, 68, 69, 72, 73, 82, 83, 8 4 CONRAD, P.: 111, 41, 4 4 , 45, 47, 50, 66 CRUON, R.: 124 CUNNINGHAM, W.: 319 CUNINGHAME-GREEN, R.A..: 96, 169, 174, 180, 184, 185, 186, 187, 204, 252, 337

GALE, D.: 307 GALIL, 2.: 255 GAREY, M.R.: 1 1 1 GARFINKEL, R.S.: 221, 225, 248, 283, 299, 3 0 0 GIFFLER, B.: 124, 155 GLICKSBERG, I.: 299 GONDRAN, M.: 120, 124, 126, 130, 137, 139, 150, 167, 169, 171, 186, 187 GRABOWSKI, W.: 299 GRXTZER, G.: 1 , 12, 14 GROTSCHEL, M .: 302 GROSS, 0.: 299

DANTZIG, G.B.: 117, 146, 184, 186, 252 DEO, N.: 147 DERIGS, U.: 189, 212, 230, 232, 292, 300, 334

369

370

Author Index

HAAG, W.: 3 3 5 L.G.: 2 1 3 , 2 1 4 AARN, H.: 111, 4 7 HAHN, W.: 2 8 2 , 2 9 8 HAMACHER, H.: 2 4 1 , 2 4 2 , 2 4 4 ,

HAEIJAN, 254,

255,

257, 258 148, 163, 204,

HAMMER, P.L.: 298,

124,

141,

142,

149

LIU, T.H.: 3 3 5 LUGOWSKI, H.: 111, 5 1 , 57,

61,

LUNTS, A.G.:

68 123,

54,

145

299

HANSEN, P.: 1 4 7 HARARY, F.: 1 6 3 HARVEY, J . : 111, 4 1 ,

44,

45,

47

HAUSMANN, D.: 3 1 0 HEIDEKRUGER, G. : 2 0 4 HERVE, P.: 1 2 4 HGLDER, 0 . : 6 8 , 6 9 HOFFMAN, A.J.: 1 6 7 , 1 8 9 , 198,

LEHMANN, D.J.:

190,

44,

45,

124,

140

47

HOPCROFT, J.E.: HU, T.C.: 1 4 8

111,

154,

155

214

HOLLAND, C.: 111, 4 1 ,

MAAESWARI, S.W.: 2 5 5 MAAR, B.: 1 2 4 , 1 4 2 MALGRANGE, Y.: 1 5 4 MALHOTRA, V.M.: 2 5 5 MARSH, A.: 3 1 9 MARTELLI, A.: 1 2 4 , 1 3 9 ,

IRI, M.: 3 1 9 ISHII, H.: 1 6 7 JANTOSCIAK, J . : 2 0 4 JOHNSON, D.B.: 1 4 7 JOHNSON, D.S.: 111 JUHNKE, F.: 2 0 3

KALABA, R . : 1 2 1 , 1 4 7 , 1 6 7 KARP, R.M.: 2 5 5 , 3'35 KARZANOV, A.V.: 2 5 5 , 2 6 2 , 2 6 8 ,

Mc NAUGHTON, R.: 1 5 2 MEIJER, P.F.J.: 3 3 7 MEYER, A.R.: 1 2 4 , 1 4 6 MINIEKA, E.: 1 2 4 , 1 3 9 , 1 6 7 MINOUX, M.: 1 2 0 , 1 2 4 , 1 3 7 , 139,

167,

169,

171,

MOISIL, G.C.: 1 2 4 MONTANARI, U.: 1 2 4 , 1 5 4 MULLER-MERBACH, H.: 1 3 9 , MUNRO, I.: 1 2 4 , 1 4 6 MURA, R.B.: 3 0 MURCHLAND, J.D.: 1 5 4 MURTY, K.G.: 2 8 3 . 2 9 7

187

149

NAAMAD. A.: 2 5 5 NORMAN, R.Z.: 1 6 3 NICAUD, J.F.: 1 3 7

OETTLI, W.: 111, 2 1 2 , ORE, 0 . : 2 7

228,

239

291

KASTNING, D.: 3 1 0 KAUFMANN, A . : 1 5 4 KLEENE, S.C.: 1 2 3 , 1 5 2 KLEIN-BARMEN, F.: 5 4 KOKORIN, A.I.: 3 0 KOPYTOV, V.M.: 3 0 KORBUT, A.A.: 2 0 4 KROGDAHL, S.: 3 2 1 KRONROD, M.A.: 2 9 8 KRUSKAL, J.B.: 3 1 0 KUHN, H.W.: 2 8 3 LAWLER, E.L.: 186, 310, 330,

257. 319, 333

111, 1 4 6 , 1 8 3 , 293, 301, 305, 320, 321, 323,

PAIR, C.: 1 2 4 PANDIT, S.N.N.: 1 2 4 PANG, C.: 1 4 7 PAPE, U.: 3 0 0 PAVLEY. R.: 1 6 7 PETEANU, P.: 1 2 4 PEZZOLI, L.: 3 1 0 PONSTEIN, J.: 2 9 1 PRADMODH KUMAR, M.: 2 5 5 PREN?WITZ, W.: 2 0 4 PUDLAK, P.: 2 5 , 2 6 , 2 7 PULLEYBLANK, W.: 2 3 0 RADO, R.: 3 1 0 RAO, M.R.: 1 8 4 , 248,

252,

186, 221, 283, 300

225,

Author Index

R H E M T U L L A , A.: 3 0 R O B E R T , P.: 1 2 4 , 1 3 9 , 1 4 8 R O Y , B.: 1 2 3 , 1 2 4 , 1 2 6 , 1 3 0 , 139,

145

R U D E A N U , S.:

37 1

U L L M A N , J.D.:

111,

169,

V O R O B J E V , N.N.: 148,

124,

140

186,

203

204

S A L O M A A , A.: 1 5 2 S A T Y A N A R A Y A N A , M.: 3 0 , S C H N O R R , C.P.: 1 4 6 S C H U N , B. : 3 0 0 S C H O L Z , K. : 3 3 8 S H I E R , D.R.: 1 2 4 , 1 3 8 ,

W A G N E R , R.A.: W A R S H A L L , S.: 51

147 124,

W E L S H , D.J.A.: 311, 139,

167

S H I L O A C H , Y.: 2 5 5 S H I M B E L , A.: 1 2 3 , 1 4 5 , 1 4 6 S H O R , N.Z.: 2 1 4 S L O M I N S K I , L.: 2 1 1 , 3 0 0 S M I T H , P.A.: 3 0 0 S R I N I V A S A N , V.: 2 9 9 , 3 0 0 S W A R C Z , W.: 2 9 8 , 2 9 9

T A R J A N , R.E.: 1 2 4 T E S C H K E , L.: 2 0 4 T H O M P S O N , G.L.: 2 9 9 , 3 0 0 T O M E S C U , I.: 1 2 4 , 1 4 8 T O M I Z A W A , N.: 2 9 2 , 3 1 9 T f M A , J.: 2 5 , 2 6 , 2 7

139,

145,

146

1,

19,

310,

320

W H I T E , L.J.: 3 1 9 , 3 3 4 W O N G S E E L A S H O T E , A.: 1 2 4 , 155,

158,

160,

Y A M A D A , H.: 1 5 2 YEN, J.Y.: 1 2 3 , 1 6 7 Y O E L I , M.: 1 2 4 , 1 4 5 ,

Z A D E H , N.: 2 5 5 Z I M M E R M A N N , K.: Z I M M E R M A N N , U.: 203, 230, 283, 300, 334,

204, 234, 292, 307, 335,

203, 124, 206, 248, 296, 310, 338

161,

130, 167

148

204 189, 212, 258, 298, 316,

190, 221, 282, 299, 330,

This Page Intentionally Left Blank

SUBJECT I N D E X

abbreviation: 1 5 3 abelian group: 3 1 absorption: 9 absorptive - matrix: 1 3 0 - network: 1 2 6 - semiring: 1 2 6 adjacency matrix: 1 4 5 admissible - transformation: 2 1 8 - incremental flow: 2 6 1 admissibility graph: 2 6 1 algebra - ,regular: 1 5 0 algebraic - assignment problems: 2 8 2 - eigenvalue problems: 1 1 5 flow problems: 2 5 7 - GAUSS-JORDAN-method: 1 3 4 - JACOBI-method: 1 3 3 k-matching problems: 3 1 9 linear programs: 1 1 5 , 1 8 9 matching problems: 1 1 5 , 3 1 9 - matroid flows: 2 4 2 --intersection problems: 1 1 5 ,

balanced: 2 6 9 base: 1 8 1 -,locally optimal: 2 3 7 basis: 1 9 best remainder set: 3 1 5 E-admissible: 2 8 0 binary operation: 3 0 block: 2 Boo 1ean lattice: 1 4 -,pseudo: 1 5 bottleneck - objective: 2 2 5 , 2 4 7 - program: 2 2 6 bound -,greatest lower: 5 -,least upper: 5 -,lower: 5 -,upper: 5 bounded - ordered set: 6 variation: 4 9

-

-

-

-

cancellation law: 3 2 cancellative - d-semigroup: 6 3 semigroup: 3 2 capacity - ,maximum : 1 2 0 -,minimum: 2 5 4 of a path: 1 4 8 of the cut: 2 5 4 chain: 3 -,elementary: 4 -,sign of a: 1 6 3 -,simple: 4 circuit: 4 , 20 mean: 1 8 5 -,negative: 1 2 1 -,elementary non-null: 1 2 6 class -,lower: 5 5 classical shortest path problem:

319

--k-intersection problems:

-

-

-

319

network flow problems: 1 1 5 path problems: 1 1 4 , 1 1 6 shortest path: 1 1 9 transportation problems: 115,

269,

-

270

algorithm -,dual greedy: 3 1 0 -,polynomial time: 111 allocation problem: 3 1 1 almost p-compatible: 2 8 4 alphabet: 1 5 1 antisymmetric digraph: 2 5 3 antisymmetry: 1 Archimedean (d-semigroup): 6 7 arc: 3 -,basic: 1 5 5 -,saturated: 2 7 2 assignment problems: 1 1 5 associativity: 9, 3 0 atom: 1 7

-

117,

120,

146

closed set: 2 2 subset: 2 9 8

373

3 74

Subject Index

closure: 1 4 1 of I: 20 operator: 2 0 -,transitive: 1 4 5 cobase: 2 2 cocircuits: 2 2 coefficient: 1 1 2 -,cost: 1 1 3 -,reduced cost: 2 1 5 combinatorial optimization:

-

-

111

commutative group: 3 1 semigroup: 3 0 - semiring: 8 5 commutativity: 9, 3 0 compatible pair: 3 2 2 , 3 3 2 complement: 1 3 complemented lattice: 1 3 comp 1e t e graph: 2 6 lattice: 7 completion -,normal: 7 2 comp 1ex ity -,computational: 111 component ,connected: 4 ,Strongly connected: 4 composition -,internal : 9 computational complexity: 111 concatenation: 1 5 1 condition -,index: 2 1 6 -,strong index: 2 1 9 conditionally complete dmonoid: 6 7 cone -,negative: 3 2 -,positive: 3 2 connected component: 4 - graph: 4 - vertices: 4 connectivity problem: 1 2 0 convex set: 4 4 corank function: 2 2 cost coefficient: 1 1 3 countable function: 1 5 7 cover: 1 4 , 4 5 , 7 7 cut: 5 4 , 2 5 4 -,capacity of the: 2 5 4 ,essential : 5 4 -,minimum: 2 5 4

-

-

-

-

-

cut set: 1 5 4 -,minimal: 1 5 4 -,separating: 1 5 4 cycle: 4 - matroid: 2 6 decomposition -,ordinal: 5 6 dense - d-monoid: 7 7 - in T: 1 2 8 dependent set: 2 0 digraph: 3 -,antisymmetric: 2 5 3 direct methods: 1 3 4 discrete semimodule: 9 3 , 9 7 distributive - lattice: 11 - lattice-ordered commutative group: 4 1 distributivity -,general law of: 1 7 -,law of: 8 5 divisible - group: 4 2 - semigroup: 4 2 divisibility: 3 6 divisor: 5 1 -,greatest common: 1 7 rule: 5 1 - semigroup: 5 1 d-monoid -,conditionally complete: 6 7 -,dense: 7 7 -,homogeneous weakly cancellative: 8 1 domain -,integral: 8 6 d-semigroup: 5 1 -,Archimedean: 6 7 -,cancellative: 6 3 -,dual: 7 4 -,irreducible: 5 2 -,ordinally irreducible: 5 2 dual d-semigroup: 7 4 extremal linear program: 1 8 8 feasible: 2 1 5 --solution: 1 8 9 - greedy algorithm: 3 1 0 - matroid: 2 2 - of a lattice: 2 7 - of P: 309 - ordering: 2

-

-

Subject Index

-

residual: 39 duality principle: 28 dually residuated: 39

edge: 3 eigenvalue: 168 eigenvector: 168 e i g e n v e r t e x : 180 -,equivalent: 180 - , n o n e q u i v a l e n t : 180 elementary chain: 4 - path: 4 e 1ement - , f e a s i b l e : 112 -,idempotent: 5 6 -,incomparable: 2 -,maximal: 5 -,minimal: 5 -,negative: 32 -,neutral: 3 0 -,self-negative: 52 -,self-positive: 1 0 8 -,strictly negative: 52 -,strictly positive: 108 elementary - chain: 4 path: 4 elimination method: 134 embedded -,order: 8 -,lattice: 7 -,semigroup: 33 e m p t y w o r d : 151 e n d p o i n t : 3, 4 -,initial: 3, 4 - , t e r m i n a l : 3, 4 equation -,linear a l g e b r a i c : 113 equivalence relation: 2 - class: 2 equivalent eigenvertices: 180 e s s e n t i a l cut: 5 4 expression -,regular: 151 extended semimodule: 105 external - c o m p o s i t i o n : 88 extremal linear program: 115, 188

-

-

-

315

f a c t o r group: 4 5 feasible e l e m e n t : 112 flow: 2 5 4 pair: 2 1 5 s o l u t i o n : 112 - , s t r o n g l y dual: 2 1 9 field: 86 of sets: 14 flat: 22 flow -,admissible incremental: 261 -,algebraic matroid: 242 -,circuit: 2 5 6 -,feasible: 2 5 4 -,incremental: 2 5 5 - , i n c r e m e n t a l path: 2 5 6 -,maximum: 2 5 4 - , m i n i m u m cost: 2 5 7 -,path: 2 5 6 value: 2 5 4 function -,corank: 22 -,countable: 157 -,height: 1 8 -,linear a l g e b r a i c : 1 1 2 -,monotonically nond e c r e a s i n g : 49 -,rank: 2 0 -,weight: 116 -,well-ordered: 1 5 7 fundamental semigroup: 7 2 monoid: 7 7

-

-

-

-

generalized J A C O B I - m e t h o d : 133 GAUSS-JORDAN m e t h o d : 1 3 4 general law of d i s t r i b u t i v i t y : 17 generated subgraph: 4 g e o m e t r i c l a t t i c e : 17 graph: 3 - , a d m i s s i b i l i t y : 261 -,complete: 2 6 -,connected: 4 -,reduced i n c r e m e n t a l : 2 8 3 -,sign-balanced: 163 -,signed: 163 -,strongly connected: 4

-

376

Subject Index

greatest lower bound: 5 common divisor: 1 7 greedy algorithm: 3 0 6 -,dual: 3 1 0 group -,abelian: 3 1 -,commutative: 3 1 -,divisible: 4 2 -,factor: 4 5

-

-,RAHN:

47

-,HAHN-type: 4 4 -,isolated: 4 1 -,isomorphism: 3 3 -,lattice-ordered: 4 1 - objective: 2 4 7 -,ordered: 3 1 -,radicable: 4 2

interval : 1 9 inverse ordering: 2 irreducible d-semigroup: 5 2 isolated group: 4 1 isomorphic - matroid: 2 4 -,order: 3 isomorphism: 3 3 -,group: 3 3 -,lattice: 7 -,order: 3 -,semigroup: 3 3 isotone binary operation: 3 1 iterative method: 1 3 4 join: 6 irreducible element: 1 2 J O R D A N - D E D E K I N D chain condition: 1 8

-

HAHN

-

-type group: 4 4 group: 4 7 height function: 1 8 homogeneous lexicographic product: 7 9 - weakly cancellative dmonoid: 8 1 Hungarian method: 2 7 2

-

ideal: 4 7 -,regular lattice: 4 4 idempotency: 9 idempotent - element: 56 semiring: 85 identity: 3 0 immediate successor: 5 4 incidence vector: 1 1 9 , 3 0 1 incomparable elements: 2 incremental flow: 2 5 5 independence system: 1 9 independent set: 2 0 index: 7 5 , 1 0 1 -,stability: 1 2 5 condition: 2 1 6 bottleneck problem: 2 2 1 inequation -,linear algebraic: 1 1 3 infimum: 5 initial endpoint: 3, 4 integral domain: 8 6 internal composition: 9 , 3 0 intersection: 3 1 8

-

-

;-complementary: 2 3 4 A-dual feasible: 2 3 4 language: 1 5 1 -,regular: 1 5 1 latin multiplication: 1 5 4 lattice: 6 -,Boolean: 1 4 -,complemented: 1 3 ,complete : 7 -,distributive: 11 -,dual: 2 7 embedded: 7 embedding: 7 -,geometric: 1 7 ideal: 4 4 isomorphic: 7 -,partition: 2 5 -,pseudo-Boolean: 1 5 -,relatively complemented: 2 4 -,root: 9 , 4 3 -,semimodular: 1 7 law -,cancellation: 3 2 -,distributivity: 85 -,weak cancellation: 6 4 least upper bound: 5 left external composition: 8 8 length of a path: 18, 1 1 6 letter: 1 5 1 lexicographically maximum vector: 3 0 2 lexicographic objective: 2 4 7

-

-

-

-

Subject Index

- o r d e r relation: 4 8 - product: 7 9 linear: 1 - algebraic equation:

113 --function: 112 --inequation: 113 --optimization problem: 112 - ordering: 2 programming: 1 1 1 locally optimal: 237 lower - bound: 5 class: 55

-

-

m-absorptive - matrix: 1 3 0 - network: 126 - semiring: 126 matching: 319 -,perfect: 2 3 0 matrix -,absorptive: 1 3 0 -,adjacency: 145 methods: 123 -,m-absorptive: 1 3 0 -,m-regular: 1 3 0 -,neutral: 94 -,oriented incidence: 241 -,totally unimodular: 239 -,unity: 9 4 -,zero-: 94 matroid: 2 0 -,cycle: 2 6 -,dual: 22 -,isomorphic: 24 -,orientable: 22 -,ret~ular: 22 -,simple: 2 2 maximal: 2 element: 5 subset: 2 maximum: 5 capacity: 1 2 0 flow: 2 5 4 value: 254 max-matroid f l o w problem: 242 m e an -, circuit: 1 8 5 meet: 6 method -,algebraic GAUSS-JORDAX: 134 -,JACOBI: 133 -,direct: 134

-

-

-

311

-,elimination: 134 -,generalized GAUSS-JORDAN: 134 -,Hungarian: 272 -,iterative: 134 minimal: 5 closed cover: 3 1 5 element: 5 positive solution: 2 9 5 minimum capacity: 254 cut: 254 module: 8 9 monoid: 3 0 -,fundamental: 7 7 monotonically non-decreasing 'function: 4 9 monotonicity: 31 m-regular matrix: 1 3 0 - network: 126 semiring: 126 p-compatible: 259 mu1 t ip 1 1 cat ion -,latin: 154

-

-

naturally ordered semigroup: 32 negative circuit: 121 - cone: 32 element: 3 2 network: 116 -,absorptive: 126 -,m-absorptive: 126 -,m-regular: 126 -,transportation: 2 6 9 neutral element: 3 0 matrix: 9 4 nonequivalent eigenvertices: 180 nonlinear parameter: 248 normal completion: 7 2 - ' i n d e p e n d e n c e system: 301 subgroup: 4 5 normally embedded: 7 2

-

-

-

objective -,bottleneck: 225, 247 -,group: 247 -,lexicographic: 2 4 7 -,p-norm: 2 4 6

378

Subject Index

-,reliability: 2 4 6 -,sum: 2 4 6 -,time-cost: 2 4 7 operator -,closure: 2 0 operat ion -,binary: 3 0 -,isotone: 3 1 optimization -,combinatorial: 111 order embedded: 3 - embedding: 3 - isomorphic: 3 - isomorphism: 3 ordered - commutative semigroup: 3 0 - group: 3 1 - semimodule: 8 9 semiring: 8 5 - semivectorspace: 8 9 ordering: 1 -,dual: 2 -,inverse: 2 -,lexicographic: 4 8 -,linear: 2 -,partial: 2 -,pre-: 1 -,pseudo-: 1 -,quasi-: 1 -,total: 2 ordinal - sum: 5 3 decomposition: 5 6 - semimodule: 101 ordinally irreducible d-semigroup: 5 2 orientable matroid: 2 2 oriented incidence matrix:

-

-

-

24 1

parameter -,nonlinear: 2 4 8 partial ordering: 2 partition: 2 - lattice: 2 5 path: 4 -,algebraic shortest: 1 1 9 -,elementary: 4 - flow: 2 5 6 -,length of: 18 -,shortest: 116 -,simple: 4

plenary set: 4 6 p-norm objective: 2 4 6 polynomial time algorithm: 111

positive cone: 3 2 positively ordered semigroup: 32

preordering: 1 primal feasible: 2 1 5 --solution: 1 8 9 principle -,duality: 2 8 problem -,algebraic assignment: 2 8 2 -,-eigenvalue: 1 1 5

-

-,-flow:

257

-,-k-matching: 3 1 9 -,-matching: 1 1 5 -,-matroid intersection: 1 1 5 , 319

-,-matroid k-intersection: 319

-,-network flow: 1 1 5 -,-path: 1 1 4 -,-transportation: 1 1 5 ,

269,

270

-,allocation: 3 1 1 -,assignment: 1 1 5 -,classical shortest path: 117,

120,

146

-,connectivity: 1 2 0 -,index bottleneck: 2 2 1 -,linear algebraic optimization: 1 1 2 -,reliability: 1 2 0 -,reduced : 1 9 6 -,-linear algebraic optimization: 3 1 8 -,transportation: 2 6 9 product -,lexicographic: 7 9 -,homogeneous lexicographic: 79 program -,algebraic linear: 1 1 5 , 1 8 1 -,bottleneck: 2 2 6 -,dual extremal linear: 1 8 8 -,extremal linear: 1 8 8 -,reduced algebraic linear: 222 programming -,extremal linear: 1 1 5 -,linear: 111 projection: 8 pseudo-Boolean lattice: 1 5 pseudoordering: 1

Subject Index

quasiordering: 1 Q-semirings: 1 4 5 radicable - group: 4 2 - semigroup: 4 2 rank function: 2 0 rational semimodule: 9 7 real bottleneck semigroup: 8 0 reduced - algebraic linear program:

-

222

cost coefficients: 2 1 5 incremental graph: 2 8 3 linear algebraic optimization problem: 3 1 8 - problem: 1 9 6 regular algebra: 1 5 0 - expression: 1 5 1 - language: 1 5 1 lattice ideal: 4 4 matroid: 2 2 reflexivity: 1 relatively complemented lattice: 2 4 reliability - objective: 2 4 6 - of a path: 1 4 7 - problem: 1 2 0 residual: 3 8 -,dual: 3 9 residuated - semigroup: 3 8 - ,- ,dually: 3 9 right external composition:

-

-

88

ring: 8 6 - of sets: 1 2 rule -,divisor: 5 1 root - system: 6 - lattice: 9 , 4 3 saturated arc: 2 7 2 self-negative: 5 2 self-positive: 1 0 8 semigroup: 3 0 -,cancellative: 3 2 -,commutative: 3 0 -,d-: 5 1 -,divisible: 4 2

319

-,divisor: 5 1 embedding: 3 3 -,fundamental: 7 2 isomorphism: 3 3 -,naturally ordered: 3 2 -,ordered: 3 0 -,positively ordered: 3 2 -,real bottleneck: 8 0 -,residuated: 3 8 -,time-cost: 8 1 semimodular: 1 7 semimodule -,discrete: 9 3 , 9 7 -,extended: 1 0 5 -,ordered: 8 9 -,ordinal: 101 -,rational: 9 7 semiring: 1 2 6 -,absorptive: 1 2 6 -,closed: 1 4 1 -,commutative: 8 5 -,complete: 1 4 0 -,countably complete: 1 4 0 -,idempotent: 8 5 -,m-absorptive: 1 2 6 -,m-regular: 1 2 6 -,ordered: 8 5

-

-,Q-:

145

set -,bounded: 6 -,closed: 2 2 -,independent: 2 0 set of all - countable functions: 1 5 1 -,elementary non-null circuits: 1 2 6 regular languages: 1 5 1 - well-ordered functions: 1 5 7 - words: 1 5 1 sets -,ring of: 1 2 -,field of: 1 4 shortest path: 1 1 7 , 1 2 0 , 1 4 6 sign balanced graph: 1 6 3 of a chain: 1 6 3 s.igned graph: 1 6 3 simple chain: 4 matroid: 2 2 path: 4 sink: 2 5 3 solution -,dual feasible: 1 8 9 -,feasible: 1 1 2

-

-

-

380

-,minimal positive: 2 9 5 -,primal feasible: 189 source: 2 5 3 spanning tree: 294 stable matrix: 1 2 5 stability index: 125 strictly positive: 3 2 - negative: 3 2 strong index condition: 219 strongly - connected components: 4 --graph: 4 d u a l feasible: 2 1 9 su bg r a p h - generated by vertices: 4 subgroup -,normal: 4 5 sublattice: 6 subsemigroup -,trivial: 104 subsemimodule: 108 srlbset -,closed: 3 1 5 -,convex: 4 4 -,plenary: 4 5 successor -,immediate: 5 4 sum objective: 2 4 6 -,ordinal : 52 Support: 3 0 1 supremum: 5 symmetry: 2 system -,root: 6 -,independence 19

-

-

-

Subject Index

t r i v i a l subsemigroup: 104

unity matrix: 9 4 upper bound: 5 value: 46 -,flow: 254 -,maximum: 254 preserving: 46 variation -,bounded: 4 9 -,total: 4 9 vector -,capacity: 2 5 4 -,coefficient: 188 -,incidence: 119, 301 -,lexicographically maximum: 302 -,!-complementary: 234 -,A-dual f e a s i b l e : 2 3 4

-

weak c a n c e l l a t i o n law: 6 4 d u a l i t y theorem: 215 w e a k l y c a n c e l l a t i v e d-semigroup: 6 4 w e i g h t function: 116 weights: 116 well -ordered - function: 157 set: 7 9 word: 1 5 1 -,empty: 151

-

-

z e r o matrix: 9 4 terminal e n d p o nt: 3, 4 theorem -,weak duality: 2 1 5 time-cost objective: 2 4 7 semigroup: 8 0 total ordering: 2 variation: 4 9 totally unimodular: 2 3 9 transitivity: I transportation - network: 2 6 9 problem: 2 6 9 tree -,spanning: 2 9 4

-

-