Discrete Convex Analysis

(/(x)) is quasi convex. Quasi-convex functions enjoy the following nice properties: • A strict local minimum of a quasi-convex function is a strict global minimum. • A local minimum of a semistrictly quasi-convex function is a global minimum. • Level sets of quasi-convex functions are convex sets. Due to these properties, quasi convexity also plays an important role in continuous optimization (see, e.g., Avriel-Diewert-Schaible—Zang [5]). The concept of quasi M-convexity is defined for a function / : Zv —> Ru{+oo} as follows. Recall the exchange axiom for M-convex functions: (M-EXC[Z]) For x,y € dom/ and u £ supp+(x — y), there exists v £ supp~ (x — y) satisfying

The sign patterns of &f(x;v,u) and A/(j/;u, v) compatible with (implied by) inequality (6.91) are as follows:

Here Q and x denote possible and impossible cases, respectively. Relaxing condition (6.91) to compatible sign patterns leads to two versions of quasi M-convex functions. We say that a function / : Zv —> R U {+00} with dom / / 0 is quasi M-convex if it satisfies the following: (QM) For x,y 6 dom/ and u G supp+(x — y), there exists v £ supp~ (a: - y) satisfying

Similarly, a function / : Zv —> R U {+00} with dom/ ^ 0 is semistrictly quasi M-convex if it satisfies the following: (SSQM) For x,y e dom/ and u € supp+(o; - y), there exists v 6 supp~ (x — y) satisfying

170

Chapter 6.

M-Convex Functions

Example 6.66. A quasi M-convex function arises from a nonlinear scaling of an M-convex function. For an M-convex function / : Zv —> R U {+00} and a function (p : R -» R U {+00}, define / : Zv -> R U {+00} by

Then / satisfies (QM) if if is nondecreasing and (SSQM) if

(SSQMW) For distinct x,y G dom/, there exist u & supp + (x — y) and v G supp~ (x — y) satisfying

The property (QM W ) can be expressed in two alternative forms below. The first (6.93) may be regarded as a variant of (6.89) with discreteness in direction, and the second (6.94) is similar to (6.50) in section 6.6. Theorem 6.67. For f : Zv —> R U {+00}, (QM W ) is equivalent to each of the following conditions:

Proof. Obviously, (6.94) implies (QM W ) and (6.93). We prove (QM W ) => (6.94) and (6.93) => (6.94) by induction on \\x — y\\\. Suppose x,y G dom/ and f ( x ) > f(y)', we may assume \\x—y\\\ > 2. If (QM W ) is true, there exist some u & supp+(x — y) and v G supp~(x — y) such that Af(x;v,u) < 0 or Af(y;u,v) < 0; in the latter case the induction hypothesis for x and y' = y + xu — Xv yields A/(x; v', u') < 0 for some u' G supp+(x — y') C supp + (x — y) and v' G supp~(x — y') C supp~(a; — y). If (6.93) is true, there exist u G supp+(a; — y) and v G supp~(x — y) such that A/(x;f,u) < 0 or f(y + Xu ~ Xv) £ f ( x } ' , m the latter case we have /(x) > f ( y ' ) for y' = y + Xu — Xv and the induction hypothesis yields A/(o;; v', u') < 0 for some u' G supp + (x — y') C supp + (x — y) and v' G supp~(x — y') C supp~(x — y). D

6.14.

Quasi M-Convex Functions

171

The relationship among various versions of quasi M-convex functions is summarized as follows. The second statement below shows that all the conditions are equivalent for / if they are imposed on every perturbation of / by a linear function. Theorem 6.68. For f : Zv —> R U {+00} the following implications hold true.

Proof. (1) The equivalence of (M-EXC[Z]) and (M-EXCW[Z]) is due to Theorem 6.5. The remaining implications are obvious. (2) Combining Theorems 6.72 and 6.74 below establishes this. D The quasi M-convexity of a set B C Zv can be defined as the quasi Mconvexity of the indicator function SB '• Zv —> {0, +00}. The properties (QM) and (QM W ) for SB correspond respectively to the following properties of B: (Q-EXC) For x, y € B and u 6 supp + (x—y), there exists v 6 supp~(xy) such that x - \u + Xv e B or y + Xu ~ Xv 6 B. (Q-EXCW) For distinct x,y € B, there exist u 6 supp+(x - y) and v € supp~ (x — y) such that x — Xu + Xv G B or y + Xu ~ Xv & B. Proposition 6.69. A set B C Zv satisfies (Q-EXCW) if and only if, for distinct x,y £ B, there exist u e supp + (x — y) and v £ supp~(x - y) with x — \u + Xv 6 B. Proof. Theorem 6.67 for f — SB reduces to this statement.

D

Proposition 6.70. For a set B C Zv satisfying (Q-EXCW), we have x(V) = y(V) for any x,y € B. Proof. The proof is easy (similar to that of Proposition 4.1).

D

Example 6.71. Whereas we have the obvious implications (B-EXC[Z]) => (QEXC) =>• (Q-EXCW), these conditions are not equivalent. For instance,

satisfies (Q-EXC) and not (B-EXC[Z]), and

satisfies (Q-EXCW) and not (Q-EXC).

•

172

Chapter 6. M-Convex Functions

The weaker version (QM W ) of quasi M-convexity for functions can be characterized by the corresponding quasi M-convexity (Q-EXCW) of level sets. For any a & R U {+00}, the level set is defined as Note that L(/, +00) = dom/ and L(/, ao) = argmin/ for ao = min/. Theorem 6.72. A function f : Zv —+ R U {+00} satisfies (QM W ) if and only if the level set L(/, a) satisfies (Q-EXCW) for all a & R. Proof, ["only if"]: Let x and y be distinct elements of L(/, a). By (QM W ), we have A/(x; v,u) < 0 or A/(y; u, v) < 0 for some u € supp+ (x — y) and v € supp~ (x — y ) . Then, x - Xu + Xv e L(/, a) or y + \u - Xv & L(/, a). ["if"]: For any distinct x,y e dom/ with /(x) > /(y), we have x — Xu + Xv & L(f,f(x)) for some u € supp + (x — y) and v € supp~(a; — y) by (Q-EXCW) and Proposition 6.69. Hence f ( x — Xu + Xv) < /0*0^ Proposition 6.73. If f satisfies (QM W ), then dom/ satisfies (Q-EXCW). Proof. In the proof of "only if" of Theorem 6.72, replace L(/, a) with dom/.

D

An M-convex function can be characterized by quasi M-convexity of level sets of perturbed functions. Theorem 6.74. A function f : Zv —» Ru{+oo} satisfies (M-EXC[Z]) if and only if the level set L(/[p], a) satisfies (Q-EXCW) for all p e Rv and a e R. Proof. The "only if" part follows from Theorem 6.72. To prove the "if" part, we first observe that Theorem 6.4 can be strengthened to a statement that (MEXC[Z]) and (M-EXCioc[Z]) are equivalent if dom/ satisfies (Q-EXCW). (This can be shown by modifying the proof of Claim 2 in the proof of Theorem 6.4.) Note that (Q-EXCW) holds for dom/ by Theorem 6.72 and Proposition 6.73. To show (MEXCi oc [Z]),takeo;,y e dom/with ||x-y||i = 4 and put y = x-Xm-Xu^+Xvi+Xvz with Ui,U2,Vi,V2 £ V and {MI,u^} D {vi,v%} = 0- In the following we assume MI ^ «2 and vi ^ V2 (the other cases can be treated similarly). Consider a bipartite graph G = (V+,V~;E) with vertex bipartition V+ — {ui,u2}, V" = {^1,^2} and arc set E = {(ui,Vj) A/(x;t; j ,u i ) < +00 (ij = 1,2)}. Claim 1: G has a perfect matching (of size 2). (Proof of Claim 1) It suffices to show that every vertex has an edge incident to it. Take p e Rv such that P(UI) + p(u2) — p(v\) — P(VZ) = f(y) — f ( x ) and P(VJ) >P(UI) -Af(x;Vj,ui) for j = 1,2 (and p(v) = 0 for v e V \ {ui,u2, Vi,v2}). Then f\p\(x) = f\p}(y) < f\p\(x - Xui + Xvj) for j = 1,2. By (Q-EXCW) for L (f\P\J\p}(x)) we nave f\p](x - Xu2 + Xv^ < f\P\(x) < +00 for some j € {1,2}. Hence there is an edge incident to U2, and similarly for other vertices. Claim 2: Inequality (6.11) is satisfied.

6.14. Quasi M-Convex Functions

173

(Proof of Claim 2) By Claim 1 we may assume {(MI, Ui), (1*2,^2)} != E. We can takepe Rv such that f\p](x) = f\p](y) and f\p](x-Xu!+Xvi) = f\p](x-Xu2+Xv2) (and p(v) = 0 for v G V \ {ui,u 2 , v\, v2}). If {(ui,u 2 ), ("2,^1)} Q E, we can choose p satisfying an additional condition f\p](x- Xm + Xv2] = f\P\(x - Xu2 + Xm); and then (Q-EXCW) for L(/[p],/[p](x)) yields (6.11). If j>i,u 2 ) € £ and (u 2 ,ui) £ £, we can choose p satisfying an additional condition f\p\(x) < f\p](x — Xui + Xv2), and then (Q-EXCW) for L(/[p],/[p](x)) yields (6.11). The remaining cases are similar. D Next we turn to the minimization of quasi M-convex functions. The following properties, respectively weaker than (SSQM) and (SSQMW), turn out to be relevant. (SSQM^) For x,y e dom/ with /(x) ^ f ( y ) and u e supp+(x - y ) , there exists v e supp~ (x — y) satisfying A/(a;;u, w) < 0 or

A/(y;w, w) < 0 or

A/(x; w , u ) = A/(y;u, u) = 0.

(SSQM^) For x, y € dom / with /(x) ^ /(y), there exist u e supp+(x— y) and u G supp~ (x — y) satisfying A/(x;w,u)<0

or A/(y;w,i>) < 0 or A/(x;v,u) = Af(y;u,v) = 0.

The property (SSQM^) can be expressed in two alternative forms below. The first (6.96) may be regarded as a variant of (6.90) with discreteness in direction, and the second (6.97) is identical to (6.50) in section 6.6. Theorem 6.75. For f : Zv ->• R U {+00}, (SSQM^) is equivalent to each of the following conditions:

Proof. The proof is much the same as that for Theorem 6.67. Global optimality (minimality) for a quasi M-convex function is characterized by local optimality. Theorem 6.76 (Quasi M-optimality criterion). (1) For f : Zv —> R U {+00} satisfying (QM W ) and x € dom/, we have

satisfying (SSQM^) and x 6 dom/, we have

174

Chapter 6. M-Convex Functions

Proof. It suffices to show <=. (1) is immediate from (6.94) in Theorem 6.67 and (2) is from (6.97) in Theorem 6.75. D The minimizer cut theorem (Theorem 6.28) for M-convex functions can be generalized for quasi M-convex functions. Theorem 6.77 (Quasi M-minimizer cut). Let f : Zv —* RU {+00} be a function with (SSQM^) 7 and assume argmin/ ^ 0. Then (1), (2), and (3) in Theorem 6.28 hold true. Proof. The proof of Theorem 6.28 is valid here when (M-EXC[Zj) is replaced with (SSQM^). D The proximity theorem for M-convex functions (Theorem 6.37) can be generalized for quasi M-convex functions. Theorem 6.78 (Quasi M-proximity theorem). Let f : Zv —> R U {+00} be a function with (SSQM^), n = \V\, and a 6 Z ++ . // xa € dom/ satisfies (6.66), then argmin/ ^ 0 and there exists x* & argmin/ with (6.67). Proof. It suffices to show that, for any 7 e R with 7 > inf/, there exists some x* e dom/ satisfying f(x*) < 7 and (6.67). Suppose that x* € dom/ minimizes | \x* —xa ||i among all vectors satisfying f(x*) < 7. In the following, we fix v 6 V and prove xa(v)—x*(v) < ( n — l ) ( a —1). (The inequality x*(v) -xa(v) < ( n — l ) ( a — 1) can be shown similarly.) We may assume xa(v) > x*(v). Put

Claim 1: For y e argmin{/(y') | y' £ S}, we have y(v) = x*(v). (Proof of Claim 1) Suppose that y(v) > x*(v). From the definition of x* we have/(y) > f ( x * ) . By (SSQM^) for y, x*, andw e supp + (y-z*) C supp+(a;Q-:r*), there exists w 6 supp~(y — x*) C supp~(o;a — x*) such that if Af(x*;v,w) > 0 then Af(y;w,v) < 0. By the choice of x*, we have Af(x*;v,w) > 0 and hence f ( y ~ Xv + Xw) < f(y)- Since y — Xv + Xw € S, this contradicts the minimality of f ( y ) . Thus Claim 1 is proved. Take any y from argmin{/(y') y' £ S}, and represent it as

We have A = xa(v) — x*(v) by Claim 1. Claim 2: For any w 6 supp"(x" — x*) with p,w > 0 and p, e [0, fj,w — l]z,

6.14.

Quasi M-Convex Functions

175

(Proof of Claim 2) We prove the claim by induction on p,. For JJL 6 [0, [iw — l]z, put x' — xa — n(xv — Xw), and assume x' 6 dom/. Note that x' 6 S and x'(v) > x*(v), and hence Claim 1 implies f ( x ' ) > f ( y ) . Since supp~(y-o:') = {v}, (SSQM^) for y, x', and w; e supp + (y-x') implies that if A/(y;i>,u>) > 0 then A/(x'; w, v) < 0. By Claim 1 we have A/(j/; v, w) > 0, from which Claim 2 follows. Claim 2 and (6.66) imply

where the third equality follows from xa(V) = y(V).

D

Theorem 6.79 (Quasi M-minimizer cut with scaling). Let f : Zv —> R U {+00} be a function satisfying (SSQM^) with argmin/ ^ 0, and assume a e Z ++ and n=\V\. Then (I) and (2) in Theorem 6.39 hold true. Proof. We prove (2), while (1) can be proved similarly. Put xa = x + a(xv ~ Xu)We may assume max{o;*(u) | x* € argmin/} < xa(v); otherwise we are done. Let x* be an element of arg min / with x* (v) maximum. The rest of the proof is the same as the proof of Theorem 6.78 (from Claim 1 until the end). D

Bibliographical Notes The concept of M-convex functions was introduced by Murota [137] and that of M^-convex functions by Murota-Shioura [151]; Theorems 6.2 and 6.3 are due to [151]. The local exchange axiom (Theorem 6.4) is given in Murota [137], and the weak exchange axiom (Theorem 6.5) is explicit in Murota [147]. Quadratic functions of the form (6.23) are treated in Camerini-ConfortiNaddef [22], and their M^-convexity is observed in Murota-Shioura [151]. Proposition 6.8 (characterization of quadratic M-convex functions) is due to MurotaShioura [155]. Quadratic functions defined by symmetric matrices of the form (6.29), or (6.30), are treated in Hochbaum-Shamir-Shanthikumar [92], and their M-convexity is noted by A. Shioura. Quasi-separable convex functions (6.32) are considered by [22], and their M^-convexity is pointed out in [151]. The M^-convexity of laminar convex functions (6.34) and minimum-value functions (6.36) is due to Danilov-Koshevoy-Murota [34], [35]. The names laminar convex functions and minimum-value functions are coined in this book. The basic operations in section 6.4 are listed in Murota [141], [144], [147]. Theorem 6.13 (8) (infimal convolution) is due to Murota [137], whereas Theorem 6.15 (2) (projection of M^-convex functions) is due to [141]. The scaling operation for M-convex functions is considered by Moriguchi-Murota-Shioura [133].

176

Chapter 6. M-Convex Functions

The supermodularity of M^-convex functions (Theorem 6.19) is observed by Murota-Shioura [153]. A special case for valuated matroids was noted earlier by Dress-Terhalle [40] and Murota [138]. Theorem 6.24 (descent direction) is observed by Murota-Tamura [160] as a generalization of its special (but essential) case with dom/ C {0,1}V due to Fujishige-Yang [69]. Proposition 6.25 is in Murota [137], [140], [142]. Theorems on minimizers are of fundamental importance. Theorem 6.26 (Moptimality criterion) and Theorem 6.30 (characterization by minimizers) are by Murota [137]. Theorem 6.28 (M-minimizer cut) is by Shioura [190]. The connection to the gross substitutes property was studied almost simultaneously by Danilov-Koshevoy-Lang [33], Fujishige-Yang [69], and Murota-Tamura [160]. The concave version of (6.60) is identical to condition GS in [33]. Propositions 6.32 and 6.33 and Theorem 6.34 are due to [160], and Proposition 6.35 and Theorem 6.36 are due to [33]. Results about minimizers under scaling are relatively new. Theorem 6.37 (M-proximity theorem) is by Moriguchi-Murota-Shioura [133]. Theorem 6.39 (Mminimizer cut with scaling) is by Tamura [197]. The convex extension of M-convex functions has been understood step by step. Convex extensibility (latter half of Theorem 6.42) and the characterization by minimizers (Theorem 6.43) are in Murota [137]. Integral convexity (Theorem 6.42) and convex extension for a pair of M-convex functions (Theorem 6.44) are by Murota-Shioura [153]. Polyhedral M-convex functions are investigated by Murota-Shioura [152], to which all the theorems in section 6.11 (Theorems 6.45, 6.47, 6.48, 6.49, 6.50, 6.51, 6.52) as well as Proposition 6.53 are ascribed. M-convexity for nonpolyhedral convex functions is considered in Murota-Shioura [156], [157]. The correspondence between positively homogeneous M-convex functions and distance functions (Theorem 6.59) is established for the case of Z in Murota [141] and generalized to the case of R in Murota-Shioura [152]. Proposition 6.56 is stated in Murota [147]. Theorem 6.61 for directional derivatives and subgradients is shown for the case of Z in Murota [140], [141] and generalized to the case of R in Murota-Shioura [152]. Theorem 6.63 (characterizations in terms of directional derivatives, subdifferentials, and minimizers) is by [152], whereas its ramification with integrality (Theorem 6.64) is stated in Murota [147]. The concept of quasi M-convex functions was introduced by Murota-Shioura [154], to which almost all major results in section 6.14 (Theorems 6.67, 6.68, 6.72, 6.75, 6.76, 6.77, 6.78) are ascribed. Exceptions are Theorem 6.74 (characterization of M-convex functions by level sets) by Shioura [191] and Theorem 6.79 (quasi M-minimizer cut with scaling) by Tamura [197]. Zimmermann [221] considers combinatorial optimization problems with quasi-convex objective functions in real variables. M-convex functions find applications in resource allocation problems (KatohIbaraki [110], Moriguchi-Shioura [134]), mathematical economics (to be treated in Chapter 11), and analysis of polynomial matrices (to be treated in Chapter 12).

Chapter 7

L-Convex Functions

L-convex functions form another class of well-behaved discrete convex functions. They are defined in terms of an abstract axiom involving submodularity and are characterized as functions obtained by piecing together L-convex sets in a consistent way or as collections of submodular set functions with some consistency. Fundamental properties of L-convex functions are established in this chapter, including the local optimality criterion for global optimality, the proximity theorem for minimizers, discrete midpoint convexity, integral convexity, and extensibility to convex functions. Duality and conjugacy issues are treated in Chapter 8 and algorithms in Chapter 10.

7.1

L-Convex Functions and iJ-Convex Functions

We recall the definitions of L-convex functions and L''-convex functions from section 1.4.1. A function g ; Zv —> R U {+00} with domg ^ 0 is said to be an L-convex function if it satisfies

(SBF[Zj) is submodularity on the integer lattice and (TRF[Zj) linearity in the direction of 1. Note that we have r G Z if g is integer valued. Also recall the submodularity inequality

We denote by £[Z —•> R] the set of L-convex functions and by £[Z —> Z] the set of integer-valued L-convex functions. Since an L-convex function g is linear in the direction of 1, we may dispense with this direction as far as we are concerned with its nonlinear behavior. Namely, instead of the function g in n = \V\ variables, we may consider the restriction g' to 177

178

Chapter 7. L-Convex Functions

an arbitrarily chosen coordinate plane, say, P(UQ) = 0 for some UQ G V, where the restriction g' is a function in n — 1 variables defined by

with the notation V = V \ {UQ} and (po,p') G Z x Zv>. A function derived from an L-convex function by such a restriction is called an L''-convex function. More formally, an L''-convex function is defined as follows. Let 0 denote a new element not in V, and put V — {0} U V. A function g : Zv —> R U {+00} is called

£^ [Z —» R] the set of L^-convex functions and by $ [Z —> Z] the set of integer-valued L''-convex functions. It turns out that L''-convexity can be characterized by a kind of generalized submodularity:

which we call translation submodularity. Note that a is restricted to be nonnegative and this inequality for a = 0 agrees with the original submodularity (SBF[Z]). Theorem 7.1. For a function g : Zv —> R U {+00} with domg ^ 0, we have

Proof. Let g be denned by (7.2). The submodularity of g, i.e.,

is translated to a condition

on g. Assuming a = q0 - p0 > 0, put p' = p - p0l and q' = q — gol- Then (P V q) - (po V q0)l = (p' - al) V q' and (p A q) - (p0 A
7.1. L-Convex Functions and L''-Convex Functions

179

Proof. By (SBF[Z]) and (TRF[Z]) we see

Theorem 7.3. An L-convex function is L^-convex. Conversely, an L^-convex function is L-convex if and only if it satisfies (TRF[Zj). Proof. This follows from Theorem 7.1 and the obvious implications (7.3) =>• (SBF»[Z]) =>(SBF[Z]). D For ease of reference we summarize the relationship between L and Lfi as

where Cn and &n denote, respectively, the sets of L-convex functions and L^-convex functions in n variables, and the expression £^ ~ Cn+i means a correspondence of their elements (functions) up to the constant r in (TRF[Zj), where (7.2) gives the correspondence under the normalization of r = 0. By the equivalence between L-convex functions and L''-convex functions all theorems stated for L-convex functions can be rephrased for L''-convex functions, and vice versa. In this book we primarily work with L-convex functions, making explicit statements for L''-convex functions when appropriate. A set function p : 2V —>• RU {+00} can be identified with a function g : Zv —> RU {+00} with domg C {0,1}V through

The following states that the submodularity of p is the same as the L^-convexity of g. Proposition 7.4. Let p : 2V —> RU {+00} be a set function with dom/9 ^ 0 and g : Zv —>• RU {+00} be the associated function in (7.5). Then we have

Proof. (SBF^Zj) for a = 0 is equivalent to the submodularity (4.9) of p. (SBF^Zj) for a > I is void for a function g with doing C {0,1}V since (p — al) V q = q and p A (q + al) = p for any p, q 6 {0,1}V and a > 1. D The proposition above shows that L^-convex functions effectively contain submodular set functions as a subclass; i.e.,

where <—> denotes the embedding by (7.5).

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Chapter 7. L-Convex Functions

We mention here the following fundamental fact, showing that submodularity (on the integer lattice) is in fact a local property. The proof is easy and omitted. Proposition 7.5 (Local submodularity). Let g : Zv —> R U {+00} be a function with domg being L^-convex. Then g satisfies the submodularity inequality (7.1) for allp,q 6 Zv if and only if it satisfies (7.1) for allp,q 6 Zv with \\p-q\\oc = 1Note 7.6. With an M^-concave function h : Zv -» R U {-00} with dom/i = {0,1} V , we may associate a function 5 : Zv —> RU{+oo} such that domg = {0,1}^ and g(p) — h(p) for p e {0, l}y. Since ft is submodular by Theorem 6.19, g is L''-convex by Proposition 7.4. In this sense we have M^-concave => L''-convex for functions on {0, l}-vectors. The converse does not hold, as is demonstrated by an lAconvex function g : Z3 —> R U {+00} with domg = {0,1}3 defined by 0(1,1,1) = -2, 0(1,1,0) = 0(1, 0,1) = -1, and g(0,0,0) = g(l, 0,0) = g(0,1,0) = g(0,0,l) = 0(0,1,1) = 0. Note that h = -oo and 0 = +00 outside {0,1} V and that any function on {0,1}V can be extended to a convex function and to a concave function. •

7.2

Discrete Midpoint Convexity

We show a characterization of L''-convexity in terms of discrete midpoint convexity

which is an obvious approximation to the midpoint convexity (1.3) of ordinary convex functions; see Fig. 7.1. We consider another property for a function 0:

where This says that the sum of the function values at a pair of points (p, q) does not increase when the pair is replaced with another pair (p — xx-,1 + Xx) of closer points. Theorem 7.7. For a function g : Zv —* R U {+00} with dom0 ^ 0, we have

Hence, each of these is a necessary and sufficient condition for g to be L^-convex. Proof. First, Theorem 7.1 shows the equivalence of (SBF^Zj) to lAconvexity. [(SBFl'[Z])=KLt|-APR[Z])]: Suppose that supp+(p - q) ^ 0 and put a = maxV£v{p(v) — q(v)} — 1. We have a > 0, (p — al) Vq — q + Xx, and p/\ (q + al) = p-Xx- Hence (L^-APR[Zj) follows from (SBF^Zj).

7.3.

Examples

181

Figure 7.1. Discrete midpoint convexity.

We have \p'(v) — q'(v)\ <1 (v 6 V), supp+(p' — q') C supp+(p — q), and supp (p'— g(p') + g(q'). Applying (L"-APR[Z]) to (p',q') yields 9(P') + 9(l'} > 9(P") + 9(q"}- Hence follows g(p) + g(q) > g(p") + g(q"). [(7.7)=^>(SBFl'[Z])]: (SBF^Z]) for g is equivalent to the submodularity of g in (7.2) (cf. proof of Theorem 7.1). Since domg is an L^-convex set by (7.7) and (5.15), dome? is an L-convex set. By Proposition 7.5, the submodularity of g is equivalent to the local submodularity of g and the latter holds if and only if

for all p 6 Zv and X, Y C V with X n Y = 0. These two conditions follow easily from discrete midpoint convexity (7.7). D

7.3

Examples

We have already seen L-convexity in network flows and in matroids (section 2.2, section 2.4). In this section we see some other examples of L-convex functions, such as linear functions, quadratic functions, and separable convex functions. First we note the following facts. Proposition 7.8. (1) The effective (2) The effective

domain of an L-convex function is an L-convex set. domain of an L^-convex function is an L^-convex set.

Proof. (1) (SBF[Z]) and (TRF[Z]) for g imply (SBS[Zj) and (TRS[Z]) for D = dom^. (2) Similarly, (SBF"[Z]) for g implies (SBS^Z]) for D = domg. D

182 Linear functions

Chapter 7. L-Convex Functions A linear (or affine) function48

with x € R™ and a G R is L-convex or L''-convex according as domg is L-convex or L''-convex. Quadratic functions

A quadratic function

with ay = Oji G R («, j = 1 , . . . , n) is L''-convex if and only if

which can be proved as in Theorem 2.7. Accordingly, g is L-convex if and only if

In Example 2.1 (Poisson equation) and Example 2.2 (electrical network), we have seen the matrices

which satisfy (7.10) and (7.11), respectively. Separable convex functions

A separable convex function

with univariate discrete convex functions gi G C[Z —> R] (i = 1 , . . . , n) is L^-convex if domgr is an L^-convex set.49 In particular, a separable convex function with a chain condition

48

In this section, V = {1,... ,n}, g denotes a real-valued function in integer variables, i.e., g : Z —> R U {+00}, and p(i) is the ith component of an integer vector p = (p(i) i = 1 , . . . , n) 6 Z". 49 It is easy to verify discrete midpoint convexity (7.7) for g. n

7.4. Basic Operations

183

is L''-convex. For g^ € C[Z —> R] (i ^ j;«, j = 1,..., n), the function

is L-convex if dom g is an L-convex set. As special cases of (7.12) and (7.14) we see the following. Proposition 7.9. Let tfj € C[Z —> R] be a univariate discrete convex function. (1) ip is L^-convex. (2) TTie function g : Z2 -> R U {+00} de/med fry 0(p) = VOK 1 ) ~ P( 2 )) «« L-confeo;. Maximum-component functions

The function

which gives the maximum value of the components of p, is L-convex. Multimodular functions A function h : Zn —> Ru {+00} is said to be multi-modular if the function g : Zn+1 —» R U {+00} defined by ff(po,p) = /i(p(l)-po,p(2)-p(l),...,p(n)-p(n-l))

(po e Z,p € Z") (7.16)

is submodular. This means that a function h : Z™ —> R U {+00} is multimodular if and only if it can be represented as for some L11-convex function g. Submodular set functions Any submodular set function may be regarded as an L^-convex function by Proposition 7.4.

7.4

Basic Operations

Basic operations on L-convex functions are presented here, whereas the most important operation, transformation by networks, is treated later in section 9.6. L-convex functions admit the following operations. See (6.41) for the definition of the projection gu. Theorem 7.10. Let g,gi,g% € £[Z —> R] be L-convex functions. (1) For A € R++, \g is L-convex. (2) For a e Zv and (3 & Z \ {0}, g(a + j3p) is L-convex in p. (3) For x e Ry, g[—x] is L-convex. (4) For U C V, the projection gu is L-convex provided gu > — oo. (5) For ipv e C[Z -f R] (w € V),

184

Chapter 7.

L-Convex Functions

is L- convex provided g > — oo. (6) The sum gi + g^ is L-convex provided dom (g\ + 52) 7^ 0Proof. (1), (2), (3), and (6) are obvious. (5) (TRF[Z]) is easy to see. For (SBF[Zj) we indicate the idea by assuming that, for each p\,pi 6 dom g, the inflmum in (7.18) is attained by some qi, q2 e Zy. Proposition 7.9 (2) and (SBF[Zj) for g respectively show

On the other hand, (7.18) implies

Adding these inequalities yields (SBF[Zj) for g. (4) A special case of (5) with if)v = <5{0} (v & U) and ij)v — Sz (v e V \ U) shows the L-convexity of g = g ^ z ^ f j , where Sfj is the indicator function of U — {p e Zv | p(v) = 0 (v G U)}. By (6.45), gu is the restriction of g to U. Then (SBF[Z]) of gu is immediate from that of g, and (TRF[Z]) of gu follows from that of g since g(p + xv) =g(p + Xu)- Q Note in Theorem 7.10 (2) that we have a scaling factor (3, in contrast to the similar statement (Theorem 6.13 (2)) for M-convex functions. Also note that g in Theorem 7.10 (5) is the infimal convolution of g with a separable convex function. Operations in Theorem 7.10 are also valid for L^-convex functions. In addition, restrictions are allowed for L''-convex functions. See (3.55) and (6.40) for the definitions of the restrictions g[a,b] and guTheorem 7.11. Let g,gi,g2 e £^[2 —> R] be L^-convex functions. (1) Operations (l)-(6) of Theorem 7.10 are valid for ifi-convex functions. (2) For a,b € (Z U {±oo})y, the restriction g^a^ to the integer interval [a,b] is L^-convex provided domg^f,] ^= 0(3) For U C V, the restriction g\j is L^-convex provided dom gu ^ 0Proof. (2), (3) It is easy to verify (SBF^Zj) for g[a
D

Note 7.12. The infimal convolution of two L-convex functions is not necessarily L-convex, and similarly for the infimal convolution of two L^-convex functions. Such functions are studied in section 8.3 under the names L2-convex functions and L2convex functions, respectively. •

7.5.

Minimizers

185

Note 7.13. The proviso gu > —oo in Theorem 7.10 (4) can be weakened to gu (po) > — oo for some po, and similarly for g > —oo in Theorem 7.10 (5). •

7.5

Minimizers

Global optimality for an L-convex function is characterized by local optimality. Theorem 7.14 (L-optimality criterion). (1) For an L-convex function g € £[Z —> R] and p 6 dom g, we have

(2) For an L^-convex function g £ &\L —> R] and p 6 domg, we have

Proof. It suffices to prove 4= in (1) and (2). We first consider (2). For any disjoint Y, Z C V, condition (7.20) together with submodularity yields

which implies the optimality criterion (3.65) for integrally convex functions. Since an L^-convex function is integrally convex (to be shown in Theorem 7.20), Theorem 3.21 establishes 4= in (2). Next, (1) follows from (2), since an L-convex function is L''-convex (Theorem 7.3) and the right-hand side of (7.19) implies g(p - \Y) =

g(p + xv\Y)>g(p)-

n

The well-known optimality criterion for a submodular set function is an immediate corollary of (2) above. Theorem 7.15. Let p be a submodular set function. A subset X 6 dom/9 is a minimizer of p if and only if p(X) < p(Y) for any Y that includes X or is included in X. Proof. Let g be the L''-convex function associated with p (see (7.5) and Proposition 7.4), and apply Theorem 7.14 (2) with p = xxD Although Theorem 7.14 affords a local criterion for global optimality of a point p, a straightforward verification of (7.19) requires O(2 n ) function evaluations. The verification can be done in polynomial time as follows. We consider a submodular set function pp defined by pp(Y) = g(p + XY) — g(p) and note that (7.19) is equivalent to saying that pp achieves its minimum at Y = 0. This condition can be verified in polynomial time by the submodular function minimization algorithms in section 10.2. The minimizers of an L-convex function form an L-convex set, a property that is essential for a function to be L-convex.

186

Chapter 7.

L-Convex Functions

Proposition 7.16. For an L-convex function g € C[Z —> R], argming is an L-convex set if it is not empty. Proof. Suppose D = arg min R U {+00} be a function with a bounded nonempty effective domain. (1) g is L-convex <^=> argmin argmin• is immediate from Theorem 7.10 (3) and Proposition 7.16. The converse is shown later in Note 7.47. D

7.6

Proximity Theorem

We show a proximity theorem for L-convex function minimization, stating that a global optimum of an L-convex function g exists in a neighborhood of a local optimum of its scaling ga defined by ga(p) = g(ap)/a. Note that ga is L-convex by Theorem 7.10 (2), and accordingly, a local minimizer of ga is a global minimizer of ga by the L-optimality criterion (Theorem 7.14). Theorem 7.18 (L-proximity theorem). Assume a € Z++ and n = \V\. (1) Let g : Zv —> R U {+00} be an L-convex function with g(p) = g(p + 1) (Vp e Zv). Ifpae domg satisfies

then arg min 5^0

and there exists p* e arg min g with

(2) Let g : Zv —>RU{+oo} be an L^-convex function. If pa € domg satisfies

then arg min
7.7. Convex Extension

187

Proof. (1) It suffices to show that, for any ft > inf g, there exists p* that satisfies g(p*) < /3 and (7.22); note that there exist only a finite number of p* satisfying (7.22). We may assume pa = 0. By (TRF[Z]) with r = 0 there exists p* such that g(p*) < ft and p* > 0; let p* be minimal (with respect to order >) among such vectors. We have p* (v) = 0 for some v G V and

We can represent p* as p* = Z)£=i MiXXi, where //$ € Z++ (i = l , . . . , f c ) , 0 ^ Xi ^2 ^ • • • ^ Xk ^ y, and 0 < k < n - 1. Claim 1:

(Proof of Claim 1) Put p = X^=i /J-iXXi+UXXj and suppose p G dom g. By JC,C supp+(p*) and (7.25) we have g(p* — XXj) > d(p*)- Since Xj = argmax^6y{p*(t;) - p ( v ) } , (LH-APR[Z]) shows that g(p* - Xx,) > 5(p*) =>ff(p+ XxJ < 5(p)- Note that g satisfies (lAAPR[Z]) by Theorem 7.7. (Proof of Claim 2) Put p — Y^i=i P-iXXi and q = ^xXj and suppose q 6 dom g. Since V" \ Xj = argmax^y{q(v) - p ( v ) } and g(p + Xv\Xj)=g(p- XXj} > g(p) by Claim 1, (Ll-APR[Z]) implies g(q) > g(q - \v\Xj) = g(q + Xxt). It follows from Claim 2 and (7.21) that ^i < a for i = 1 , . . . , k, and hence

(2) This follows from (1) applied to g in (7.2).

D

The algorithmic use of the above theorem is shown in sections 10.3.2 and 10.4.5.

7.7

Convex Extension

This section establishes one of the major properties of L-convex functions, which is that they can be extended to convex functions in real variables. The extensibility to convex functions is by no means obvious from the definition of L-convex functions in terms of axioms referring only to function values on integer points. The convex extension of an L-convex function can be obtained by piecing together the Lovasz extensions. With a function g : Zv -* R U {+00} and a point p e domg we associate a set function pg^p : 2,v —> R U {+00} defined by

If g is L-convex, the associated set function pg,p belongs to <S[R] (i.e., p9tp is submodular, /?g,p(0) = 0, and pg,p(V) < +00).

188

Chapter 7. L-Convex Functions

The next theorem shows that an L-convex function g can be extended to a convex function, and that the convex extension can be constructed from the Lovasz extension of pg>p for varying p € Zv. Theorem 7.19. Let g € £[Z —> R] be an L-convex function and ~g be its convex closure. (1) For p 6 domg and q £ [0, I]R, we have

where p3tp denotes the Lovasz extension (4.6) of the associated set function pg>p in (7.26), q\ > 92 > • • • > qm are the distinct values of the components of q, and

(2) For p e Zv and q e [0,1]R, we have50

Proof. (2) For each p € Zv, let hp(q) denote the function in q e [0, I]R defined by the right-hand side of (7.29). If p € domg, p9jp belongs to <S[R], and hence hp is a polyhedral convex function by Theorem 4.16. With the representation of a real vector s = p + q with p = [s\ and q = s — [s\ we define a function h : Hv -* R U {+00} by h(s) = hp(q). We have

where C/o — 0 and t/j = Ui(q) for i = 1 , . . . , m, as in (7.28). By construction, /i is convex in \p,p + I]R for each p G Zy. Furthermore, it is convex in the entire space Ry, because h(s - al) = h(s) - ar for s e Rv and a e R by (TRF[Z]) of g, and for each s € Rv there exist a € R and p € Zv such that s — al is an interior point of \p,p + I]R. Obviously, we have h(p) = g(p) for every p e Zv, and /i is the maximum among convex functions with this property. Therefore, h = ~g. (I) If p G doing, the right-hand side of (7.29) can be rewritten as (7.27). 50

Expression (7.29) does not involve oo — oo even for p f domg.

7.8.

Polyhedral L-Convex Functions

(3) This is a special case of (7.29) (4) This is immediate from (7.29) (5) For q e N0 we put a — min^ey We can apply (7.27) to g(p + (q - al))

189

with q — 0. and (TRF[Z]). q(v). By (4), g(p+q) = g(p+(q-al))+ar. since q - al e [0,1]R. D

The above theorem implies the integral convexity of an L-convex function and hence that of an L''-convex function. Theorem 7.20. An L^-convex function is integrally convex. In particular, an ifi -convex function is convex extensible. An integrally convex function with submodularity (SBF[Zj) is called a submodular integrally convex function. This turns out to be a synonym of L''-convex function. Theorem 7.21. For a function g : Zv —> R U {+00} with doing ^ 0, g is L^-convex ^=> g is submodular integrally convex. Proof. The implication =>• follows from (SBF[Zj) and Theorem 7.20. The converse can be shown as follows. By the integral convexity of g, the convex closure g coincides with the local convex extension, and, by the submodularity of g, the latter is obtained as the Lovasz extension (7.27) of pg

for any p, q G Zv. On the other hand, we have

from convex extensibility and midpoint convexity (1.3). From these follows the discrete midpoint convexity (7.7) of g, which means L^-convexity by Theorem 7.7. D Note 7.22. Submodularity (SBF[Zj) alone does not guarantee convex extensibility. For example, any function g : Z2 —> Ru{+oo} with doing = {p € Z2 | p ( l ) = p(2)} is submodular. •

7.8

Polyhedral L-Convex Functions

As we have seen, L-convex functions on the integer lattice can be extended to convex functions in real variables. The convex extension of an L-convex function is a polyhedral convex function when restricted to a finite interval. Motivated by this

190

Chapter 7. L-Convex Functions

we define here the concept of L-convexity for polyhedral convex functions in general and show that major properties of L-convex functions survive in this generalization. A polyhedral convex function g : Rv —>.R U {+00} with doniRg ^ 0 is said to be L-convex if it satisfies

(SBF[Rj) is submodularity on the real-vector lattice and (TRF[Rj) is linearity in the direction of 1. We denote by £[R —> R] the set of polyhedral L-convex functions. Polyhedral L-concave functions are defined in an obvious way. Submodularity (SBF[Rj) is in fact a local property. Under auxiliary conditions on the effective domain, it is implied by submodularity for a certain set of local pairs of p and q. The following two propositions are mentioned here; the proofs are straightforward and omitted (see Theorems 4.26 and 4.27 of Murota-Shioura [152]). Proposition 7.23. Let g : R^ —> R U {+00} be a function with dom^g a closed set. Then g is submodular (SBF[Rj) if, for each po € domR<7; there exists e = e(po) > 0 such that inequality (7.1) is satisfied for all p, q with \\p-po\\oo < £ ana Ik-Polloo <£• Proposition 7.24. Let g : R^ —> RU{+oo} be a function withdom^g an interval. (1) g is submodular (SBF[Rj) if

for all p G doniRg; u, v 6 V with u ^ v; and A, p, & R+. (2) g is submodular (SBF[Rj) if (7.30) holds for allp e doniRg; u, v £ V with u / v; A e [0,j3i_i -PJ]R; and p, e [0,Pj-i - PJ]R, where pl > p2 > • • • > pm denote the distinct values of the components of p, the indices i and j are such that p(u) = pi and p(v) =pj, and [0,po —pi\R means [0, +OO)R by convention. The Lovasz extension p of a submodular set function p is a polyhedral L-convex function. Proposition 7.25. For p e <S[R] we have p 6 £[R -> R]. Proof. (TRF[Rj) for g — p is obvious, and we show (SBF[Rj) below. First, assume p is finite valued. Then doniR/5 = Rv. By Proposition 7.24 (2), (SBF[R]) follows from inequality (7.30) for u 6 Ui \ f/i_i, v € Uj \ Uj-i, A € [0,pi_i — pi\R, and p, & [0,pj_i — PJ]R, where Ui (i = 1 , . . . , m) are defined in (4.4) and UQ = 0- Expression (4.6) shows

7.8. Polyhedral L-Convex Functions

191

If i =£ j, we have

and, if i = j, we may assume A > // and then we have

from the submodularity of p. Hence follows (7.30). The general case where p may possibly take the value of +00 can be reduced to the finite-valued case. For each k £ Z++, we define pk by

and put R] on the integer lattice is a polyhedral L-convex function, i.e., ~g G £[R —> R], provided that g is polyhedral. Proof. We use (7.27) in Theorem 7.19. (TRF[R]) is obvious. By Proposition 7.25, pgtp is submodular on Rv, and hence g is submodular on [P,P+!]R for eachp G Zv. The submodularity (SBF[Rj) of g on R^ follows from this. D Example 7.27. The convex extension g of an L-convex function g G £[Z —» R] may consist of an infinite number of linear pieces, in which case ~g is not polyhedral convex. For example, we have g G £[Z —* R] and g ^ £[R —» R] for g : Z2 —> R defined by g(p) = (p(l) — p(2)) 2 . It is worth noting that, if donizgr is bounded, g is polyhedral and therefore ~g G £[R —> R] by Theorem 7.26. • Polyhedral L-convex functions with integrality (6.75) are referred to as integral polyhedral L-convex functions, the set of which is denoted by £[Z|R —» R]. Polyhedral L-convex functions with dual integrality (6.76) are referred to as dual-integral polyhedral L-convex functions, the set of which is denoted by £[R —* R|Z].

192

Chapter 7.

L-Convex Functions

By Theorem 7.26 and Proposition 7.16 an integral polyhedral L-convex function is nothing but a polyhedral L-convex function that can be obtained as the convex extension of an L-convex function on integer points. Therefore, we have

where the second expression means that there exists an injection from £[Z|R —> R] to £[Z -> R], representing an embedding of £[Z|R -> R] into £[Z -» R]. The effective domain of a polyhedral L-convex function is an L-convex polyhedron, which is homogeneous in direction 1. Hence, polyhedral L^-convex functions can be defined as the restriction of polyhedral L-convex functions, just as L''-convex functions on integer points are defined from L-convex functions via (7.2). We denote by £^[R —> R] the set of polyhedral L''-convex functions and by £ ll [Z|R —> R] the set of integral polyhedral L^-convex functions. The relationship between L and Lfi is described by Cn C £„ ~ £ r a +ij where £ra and C\ denote, respectively, the sets of polyhedral L-convex functions and polyhedral L''-convex functions in n variables. The R-counterpart of (SBF^Z]) is the following:

Theorem 7.28. For a polyhedral convex function g : Rv —> R U {+00} with domR<7 7^ 0, (SBF^Rj) is equivalent to polyhedral L^-convexity. The condition (7.32) below is stronger than (SBF^Rj) in that it requires the inequality not only for nonnegative a but also for negative a. Theorem 7.29. A polyhedral L-convex function g e £[R —> R] satisfies

Theorem 7.30. A polyhedral L-convex function is polyhedral L^-convex. Conversely, a polyhedral L^-convex function is polyhedral L-convex if and only if it satisfies (TRF[Rj). Almost all properties of L-convex functions on integer points carry over to polyhedral L-convex functions. To be specific, Theorems 7.10, 7.11, and 7.14 and Proposition 7.16 are adapted as follows. Note, however, that the proofs are not straightforward adaptations; see Murota-Shioura [152]. Theorem 7.31. Let g,gi,g2 € £[R —>• R] be polyhedral L-convex functions. (1) For A € R++, A<7 is polyhedral L-convex. (2) For a £ Tiv and (3 € R \ {0}, g(a + ftp) is polyhedral L-convex in p. (3) For x & Hv, g[—x\ is polyhedral L-convex. (4) For U C V, the projection gu is polyhedral L-convex provided gu > — oo.

7.9.

Positively Homogeneous L-Convex Functions

193

(5) For V>« e C[R -> R] (v 6 V),

is polyhedral L-convex provided g > — oo. (6) The sum g\ + g% is polyhedral L-convex provided dom (g\ + $2) ^ 0Theorem 7.32. Let g,gi,g% G &[R —> R] be polyhedral 1$-convex functions. (1) Operations (l)-(6) of Theorem 7.31 are valid for polyhedral Lfi-convex functions. (2) For a,b e (R U {±oo})v, i/ie restriction g^a^ to the real interval [a,b] is polyhedral Lfi-convex provided dom.g[a^ ^ 0. (3) ForU C I/, the restriction gu is polyhedral l}-convex provided dom gu ^ 0. Theorem 7.33 (L-optimality criterion). (1) For a polyhedral L-convex function g & £[R —> R] and p e domRg, we have

(2) For a polyhedral L^-convex function g e /^[R —> R] and p e doniR^, we have

Proposition 7.34. Let g € £[R —> R] be a polyhedral L-convex function. For any x € R^; argming[—x] is an L-convex polyhedron if it is not empty. The property in Proposition 7.34 characterizes polyhedral L-convexity, to be shown in Theorem 7.45. Note 7.35. The proviso gu > -oo in Theorem 7.31 (4) can be weakened to 9U(po) > -°° for some po, and similarly for g > -oo in Theorem 7.31 (5). • Note 7.36. For L^-convex functions on integer points we have seen characterizations in terms of discrete midpoint convexity and submodular integral convexity (Theorems 7.7 and 7.21). These characterizations, however, do not carry over to polyhedral L^-convex functions. In contrast, translation submodularity (SBF^Z]) is generalized to (SBF^R]), as stated in Theorem 7.28. •

7.9

Positively Homogeneous L-Convex Functions

Positively homogeneous L-convex functions coincide with the Lovasz extensions of submodular set functions. We denote by o£[R ~* R] the set of polyhedral L-convex functions that are positively homogeneous in the sense of (3.32) and by o£[Z|R —> R] the set of

194

Chapter 7.

L-Convex Functions

integral polyhedral L-convex functions that are positively homogeneous. Also we denote by o£[Z —> R] the set of L-convex functions g e £[Z —> R] on integer points such that the convex extensions ~g are positively homogeneous. These three families of functions can be identified with each other, i.e.,

by the following proposition. We introduce yet another notation, oC[L —» Z], for the set of integer-valued functions belonging to o£[Z —> R]. Proposition 7.37.

(1) 0 £[Z|R->R] = 0 £[R->R].

(2) The convex extension of a function in o£[Z —> R] belongs to o£[R —» R]. Proof. (1) Take g e o£[R -* R]- For any x e Ry, argming[-z] is a cone that is an L-convex polyhedron (or empty) by Proposition 7.34. Hence, argmingf—x] = D(7) for a {0, +oo}-valued distance function 7; see section 5.6. This shows the integrality of argmin R]. (2) Take g e o£[Z —> R]. Since g is integrally convex and g is positively homogeneous, g can be represented as the maximum of a finite number of linear functions. Hence ~g is polyhedral, and <; e £[R —> R] by Theorem 7.26. D A positively homogeneous L-convex function g induces a submodular set function pg by More precisely, we have the following statements, where <S[R] and <S[Z] denote respectively the sets of real-valued and integer-valued submodular set functions defined in (4.10) and (4.11). Proposition 7.38. (1) For g e 0£[R -» R], we have pg <E <S[R]. (2) For 5 e 0£[Z -> Z], we ftave ps € <S[Z]. Proof. The submodularity of /9S is immediate from that of 3. Note also that Pg(9) = 5(0) = 0 and pg(V) = g(l) < +00. D Conversely, the Lovasz extension /3 of a submodular set function p e <S[R] is a positively homogeneous L-convex function. We recall from (4.6) the definition

where pi > £>2 > • • • > pm denote the distinct values of the components of p, and Ui = Ui(p) = {v e V I p(v) > pi} for i = 1 , . . . , m. Denote by pz the restriction of p to Zv, and note that we have pz : Zy —> Z U {+00} for integer-valued p.

7.9. Positively Homogeneous L-Convex Functions

195

Proposition 7.39. (1) For p e 5[R], we have p &0£[R -> R](2) For p £ S[Z], we have pz € 0C\Z -> Z]. Proof. (1) We have p € £[R —> R] by Proposition 7.25, whereas the positive homogeneity of p is obvious. (2) follows easily from (1). D The next theorem shows a one-to-one correspondence between positively homogeneous L-convex functions and submodular set functions. Theorem 7.40. For0£ - 0£[R ->• R] and S — <S[R], the mappings $ : 0£ -+ 5 and \l/ : 5 —> Q£ defined by are inverse to each other, establishing a one-to-one correspondence between $C and S. The same statement is true for Q£ = o£[Z —> Z] and <S = «S[Z] wiift $ : g ^> pg and $ : p i—> />z • Proof. For p e S we have *(p) 6 0£ by Proposition 7.39 and $ o ty(p) = p by p(JQ = p(xx) m (4.7). For 5 e Q£ we have p = $(5) £ 5 by Proposition 7.38. Denote by gz the restriction of g to Z^. By Theorem 7.19 (5) we have

for q G NO, which remains valid for all q & Ry since g is positively homogeneous and the origin 0 is an interior point of -/Vo- Hence g — \P o 3>(g). D The above argument leads to the following proposition, to be used in section 7.10. Proposition 7.41. For a positively homogeneous polyhedral convex function g : Rv —> R U {+00} with domR(? ^ 0, conditions (a) and (b) below are equivalent.

( a )c?eo£[R->R].

(b) argmin — oo. Proof, (a) =J> (b) is immediate from Proposition 7.34. For (b) =>• (a), define p by p(X) = g(xx) (X C V) and denote by p its Lovasz extension (7.36). Claim 1: g(p) = p(p) for all p 6 Ry. (Proof of Claim 1) The positively homogeneous convexity of g as well as (7.36) yields p(p) > g(p). We may assume p e doniR.<7, since otherwise /3(p) — g(p) = +00. Take x such that p € argming[—x , put Z) = argmin R]- A set function p. defined by fi(X} — SD(XX) (X C V) belongs to <S[R] and its Lovasz extension coincides with SD by Theorem 7.40. From this and (4.8) we see

196

Chapter 7. L-Convex Functions

In view of the expression (4.6) and the linearity of g on D we obtain g(p) = p(p). By Claim 1 and the convexity of g, p is submodular by Theorem 4.16. In addition we have p(0) = g(0) = 0 and p(V) = p(l) = R] by Proposition 7.39 (1). D

7.10

Directional Derivatives and Subgradients

Directional derivatives and subgradients of L-convex functions are considered in this section. For a polyhedral L-convex function g, the directional derivative g'(p~, d) is a positively homogeneous L-convex function in d and the subgradients of g at a point form an M-convex polyhedron. Furthermore, each of these properties characterizes L-convexity. We start with directional derivatives of a polyhedral L-convex function g G £[R —> R]. Recall from (3.25) that, for p G domR<7, there exists £ > 0 such that

Proposition 7.42. If g & £[R —> R] and p G domRg, then g'(p', •) G o£[R —> R]. Proof. By (7.37), g'(p\ •) satisfies (SBF[R]) and (TRF[R]) in the neighborhood of d = 0. Then the claim follows from the positive homogeneity of g'(p; •). D Directional derivatives and subdifferentials of L-convex functions are given as follows. It is recalled that .Mo[R], -Mo[Z|R], M 0 [ Z ] , and £[R -> R|Z] denote, respectively, the sets of M-convex polyhedra, integral M-convex polyhedra, M-convex sets, and dual-integral polyhedral L-convex functions. See (3.23), (6.86), and (6.88) for the notation OR and dz, and note that ~g'(p', •) in Theorem 7.43 (2) denotes the directional derivative of the convex extension ~g of g at p. Theorem 7.43. (1) For g e £[R -» R] and p G dom Rff , define pg,p(X) = g'(p; Xx) Then

(XCV).

and dfig(p) ^ 0 in particular. If g G £[R —> R Z], then

(2) For g G £[Z -> R] and p G domzg, define pg,p(X) = g(p + Xx) ~ g(p) (XCV). Then

and &Rg(p) ^= 0 in particular. If g & £[Z —» Z], iften

7.10.

Directional Derivatives and Subgradients

197

and dzg(p) ^ 0 in particular. Proof. (1) Proposition 7.42 shows g'(p;-) e o£[R —> R], from which follows Pgtp € <S[R], by Proposition 7.38. By Theorem 7.33 (1) (L-optimality criterion),

We have B(p9,p) € M0[H} by (4.39) and g'(p; •) = p 9 , p (-) by (3.31), (3.33), and (4.14). If g e £[R -»• R|Z], dn,g(p) is an integral polyhedron by (6.76) and pg,p(X) = sup{x(X) | x e dRg(p)} € Z. (2) It is easy to see p9tp e <S[R]. The rest of the proof is similar to (1), where we use Theorem 7.14 (1) instead of Theorem 7.33 (1) and Theorem 4.15 instead of (4.39). D We have consistency between (1) and (2) in Theorem 7.43. Proposition 7.44. For g e £[Z|R —» R] andp e domR#nZ v , we have g'(p; xx) =

g(p + xx)-g(p) forX

cv.

Proof. This follows from integrality (6.75) and (5.19).

D

The next theorem affords characterizations of polyhedral L-convex functions in terms of the L-convexity of directional derivatives, the M-convexity of subdifferentials, and the L-convexity of minimizers. Theorem 7.45. For a polyhedral convex function g : Hv —> R U {+00} with dom.R<7 ^ 0, the four conditions (a), (b), (c), and (d) below are equivalent. (a) 0 e £ [ R - > R ] . (b) g'(p; •) € 0£[R -»• R] for every p e doniR^. (c) dn.g(p) e A^0[R] for every p e doniR^. (d) argming[—x] € £o[R] for every x 6 Hv with mfg[—x] > — oo. Proof, (a) =^> (b) is by Proposition 7.42, (a) => (c) by Theorem 7.43, and (a) => (d) by Proposition 7.34. (b) =>• (a): By (7.37), g is submodular and linear in the direction of 1 in a neighborhood of each p e Hv. This implies (SBF[R]) and (TRF[R]) (see Proposition 7.23). (b) •& (c): This follows from the relation (<5aRfl(p))" = g'(p; •) in (3.33) and the one-to-one correspondence between .Mo[R] and o£[R —> R], which is a consequence of (4.39) and Theorem 7.40. (d) =>• (b): To use Proposition 7.41 let x € Rv be such that inf g'(p; -)[-x] > —oo. Then inf g[—x] > —oo and argmingi[—x] E £o[R] by (d). By (5.18) we have argmingf[—x] = {q € R^

q(v) — q(u) < j(u,v) (u,v € V)}

198

Chapter 7.

L-Convex Functions

for some 7 € 3"[R]- Since argmin(g'(p; •)[~ a; ]) is a cone, we see

with Ap = {(u,v) | p(v) — p(u) = 7(u,u)}. This shows that argmin((/(p; •)[~:r]) is an L-convex cone. Then (b) follows from Proposition 7.41. D An integrality consideration in the equivalence of (a) and (d) in the above theorem yields a characterization of integral polyhedral L-convex functions. Theorem 7.46. For a polyhedral convex function g : Rv —> R U {+00} with doniRg ^ 0, the two conditions (a) and (d) below are equivalent. (a)«7e£[Z|R->R]. (d) argmin<7[—x] G £o[Z|R] for every x 6 R17 wz£/i inf g[—x] > — oo. Note 7.47. We prove <£= of Theorem 7.17 (1). We have argmin#[-x] e £o[Z] for every x € R^ by the assumption. Since an L-convex set is integrally convex (Theorem 5.10), g is an integrally convex function by Theorem 3.29. By the boundedness of dome/, the convex closure ~g of g is a polyhedral convex function and argming[—x] = argmin R] by Theorem 7.46 and therefore g e £[Z -> R]. •

7.11

Quasi L-Convex Functions

Quasi L-convex functions are introduced as a generalization of L-convex functions. The optimality criterion and the proximity theorem survive in this generalization. To define quasi L-convexity, we relax the submodularity inequality

to sign patterns of g(p A q) — g(p) and g(p V q) — g(q) compatible with (implied by) this inequality, which are given as follows:

Here Q and x denote possible and impossible cases, respectively. We call a function g : Zv —> R U {+00} quasi submodular if it satisfies the following:

(QSB) For any p, q e Zv, g(p A g) < g(p) or g(p V q) < g(q). Since p and q are symmetric, (QSB) implies also that g(p A q) < g(q) or g(p V q) < g(p). Similarly, we call g semistrictly quasi submodular if it satisfies the following property:51 51 The condition (SSQSB) was introduced by Milgrom-Shannon [129], in which g : Zv —> R U {—00} is called quasi supermodular if — g satisfies (SSQSB).

7.11. Quasi L-Convex Functions

199

(SSQSB) For any p.q e Zv, both (i) and (ii) hold:

Furthermore, a function g : Zv —> R U {+00} with domg ^ 0 is called quasi L-convex if it satisfies (QSB) and (TRF[Zj) and semistrictly quasi L-convex if it satisfies (SSQSB) and (TRF[Z]). Example 7.48. A quasi L-convex function arises from a nonlinear scaling of an L-convex function. For a submodular function g : Zv —> R U {+00} and a function if : R -» R U {+00}, define g : Zv -> R U {+00} by

Then g satisfies (QSB) if ip is nondecreasing and (SSQSB) if tp is strictly increasing. If g satisfies (TRF[Zj) with r = 0, this property is inherited by g. • Weaker variants of (QSB) and (SSQSB) can be conceived by considering possible sign patterns of the four values g(p A q) — g(p), g(p A q) — g(q), g(p V q) — g(p), &ndg(pVq)-g(q).

(QSBW) For a,nyp,q e domg, m&x{g(p),g(q)} >

mm{g(p/\q),g(pVq)}.

(SSQSBW) For any p, q e domg, either (i) or (ii) holds: (i) max{g(p),g(q)} > min{g(p A q),g(p V q)}, (ii) g(p) = g(q) = g(p /\q)= g(p V q). The relationship among various versions of quasi submodularity is summarized as follows. The second statement below shows that all the conditions are equivalent for g if they are imposed on every perturbation of g by a linear function. Recall the definition of g[x]; i.e., g[x](p) — g(p) + ( p , x ) . Theorem 7.49. For g : "Zv —> RU {+oc}, the following implications hold true.

Proof. (1) This is immediate from the definitions. (2) Combining Theorems 7.51 and 7.52 below establishes this. As is easily seen from the definitions of quasi L-convexity, most of the properties of quasi-submodular functions can be restated naturally in terms of quasi L-convex functions, and vice versa. We will work mainly with quasi-submodular functions. The following are quasi versions of Theorem 7.2 for L-convex functions.

200

Chapter 7.

L-Convex Functions

Proposition 7.50. Assume that g : Zv —> RU {+00} satisfies g(p) = g(p + 1) for allpeZv. (1) For g satisfying (QSBW) and for p, q € Zv and a & Z, we have

In particular, for p, q e domg and a e [0,ai - a 2 ]z, we have

where X C V, a\ € Z, and a<2 G Z U {—00} are defined by

(2) For g satisfying (SSQSBW) andforp,q 6 Z17 with g(p) ^ (/(g) and a € Z, we /ia«e inequality (7.39) wzi/i sinci inequality. In particular, for p, q G domg with g(p) 7^
7n particular, for p, q € domg ana7 a € [0, a\ — a-2\z, we have

Proof. Inequality (7.39) follows from max{#(p), g(q)} = max{g(p),g(q - al)} > min{g(p V (q - al)),g(p A (q - al))}, in which g(p/\(q — al)) = g((p/\(q — al)) + al) = ^((p + al) Ag). Inequality (7.40) is obvious from (7.39) since pV{o— (c {0, +CXD}. (QSB) for Srj is equivalent to (QDL) p, q e D ==> p/\qeDoipVqeD for D, whereas (SSQSB) for 6D is equivalent to (SBS[Z]) for D. Level sets of quasi-submodular functions have quasi submodularity. Furthermore, the weaker version (QSBW) of quasi submodularity for functions can be characterized by the property (QDL) of level sets; recall the notation L(g, a) from (6.95). Theorem 7.51. A function g : Zv —> R U {+00} satisfies (QSBW) if and only if the level set L(g, a) satisfies (QDL) for every a e R.

7.11.

Quasi L-Convex Functions

201

Proof. For the "if" part, take p,q & doing and put a = max{g(p),g ( min{g(p A q),g(p V 5)}. The "only if" part is even easier. D A submodular function over the integer lattice can be characterized by using level sets of functions perturbed by linear functions. Theorem 7.52. A function g : Zv —>• RU {+00} satisfies (SBF[Z]) if and only if the level set L(g[x], a) satisfies (QDL) for all x € Rv and a £ R. Proof. The "only if" part follows from Theorem 7.51 and the submodularity of g[x\. For the proof of the "if" part, takep,q G domg. By (QDL) for L(g, max{g(p),g ( 0, we can choose some x G Ry such that g[x\(p) = g[x](q) = g[x}(p A q) - e. (QDL) for L(g[x],o) with a = g(x\(p) shows pVq G L(g[x],a), which implies g[x](p) + g[x](q) = 1a> g[x](p/\q)+g[x](pVq)-e. Since e > 0 is arbitrary, this means (SBF[Zj). D Next we turn to the minimization of a quasi L-convex function. We assume r = 0 in (TRF[Zj) since otherwise no minimizer exists. Global minimality is characterized by local minimality. Theorem 7.53 (Quasi L-optimality criterion). Assume that g : Zv —> R U {+00} satisfies g(p) = g(p + 1) (Vp G Zv). (1) For g satisfying (QSBW) and p G domg, we have: g(p) < g(q) for all q e Zv such that q -p is not a multiple of 1. <$=>• g(p) < g(p + Xx) for all X C V withX i {0,V}. (2) For g satisfying (SSQSBW) and p G domg, we have: g(p) < g(q) (Vq G Proof. We prove •<= of (1) by contradiction. Suppose that g(q) < g(p) for some q G domg such that q — p is not a multiple of 1. We may assume that q > p by (TRF[Zj) with r = 0 and that q minimizes maxt,ey{q'(t;) — p(v)} among such vectors. Put X = argmax.vev{q(v) - p ( v ) } , where X ^ V. By Proposition 7.50 (1), we obtain I

whereas g(p) < g(q — Xx) by the choice of q. Hence follows g(p) > g(p + xx), a contradiction to the strict local minimality of p. The other direction => of (1) is obvious, and (2) can be shown similarly by Proposition 7.50 (2). D The proximity theorem for L-convex functions (Theorem 7.18) can be generalized for quasi L-convex functions. Theorem 7.54 (Quasi L-proximity theorem). Let g : Zv —» R U {+00} be a function satisfying (SSQSB) and g(p) — g(p+ 1) (Vp G Z y ), and assume n = \V\

202

Chapter 7.

L-Convex Functions

and a € Z ++ . If pa 6 doing satisfies (7.21), then argming ^ 0 and there exists p* e argming with (7.22). Proof. The proof of Theorem 7.18 works with (7.43) and (7.44) in place of (L*-APR[Z]).

Bibliographical Notes The concept of L-convex functions was introduced by Murota [140]. L^-convex functions are defined by Pujishige-Murota [68] as a variant of L-convex functions, together with the observation that they coincide with the submodular integrally convex functions considered earlier by Favati-Tardella [49]. Theorems 7.1 and 7.3 are due to [68], and Theorem 7.2 is stated in Murota [147]. Discrete midpoint convexity is considered by Favati-Tardella [49] with an observation of its equivalence to submodular integral convexity. The equivalence of discrete midpoint convexity to translation submodularity (SBF^Zj) in Theorem 7.7 is by Fujishige-Murota [68], whereas that to (L^-APRfZ]) is noted in Murota [147]. Condition (7.11) for quadratic L-convex functions is given in Murota [141]. Separable convex functions with chain conditions (7.13) are considered in BestChakravarti-Ubhaya [11]. Multimodular functions are treated in Hajek [85]. The basic operations in section 7.4 are listed in Murota [141], [144], [147]. The theorems on minimizers of L-convex functions are of fundamental importance. Theorem 7.14 (L-optimality criterion) is stated in Murota [145]. Theorem 7.15 (optimality for submodular set functions) can be found as Theorem 7.2 in Fujishige [65]. Theorem 7.17 (characterization by minimizers) is a corollary of Theorem 7.45 due to Murota-Shioura [152]. A thorough study of minimizers of submodular functions is made in Topkis [202]. The L-proximity theorem (Theorem 7.18) is due to Iwata-Shigeno [105]. The present proof based on (L^-APRfZ]) is by Murota-Shioura [154]. The construction of the convex extension of an L-convex function by means of the Lovasz extensions (Theorem 7.19) is due to Murota [140]. The same idea, however, was used earlier by Favati-Tardella [49] for submodular integrally convex functions. The equivalence of L''-convexity to submodular integral convexity (Theorem 7.21) is due to Fujishige-Murota [68]. Polyhedral L-convex functions are investigated by Murota-Shioura [152], to which all the theorems in section 7.8 (Theorems 7.26, 7.28, 7.29, 7.30, 7.31, 7.32, and 7.33) as well as Proposition 7.34 are ascribed. L-convexity for nonpolyhedral convex functions is considered in Murota-Shioura [156], [157]. The correspondence between positively homogeneous L-convex functions and submodular set functions (Theorem 7.40) is established for the case of Z in Murota [140] and generalized to the case of R in Murota-Shioura [152]. Proposition 7.37 is stated in Murota [147]. Theorem 7.43 for directional derivatives and subgradients is shown for the case of Z in Murota [140], [141], and generalized to the case of R in Murota-Shioura [152]. Theorem 7.45 (characterizations in terms of directional derivatives, subdifferentials,

7.11.

Quasi L-Convex Functions

203

and minimizers) is by [152], whereas its ramification with integrality (Theorem 7.46) is stated in Murota [147]. The concept of quasi L-convex functions was introduced by Murota-Shioura [154] on the basis of the idea of Milgrom-Shannon [129]. Theorem 7.52 is due to [129], and the other theorems in section 7.11 (Theorems 7.49, 7.51, 7.53, and 7.54) are in [154].

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Chapter 8

Conjugacy and Duality

By addressing the issues of conjugacy and duality, this chapter provides the theoretical climax of discrete convex analysis. Whereas conjugacy in convex analysis gives a symmetric one-to-one correspondence within a single class of closed convex functions, conjugacy in discrete convex analysis establishes a one-to-one correspondence between two different classes of discrete functions with different combinatorial properties distinguished by "L" and "M." The conjugacy between L-convexity and M-convexity is thus one of the most remarkable features of discrete convex analysis. Discrete duality is another distinguishing feature. It is expressed in a number of theorems, such as the separation theorems for M-convex/M-concave functions and for L-convex/L-concave functions (M- and L-separation theorems) and the Fenchel-type duality theorem. Besides formal parallelism with convex analysis, these discrete duality theorems carry deep combinatorial facts, implying, for example, Edmonds's intersection theorem and Prank's discrete separation theorem for submodular set functions as special cases.

8.1

Conjugacy

M-convex functions and L-convex functions form two distinct classes of discrete functions that are conjugate to each other under the Legendre-Fenchel transformation. This stands in sharp contrast with conjugacy in convex analysis, which is a symmetric one-to-one correspondence within a single class of closed convex functions. The conjugacy correspondence between M-convexity and L-convexity is in fact a translation of two different combinatorial properties, exchangeability and submodularity, on top of convexity. The relationship between submodularity and supermodularity with respect to conjugacy is discussed first in section 8.1.1. Conjugacy for polyhedral M-/L-convex functions is established in section 8.1.2 and that for integer-valued M-/L-convex functions on integer points in section 8.1.3.

205

206

8.1.1

Chapter 8. Conjugacy and Duality

Submodularity under Conjugacy

Submodularity and supermodularity are not symmetric under the Legendre-Fenchel transformation. The conjugate of a submodular function is always supermodular, whereas the conjugate of a supermodular function is not necessarily submodular. In this subsection we assume that / is a function in real variables, / : THV RU {+00}, with a nonempty effective domain. Recall that / is submodular if

and supermodular if

Also recall from (3.26) that the Legendre-Fenchel transform /* : Ry -> RU {+00} is denned by

Theorem 8.1. For a submodular function f, the Legendre-Fenchel transform /* is supermodular. Proof. For x, y € Hv and p, q e Rv, we have

From this inequality, Submodularity (8.1), and the definition (8.3), we see that

Taking the supremum over x and y we obtain

which shows the supermodularity of /*.

D

In contrast to Theorem 8.1, the Legendre-Fenchel transform of a supermodular function is not necessarily submodular. For example, consider a pair of convex quadratic functions f(x) = ^ X T Ax and g(p) = ^pTA~lp with

We have g = f by Proposition 2.9, whereas / is supermodular and g is not submodular by Proposition 2.6.

8.1.

Conjugacy

207

If n = 2, however, supermodularity does imply submodularity of the LegendreFenchel transform. Proposition 8.2. For a supermodular function f in two variables, the LegendreFenchel transform /* is submodular. Proof. It suffices to show that

for p = (p(l),p( 2 )) e R2 and q = ( q(l) and p(2) > q(2). We claim that

for any x = (x(l),x(2)) e R2 and y = (j/(l),y(2)) e R2. The inequality (8.4) is an immediate consequence of (8.5), since the supremum of the left-hand side of (8.5) over x and y coincides with the left-hand side of (8.4). Proof of (8.5): If x(l) > y(l) and z(2) > j/(2), we have

and, therefore,

If x(l) < y(l), we have

and, therefore,

A similar argument holds for the case of x(2) < j/(2). By Theorem 8.1, submodularity is preserved under the transformation / H-> —/*. However, this does not establish a symmetric one-to-one correspondence within the class of submodular functions. It is not true, either, that the mapping / i — > / * gives a one-to-one correspondence between the class of submodular functions and the class of supermodular functions.

208

8.1.2

Chapter 8. Conjugacy and Duality

Polyhedral M-/L-Convex Functions

Conjugacy for polyhedral M-convex and L-convex functions is considered here. We start with a technical lemma. Proposition 8.3. Let g £ £[R —> R] be a polyhedral L-convex function. For x,y £ R^ with inf g[—x] > — oo and inf g[—y] > — oo and for u £ supp+(x — y), there exists v £ supp~(x — y) such that

Proof. We may assume argming[—x] j^ 0 and argmin<ji[—y] ^= 0. By Proposition 7.34, we have argmingf—x] £ A)[R] and argming[—y] £ £o[R]- It suffices to demonstrate the existence of v £ supp~(x — y) such that p(v) < q(v) for all p £ Dx and q £ Dy, where

To prove this by contradiction, suppose that for every v £ supp (x — y) there exist pv £ Dx and qv £ Dy with pv(v) > qv(v). Then, for

we have p* £ Dx, q* £ Dy, and p*(v) > q*(v) (Vu e supp (x - y ) ) . By denning

with A = mm{p*(v) — q*(v) \ v £ supp+(p* — g*)} > 0, we obtain

from Theorem 7.29. By supp (x — y) C supp+(p* — ?»), on the other hand, we see

where the last equality is due to x(V) = y(V) — r (the constant in (TRF[R])). Combining (8.7) and (8.8) results in

8.1.

Conjugacy

which is a contradiction to p* G argming[—x] and q* € argming r [—y].

209

D

The conjugacy theorem for polyhedral M-convex and L-convex functions is now stated. Theorem 8.4 (Conjugacy theorem). (1) The classes of polyhedral M-convex functions and polyhedral L-convex functions, M. = .M[R —> R] and £, = £[R —> R], are in one-to-one correspondence under the Legendre-Fenchel transformation (8.3). That is, for f G M. and g G £, we have f £ £, g* G M, f" = f, and g9' = g. (2) The classes of polyhedral A/fl-convex functions and polyhedral L^-convex functions, Ai^R —» R] and $\R —> R], are m one-to-one correspondence under (8.3) m a similar manner. Proof. (1) and (2) are equivalent, so we prove (1). We first note that /** = / for any polyhedral convex function /. For / G M we have $R/*(P) = argmin/[—p] G .Mo[R] (Vp £ dom/*) by Proposition 6.53. Then /* G £ by (c)=^(a) in Theorem 7.45. (An alternative proof is described in the proof of Theorem 8.6.) Conversely, take g € C, and x,y G domg*. Since inf<7[—x] > —oo and inf g[—y] > —oo, Proposition 8.3 shows that for every u € supp+(x — y) there exists v G supp-(z - y) satisfying (8.6). Noting that (£ a r g min g [-x])* = (g')'(x', •)> which follows from (3.30) and (3.33), we obtain

This shows (M-EXC'[R]) for g*, and hence g' & M. Theorem 8.4 (2) states that L^-convex functions and M^-convex functions are transformed to each other, where L^-convex functions are submodular (Theorem 7.28) and M^-convex functions are supermodular (Theorem 6.51). It is noted that Theorem 8.4 (2) does not imply, nor is it implied by, Theorem 8.1, which shows the supermodularity of the conjugate of a submodular function. Recalling the basic fact that the conjugate of the indicator function of a convex set is a positively homogeneous convex function, and vice versa, we see from Theorem 8.4 above that M-convex polyhedra ./Vfo[R] and positively homogeneous L-convex functions o£[R —»• R] are conjugate to each other and also that L-convex polyhedra £o[R] and positively homogeneous M-convex functions o.M[R —> R] are conjugate to each other. On the other hand, we can identify positively homogeneous L-convex functions o£[R —* R] with submodular set functions <S[R] (Theorem 7.40) and positively homogeneous M-convex functions oA^[R —> R] with distance functions with the triangle inequality T[R] (Theorem 6.59). We can summarize these

210

Chapter 8. Conjugacy and Duality

one-to-one correspondences in the following diagram:

In addition, the polarity between M-convex cones and L-convex cones follows from (8.9). This is because two convex cones are polar to each other if and only if their indicator functions are conjugate to each other. Thus we obtain the following theorem. Theorem 8.5. A polyhedral cone is M-convex if and only if its polar cone is Lconvex. Hence, the classes of M-convex cones and L-convex cones are in one-to-one correspondence under polarity (3.34). Taking integrality into account in diagram (8.9), we obtain

where ,M[Z|R —> R] and £[Z|R —> R] denote the sets of integral polyhedral Mconvex and L-convex functions, respectively; .M[R —> R|Z] and £[R —> R|Z] denote the sets of dual-integral polyhedral M-convex and L-convex functions, respectively; and o.M[R —> R|Z] and o£[R —> R|Z] are their subclasses with positive homogene-

ity.52

It is known that the conjugacy relationship between M-convexity and Lconvexity holds more generally for closed proper convex functions. Recall that the Legendre-Fenchel transformation gives a symmetric one-to-one correspondence in the class of all closed proper convex functions (Theorem 3.2). Theorem 8.6. A closed proper convex function f satisfies (M-EXC[R]) if and only if f = g* for a closed proper convex function g that satisfies (SBF[Rj) and (TRF[Rj). Proof. The proof of the "if" part is essentially the same as the latter half of the proof of Theorem 8.4. The "only if" part needs a new approach, since the implication (c)=^(a) in Theorem 7.45 does not carry over to nonpolyhedral convex functions. Suppose that / satisfies (M-EXC[R]) and put g = /*. It is easy to show (TRF[Rj) for g. The proof of the submodularity (SBF[Rj) for g consists of the following steps. 52 The notation for dual integrality extends naturally to other classes of functions. For example, .M^R -+ R|Z] and /^[R —> R|Z] denote the sets of dual-integral polyhedral M^-convex and L^-convex functions, respectively.

8.1. Conjugacy

211

1. We may assume that dom/ is bounded, so that domg — Rv. 2. For po e Tiv and U C V with \U\ = 2, denote by / : Hu -> R U {+00} the projection of /[—po] to U and by 5 : Ru —* R the restriction of g(po + p) to U. Then we have g = (/)*. 3. (M-EXC[R]) of / implies the supermodularity of /. 4. The supermodularity of / implies the submodularity of (/)* by Proposition 8.2. 5. The submodularity of g for any po and any U implies the submodularity of g. The details are given in Murota-Shioura [156], [157]. Note 8.7. With Theorem 8.4 we complete the proof of Theorem 6.63 (characterizations of polyhedral M-convex functions), (b) ^ (c) follows from <59R/(X)* = f'(x; •) in (3.33) and the correspondence between £o[R] and o-M[R —> R], which is a special case of Theorem 8.4. To show (a) •&• (c) •& (d), put g = /* and note that d R f ( x ) = argmin R] <^ argming[-z] 6 C0[R] & ^(p) & M0[R}. Note 8.8. We complete the proof of Theorem 6.45 using Theorem 6.63 ((d) (a)) established in Note 8.7. Let / be the convex extension of / 6 M[Z —> R]. For any p € Ry, argmin/[—p] is an M-convex polyhedron if it is not empty (Theorem 6.43). Since / is polyhedral by the assumption, Theorem 6.63 ((d) =>• (a)) sho /e.M[R^R]. Note 8.9. Using Theorem 8.5, we complete the proof of (4.42), the representation of an M-convex cone in terms of vectors Xu ~ Xv (u>v £ V). Let B be an Mconvex cone and D be the polar of B. By Theorem 8.5, D is an L-convex cone, and by (5.18) it can be represented as D — {p G R^ (p,a») < 0 (i = 1,... ,m)} for some a^ = \Ui — \Vi (i = 1,... ,m). Since B is the polar of D, this implies B = {x 6 Rv | x is a nonnegative combination of a* (i = 1 , . . . , m)} by (3.36). Conversely, a convex cone of this form is M-convex, as can be shown by reversing the above argument. Note 8.10. Using Theorem 8.5, we complete the proof of (5.21), the representation of an L-convex cone in terms of a ring family T> C 2V. Let D be an L-convex cone and B be the polar of D. By Theorem 8.5, B is an M-convex cone, and, by (4.39), it can be represented as B = B(p) using p € <S[R] with p : 2V —> {0, +00}; i.e.,

with T> = dom/5, which is a ring family. Since D is the polar of B, this implies

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Chapter 8. Conjugacy and Duality

by (3.36). Conversely, a convex cone of this form is L-convex, as can be shown by reversing the above argument. •

8.1.3

Integral M-/L-Convex Functions

We turn to functions defined on integer points. For functions / : Zv —> RU{+oo} and h : Zv —> RU{—oo}, discrete versions of the Legendre-Fenchel transformations are defined by

We call (8.11) and (8.12), respectively, convex and concave discrete LegendreFenchel transformations. The functions /* : R^ —* R U {±00} and h° : R^ —> R U {±00} are called the convex conjugate of / and the concave conjugate of h, respectively. Note that h°(p) = — (—h)'(—p). For an integer-valued function /, f'(p) is integral for an integer vector p. Hence, (8.11) with p 6 Zv defines a transformation of / : Zv —> Z U {+00} to /* : Zv -> Z U {±00}; we refer to (8.11) with p e Zv as (8.11)z. We call (/*)* using (8.11)z the integer biconjugate of / and denote it by /**. Similarly, (8.12) with p e Zv is designated by (8.12)z and we define h°° = (h°)°. The following fact is fundamental for the conjugacy of discrete functions. Proposition 8.11. For a function f : Zv —> Z U {+00} and a point x 6 dom z /, we have f"(x) = f ( x ) i f d z f ( x ) ^ 0, where f" means the integer biconjugate with respect to the discrete Legendre-Fenchel transformation (8.11)z. Proof. For p e d z f ( x ) , we have f ' ( p ) = (p,x) - f ( x ) (cf. (3.30)), and therefore f"(x] = sup{(q,x) -/'((?) q£Zv}> (p,x) - /'(p) = f ( x ) . On the other hand, f"(x) < f ( x ) for any / and x. D The conjugacy theorem for discrete M-convex and L-convex functions reads as follows. Theorem 8.12 (Discrete conjugacy theorem). (1) The classes of integer-valued M-convex functions and integer-valued Lconvex functions, M — M[Z —> Z] and C = £[Z —> Z], are in one-to-one cor respondence under the discrete Legendre-Fenchel transformation (8.11)z- That is, for f e M and g 6 £, we have /* 6 C, g* e M, f" — f, and g" = g. (2) The classes of integer-valued M^-convex functions and integer-valued L$convex functions, M^[Z —> Z] and &[Z —> Z], are in one-to-one corresponden under (8.11)z in a similar manner. Proof. The basic idea of the proof is to apply Theorem 8.4 to the convex extensions of / and g with additional arguments for discreteness. Since (1) and (2) are equivalent, we deal with (2).

8.1.

Conjugacy

213

(i) Take / e M.^[Z —> Z]. Let / be the convex extension of / and / be the conjugate of / in the sense of (8.3). We have f ' ( p ) = f (p) for p e Zv. If doniz/ is bounded, / is polyhedral convex, and therefore / e .M^R —> R] by Theorem 6.45. Then Theorem 8.4 shows /* € &[H -> R]. Since /*(p) = /*(p) for p € Zv, (SBF^R]) for /* implies (SBF^Z]) for /*. Hence /• e C\Z -> Z]. If domz/ is unbounded, we consider the restriction /& of / to integer interval [—fcl,fcl]z for A; E Z large enough to ensure domz/ fl [—fcl,fcl]z ^ 0- Then we have fk & M^Z -> Z} and /£ e -C^Z -» Z] by the argument above. For each p € domz/*, there exists kp such that /*(p) — /*(p) for all k > kp (cf. Theorem 6.42 and Proposition 3.30). Therefore, (SBF^Z]) for ft implies (SBF^Z]) for /*. Hence/* € &[Z -> Z]. (ii) Take # € /^ [Z —> Z]. Let ^ be the convex extension of g and g* be the conjugate of ~g in the sense of (8.3). We have

and, in particular, g*(x) = g*(x) (x e Zv). If domz5 is bounded, ~g is polyhedral convex, and therefore ~g 6 £tl [R —» R] by Theorem 7.26. Then g* 6 A^^R -> R] by Theorem 8.4, and Z] by Theorem 6.2. To show a0 — 1, fix x,y e doniz*?*, •u e supp+(x - y), and u e supp~(x - y) U {0} in (M^-EXCfRj). By the assumed boundedness of domzg, the supremum in (8.13) is attained by some p. Moreover, there exist po e Zv and a\ > 0 such that

for all a e [0, QIJR. Condition (8.14) can be written as

which is equivalent, by the L-optimality criterion (Theorem 7.14 (2)), to

Note that the right-hand side is an integer and the coefficient of a on the left is either ±1 or 0. By virtue of this integrality, the above inequality is satisfied by all a e [0, I]R, if it is satisfied by some a > 0. Therefore, (8.14) holds for all a 6 [0, I]R. Similarly, there exists qo G Zv such that

for all a 6 [0,1]R. Combining (8.14) and (8.16) shows

214

Chapter 8. Conjugacy and Duality

for all a e [0, I]R. Hence ao = 1 is valid.53 If donizgr is unbounded, we consider the restriction g^ of g to integer interval [—fcl,fcl]z for k £ Z large enough to ensure domzg n [—fcl,fcl]z ^ 0. Then we have gk e ^[Z —> Z] and g* e .M^Z —> Z] by the argument above. For each x € doniz kx (cf. Theorem 7.20 and Proposition 3.30). Therefore, (M^-EXC[Z]) for g*k implies (M&-EXC[Z]) for g*. Hence 5* 6 A^[Z ^ Z]. (iii) Finally, /" = / and g" = g follow from Proposition 8.11, Theorem 6.61 (2), and Theorem 7.43 (2). D As the discrete counterpart of diagram (8.9), we obtain the following:

This follows from the discrete conjugacy theorem (Theorem 8.12) in combination with Theorems 7.40, 6.59, 4.15, and 5.5. In addition, we can obtain the M^-fLfiversion of (8.17). The conjugacy relationship among discrete convex functions is schematized in Fig. 8.1, where M^-convex and L^-convex functions are defined in section 8.3. This is the ultimate picture for the discrete conjugacy relationship, which originated in the equivalence between the base family and the rank function of a matroid (section 2.4). In other words, the exchange property and submodularity are conjugate to each other at various levels. Examples of mutually conjugate M-convex and L-convex functions are demonstrated below for integer-valued functions defined on integer points. In the network flow problem in section 2.2, if /„ e C[Z —> Z] and ga e C[Z —> Z] are conjugate for each arc a € A, then

in (2.42) and (2.43) are conjugate to each other. We will dwell on the conjugacy in network flow in section 9.6. In a valuated matroid (V,B,w), which arises, e.g., from a polynomial matrix (section 2.4.2),

53 An alternative proof is possible on the basis of (i)—(iii) below if (iii) is accepted as a known fact: (i) (8.15) is equivalent to x — a(xu — Xv) 6 dn9(po), (ii) dRg(po) is an integral M^-convex polyhedron (Theorem 7.43 (2)), and (iii) for an integral M^-convex polyhedron Q and x, y 6 Qr\Zv, a0 = 1 is valid in (B^EXCp,]).

8.1.

Conjugacy

215

(FNC = function, SET = set, PHF = positively homogeneous function) Figure 8.1. Conjugacy in discrete convex functions.

in (2.77) and (2.78) are conjugate to each other. If / e M^[Z —> Z] and g 6 &\L —> Z] are conjugate, then the restriction ju and the projection gu to a subset U C V are conjugate to each other and the projection fu and the restriction gu are conjugate to each other. If / e M[Z -» Z] and g e £[Z ->• Z] are conjugate and ^ e C[Z ->• Z] and V>t) € C[Z —> Z] are conjugate for each v e V, then

216

Chapter 8. Conjugacy and Duality

in (6.46) and (7.18) are conjugate to each other. If fi e M\7, ->• Z] and 3, e £ h [Z -> Z] are conjugate for i = 1,2, then /i n z /2 € .M^Z —> Z] and gi + g2 £ ^[Z —> Z] are conjugate to each other. Note 8.13. In section 2.1 we saw the conjugacy between M^-convex and L''-convex quadratic functions in real variables (Theorems 2.11 and 2.16). This conjugacy relationship does not have a discrete counterpart. Let / : Zn —> Z be a quadratic function represented as f ( x ) = XT Ax, with a positive-definite symmetric matrix A with integer entries, and /* : Z" —> Z be the discrete Legendre-Fenchel transform (8.11) of /. If A satisfies (6.26) through (6.28), then / is M^-convex and /* is L^-convex, but /* is not necessarily a quadratic function. Likewise, if A satisfies (7.10), then / is L^-convex and /* is M^-convex, but /* is not necessarily a quadratic function. For instance, f ( x ) = x2, where x G Z, is an M^-convex function with

which is not quadratic since /'(-I) = /*(0) = /'(I) = 0.

8.2

Duality

Discrete duality theorems lie at the heart of discrete convex analysis. Major theorems presented in this section are the separation theorem for M-convex/M-concave functions (M-separation theorem), the separation theorem for L-convex/L-concave functions (L-separation theorem), and the Fenchel-type duality theorem. These theorems look quite similar to the corresponding theorems in convex analysis, but they express, in fact, some deep facts of a combinatorial nature. Almost all duality results in optimization on matroids and submodular functions are corollaries of these theorems.

8.2.1

Separation Theorems

We start by reviewing the preliminary general discussion in section 1.2. A discrete separation theorem is a statement that, for / : Zv —> Z U {+00} and h : Zv —> Z U {—00} belonging to certain classes of functions, if f(x) > h(x] for all x £ Zv, then there exist a* £ Z and p* £ Zv such that

Denoting by / the convex closure of / and by h the concave closure of h (i.e., —h is the convex closure of —ft), we observed the following phenomena in Examples 1.5 and 1.6:

8.2. Duality

217

2. f ( x ) > h(x) (Vx e Zv) =^ existence of a* £ R and p* 6 R^, 3. existence of a* e R and p* £ Ry =^=> existence of a* 6 Z and p* 6 Zv. We will see below that all three implications hold true for M-convex and Lconvex functions. The following proposition addresses the first. Proposition 8.14.

(1) 7//,-/ie.Ml[Z->R], then

Proof. (1) Theorem 6.44 with /i = / and /2 = —h shows that fi + f% > 0 implies A+]2>o. (2) It suffices to prove the claim when g, — k e £[Z —> R]. Theorem 7.19 applied to g and — k shows this. D The separation theorem for M-convex/M-concave functions reads as follows. It should be clear that /* and h° are the convex and concave conjugate functions of / and h defined by (8.11) and (8.12), respectively. In the proof we use the notations <9R and d'z for the concave version of subdifferentials denned as

Theorem 8.15 (M-separation theorem). Let f : Zv —> R U {+00} be an Aflconvex function and h : Zv —> R U {—00} be an Afl-concave function such that domz/ n domz/i ^ 0 or domR/* n domR/i° ^ 0. // /(x) > h(x) (Vx 6 Zv), there exist a* e R and p* & Ry such that

Moreover, if f and h are integer valued, there exist integer-valued a* 6 Z and p* 6 Zv. Proof. We may assume /, —h e M\L —> R]. (i) Suppose that domz/ n domz/i ^ 0. For the convex closure f of f and the concave closure h of h, we have f(x] > h(x) (Va; 6 R^) by Proposition 8.14 (1). Since doniR/ n dorriR/i ^ 0, the separation theorem in convex analysis (Theorem 3.5) gives a* e R and p* e Rv such that/(x) > o^ + (p*,ar) > /i(x) (Vx e Ry) (see Note 8.19). This implies (8.20) since / = / and h = ft on Zv by Theorem 6.42. The integrality assertion is proved from the facts that the integer subdifferential of an integer-valued M-convex function is an L-convex set and that Lconvex sets have the property of convexity in intersection. We may assume that inf{/(x) - h(x) | x 6 Zv} - 0. Then there exists x0 e Zv with /(x 0 ) - h(x0) = 0

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Chapter 8. Conjugacy and Duality

(by the integrality of the function value). By (6.87) and Theorem 6.61 (2) we have

which is nonempty since p* € dfif(xo) n d^h(xo). Since d z f ( x o ) and d'zh(x0) above are L-convex, convexity in intersection for L-convex sets (5.9) guarantees the existence of an integer vector p** e d z f ( x 0 ) n d'zh(xo). With this p** and a** = h(x0) - (p**,x0) e Z, the inequality (8.20) is satisfied. (ii) Next suppose that domz/ n dom z ft = 0 and domR/* n dom R ft° ^ 0. For a fixed po G domR/* n domR,ft° and for any p € Rv, we have

from which follows

Since domz/ and domzft are disjoint M-convex sets, the separation theorem for Mconvex sets (Theorem 4.21) gives p* G Hv such that the right-hand side of (8.21) with p = p* is nonnegative. With this p* and a* G R such that f ' ( p * ) < -a* < h°(p*), the inequality (8.20) is satisfied. For integer-valued / and ft, we have /•,—ft 0 G £[Z|R —> R] and, hence, dom R /*,dom R ft° G £0[Z|R]. We may assume p0 G Zv by (5.9) and p* G Zv by Theorem 4.21. Then f*(p*) and h°(p*) are integers, and therefore we can take an integer a* e Z. D Next we state the separation theorem for L-convex/L-concave functions. Theorem 8.16 (L-separation theorem). Let g : Zv —» R U {+00} be an Lficonvex function and k : Zv —> R U {—00} be an L^ -concave function such that domzg n dom z fc ^ 0 or domRg° n domRfc° ^ 0. If g(p) > k(p) (Vp e Zv), there exist (3* € R and x* e Rv such that

Moreover, if g and k are integer valued, there exist integer-valued j3* G Z and

x* e zv.

Proof. We may assume g, —k & £[Z —> R]. (i) Suppose that domzg n domzfc j^ 0. For the convex closure ~g of g and the concave closure A; of k, we have (?(p) > fc(p) (Vp e R17) by Proposition 8.14 (2). Since domR<7 n domRfc ^ 0, the separation theorem in convex analysis (Theorem 3.5) gives /?* e R and x* e Ry such that g(p) > /3_* + (p,x*) > Jfc(p) (Vp e Ry) (see Note 8.19). This implies (8.22) since g = g and k = k on Zv by Theorem 7.20.

8.2. Duality

219

The integrality assertion is proved from the facts that the integer subdifferential of an integer-valued L-convex function is an M-convex set and that Mconvex sets have the property of convexity in intersection. We may assume that inf{0(p) - k(p) | p 6 Zv} = 0. Then there exists p0 e Zv with g(pQ) - k(p0] = 0 (by the integrality of the function value). By (6.87) and Theorem 7.43 (2), we have

which is nonempty since x* € dRg(po) H c^fc(Ri)- Since dzg(po) and d'zk(po) above are M-convex, convexity in intersection for M-convex sets (4.34) guarantees the existence of an integer vector x** £ dzg(po) n d'zk(po). With this x** and /3** = k(p0) - (po,x**) e Z, the inequality (8.22) is satisfied. (ii) Next suppose that domzg n domzfc = 0 and doniR,
from which follows

Since domz
if and only if there exists p* e Rv such that

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Chapter 8. Conjugacy and Duality

Conditions (8.25) and (8.26) are equivalent, respectively, to

with the notation xo = 0, and for such a p* we have

Moreover, if fi and fa are integer valued, i.e., /i,/2 G M^[Z —> Z], we can choose integer-valued p* G Zv. Proof. The sufficiency of (8.25) and (8.26) is obvious. Conversely, suppose that (8.24) is true and apply the M-separation theorem (Theorem 8.15) to f ( x ) = f \ ( x ) and h(x) = f i ( x * ) + h(x*) — fa(x) to obtain a* and p* satisfying (8.20). We have a* = fi[—p*](x*) from (8.20) with x = x* and hence

This implies (8.25) and (8.26), which are equivalent to (8.27) and (8.28), respectively, by the M-optimality criterion (Theorem 6.26 (2)). To prove (8.29) take x G argmin(/i + / 2 ). Then

which, along with (8.25) and (8.26), implies x G argmin/i[— p*] n argmin/2[+p*]. Hence follows C in (8.29), whereas D is obvious. Finally, the integrality of p* is due to the integrality assertion in Theorem 8.15. D Note 8.18. The assumptions on the effective domains are necessary in the separation theorems (Theorems 8.15 and 8.16). For instance, for an M-convex function / : Z2 —> Z U {+00} and an M-concave function h : Z2 —> Z U {—00} denned by

we have

domz/ndomzft = 0, and domz/* ndomzft 0 = 0. There exists no separating affine function for (/, ft) or (/*, Note 8.19. This is a technical supplement to the proof of the M-separation theorem (Theorem 8.15). (A similar remark applies to the proof of the L-separation theorem.) We applied the separation theorem to / and ft, the convex and concave

8.2. Duality

221

extensions of / and h, without verifying the assumption in Theorem 3.5. If / and h are polyhedral, the assumption (a2) of Theorem 3.5 is met and the theorem is literally applicable. If inf{/(x) — h(x) \ x e Zv} is attained by some x = x0 e Zy, we have

in which the directional derivatives / (XQ;-) and h (XQ; •) of / and h at XQ are polyhedral and Theorem 3.5 may be used for the pair of /(XQ) + f (XQ; x — XQ) and h(xo) + h (XQ; x — XQ). Otherwise we have to resort to a variant of the separation theorem such as the following: Let / : Zv -> R U {+00} and h : Zv -» R U {-00} be integrally convex and concave functions with domz/ fl doniz/i ^ 0, and denote their convex and concave extensions by / and h. If f(x) > h(x) (\/x € Rv), then there exist a* <E R and p* e Hv such that J(x) > a* + (p*,x) > ~h(x) (Vx e Hv). *

Note 8.20. The original proof of the M-separation theorem is based on an algorithmic argument for a generalization of the submodular flow problem involving an M-convex cost function (Murota [142]). In particular, the argument is purely discrete, not relying on the separation theorem in convex analysis. See sections 9.1.4 and 9.5. • 8.2.2

Fenchel-Type Duality Theorem

The Fenchel-type duality theorem is discussed here. Before giving a precise statement of the theorem we explain the essence of the assertion. For any functions / : Zv —> R U {+00} and h : Zv —> R U {—oo}, we have a chain of inequalities

from the definitions (8.11) and (8.12) of conjugate functions, where / and h are the convex and concave closures of / and h, respectively. We observe the following: 1. The second inequality is in fact an equality (under certain regularity assumptions) by the Fenchel duality theorem in convex analysis (Theorem 3.6). 2. The first inequality can be strict even when / is convex extensible and h is concave extensible, as is demonstrated by Example 1.6. A similar statement applies to the third inequality. The following theorem asserts that the first and second inequalities in (8.30) turn into equalities for Mb-convex/M^-concave functions and L^-convex/L^-concave functions and that all three inequalities are equalities for such integer-valued functions.

222

Chapter 8. Conjugacy and Duality

Theorem 8.21 (Fenchel-type duality theorem). (1) Let f : Zv -> RUJ+oo} be an Afl-convex function and h : Zv -> Ru{-oo} be an M^-concave function, i.e., /, —h e M.^[Z —> R], sucft t/iaidomz/ndomz/i 7^ 0 or doniR/* n doniR/i0 ^ 0. Then we have

If this common value is finite, the supremum is attained by some p € domR/* n doniR/i°. (2) Let g : Zv —» RU{+oo} be an L^-convex function and k : Zv —> RU{—00} be an L^-concave function, i.e., g, —k € fi[Z —» R], suc/i i/ia£ domz
If this common value is finite, the supremum is attained by some x € doniRg* n domRfc0. (3) Let f : Zv —> Z U {+00} 6e an integer-valued Afi-convex function and h : v Z —> Z U {—(X)} 6e an integer-valued Afi-concave function, i.e., /, — h £ M^[Z —>• Z], such that domz/ fl domz/i ^ 0 or domz/* PI domz/i° 7^ 0. T/ien we Aave

If this common value is finite, the infimum is attained by some x € domz/ndomz/i and the supremum is attained by some p & domz/* H domz/i 0 . (4) Let g : Zv —> Z U {+CXD} 6e an integer-valued L^-convex function and k : v Z —> ZU{—00} 6e an integer-valued L^-concave function, i.e., g, —k £ &[Z —> Z], swcft i/iai domzgi n domz/c 7^ 0 or domZ5* H dom z fc 0 7^ 0. T/ien we have

If this common value is finite, the infimum is attained by somep e domzgndomzfc and the supremum is attained by some x € domzg* n domzfc 0 . Proof. (1) Suppose that domz/ H dornzft. ^ 0. By (8.30) we may assume that A = mf{f(x) — h(x) x e Zv} is finite. By the M-separation theorem (Theorem 8.15) for (/ - A, h), there exist a* 6 R and p* & Hv such that

for all x e Zv, which implies ft°(p*) - f ' ( p " ) > A. Combining this with (8.30) shows (8.31) as well as the attainment of the supremum by p*. Next suppose that domz/ n domz/i = 0 and doniR/* n doniR/i0 ^ 0. The separation theorem for M-convex sets (Theorem 4.21) applied to BI = domz/i and B2 = domz/ gives p* e {0, ±1}V satisfying (4.33). Putting p = p0 + cp* in (8.21) (within the proof of the M-separation theorem) and letting c —> +00, we obtain sup — +00 in (8.31), whereas inf — +00 by domz/ n domz/i = 0.

8.2. Duality

;

223

(2) (The proof goes in parallel with (1).) Suppose that domzg fl domzA; ^ 0. By (8.30) we may assume that A = inf{
for all p & Zv, which impliesfc°(a;*)- ff*(x*) > A. Combining this with (8.30) shows (8.32) as well as the attainment of the supremum by x*. Next suppose that domzg n domzfc = 0 and domR, +00, we obtain sup — +00 in (8.32), whereas inf = +00 by domzg fl donizfc = 0. (3) In the proof of (1) we can take a* € Z, p* e Zv, and c € Z. The supremum and infimum for finite (8.33) are attained since the functions are integer valued. (4) In the proof of (2) we can take j3* e Z, x* £ Zv, and c e Z. The supremum and infimum for finite (8.34) are attained since the functions are integer valued. 0 The M-separation and L-separation theorems are parallel or conjugate in their statements as well as in their proofs. In contrast, the Fenchel-type duality theorem for integer-valued functions is self-conjugate in that the substitution of / = g* and h = k° into (8.33) results in (8.34) by virtue of g = g" and k = k°°. To emphasize the parallelism we have proved the M-separation theorem and the L-separation theorem independently and derived the Fenchel-type duality theorem therefrom. It is noted, however, that, with the knowledge of M-/L-conjugacy, these three duality theorems are almost equivalent to one another; once one of them is established, the other two can be derived by relatively easy formal calculations. Note 8.22. In Theorem 8.21 (1) the infimum is not necessarily attained by any x G Zv (and similarly for (2)). For example, consider / : Z —» R U {+00} and h : Z -^ R U {-00} defined by

which are M^-convex and M^-concave, respectively. We have domz/ = domz/i = Z+, domR/* = (-oo,0]R, domR/i0 = [0,+oo)R, and inf = sup = 0 in (8.31). However, no x attains the infimum, whereas the supremum is attained by p = 0. • Note 8.23. The assumptions on the effective domains are necessary in Theorem 8.21. For the M-convex and M-concave functions / and h in Note 8.18, we have domz/ n domz/i = 0 and domz/* n domz/i0 = 0. The identity (8.33) fails with infimum = +00 and supremum = — oo. •

Chapter 8. Conjugacy and Duality

224

Figure 8.2. Duality theorems (/: M^-convex function, h: Aft-concave function). 8.2.3

Implications

In spite of the apparent similarity to the corresponding theorems in convex analysis, the discrete duality theorems established above convey deep combinatorial properties of M-convex and L-convex functions. We now demonstrate this by deriving major duality results in optimization on matroids and submodular functions as immediate corollaries of these theorems (see also Fig. 8.2). The connection to the duality in network flow problems is discussed in Chapter 9. Example 8.24. Frank's discrete separation theorem (Theorem 4.17) is a special case of the L-separation theorem (Theorem 8.16). By Proposition 7.4, the submodular and supermodular set functions p and // can be identified, respectively, with an L''-convex function g : Zv —> R U {+00} with donizg C {0,1} V and an lAconcave function k : Zy -> RU {-00} with domz/fc C {0,1}V by p(X) = g(xx) and n(X) = k(xx) for X C V. The L-separation theorem applies to (g,k) since the first assumption, domz^ndonizA; ^ 0, is met by g(0) = fc(0) = 0, which follows from p(0) = ^(0) = 0. We see /?* = 0 from the inequality (8.22) for p = 0, and then the desired inequality (4.27) is obtained from (8.22) with p = xx for X C V. When p and p, are integer valued, g and k are also integer valued, and the integrality assertion in the L-separation theorem implies the integrality assertion in Theorem 4.17. • Example 8.25. Edmonds's intersection theorem (Theorem 4.18) in the integral case is a special case of the Fenchel-type duality theorem (Theorem 8.21 (3)). This is explained in Example 1.20. • Example 8.26. The Fenchel-type duality theorem for submodular set functions is a special case of the Fenchel-type duality theorem for L''-convex functions (Theorem 8.21 (2), (4)). The conjugate functions of a submodular set function p : 2V —> RU{+oc} and a supermodular set function \JL : 2V —» R(j{—00} (i.e., p, —/it G <S[R])

8.2. Duality

225

are defined by

The Fenchel-type duality theorem for submodular set functions is an identity

with an additional integrality assertion that, for integer-valued p and n, the maximum on the right-hand side of (8.35) can be attained by an integer vector x 6 Zv. As in Example 8.24, we consider an L''-convex function g and an L''-concave function k associated with p and p.. We have g* = p°, k° = /it 0 , and domz R such that (i) w = w* + toj-I, (ii) B* is a minimum-weight base of (V, B\) with respect to w"(, and (iii) B* is a minimum-weight base of (V, BZ) with respect to w%In addition, the theorem states that, if w is integer valued, the vectors w^ and w\ can be chosen to be integer valued. The combinatorial content of this theorem lies in the assertion about the existence of an integer weight splitting in the case of integer-valued weight. Applying Theorem 8.17 to a pair of M-convex functions

yields p* satisfying (8.25) and (8.26) with additional integrality in the case of integervalued w. A weight splitting constructed by

has the properties (ii) and (iii) because of (8.25) and (8.26). In Example 1.21 we derived the weight-splitting theorem (integer-weight case) from the M-separation theorem (Theorem 8.15). • Example 8.28. Suppose we are given two valuated matroids (V, a>i) and (V,^) as well as a weight vector w : V —> R. The valuated matroid intersection problem is to find B C V that maximizes w(B) + u\(B) + w 2 (B). The weight-splitting theorem for valuated matroid intersection says that a common base B* maximizes

Chapter 8. Conjugacy and Duality

226

w(B) + LOi(B) + W2(S) if and only if there exist real vectors w*,w^ '• V —> R such that (i) w = wl + w^, (ii) B* maximizes u>i[w;*], and (iii) B* maximizes u^fu^L where wiftu*] and u^^] are defined by (2.76). In addition, the theorem states that, if wi, uj2, and w are all integer valued, the vectors w* and w^ can be chosen to be integer valued. Let /i and /2 be the M-convex functions associated, respectively, with LUi and o>2 by (2.77). Maximizing w(B) + uJi(B) + ^(B) is equivalent to minimizing f \ ( x ) + / 2 [ — w ] ( x ) , and a desired weight splitting can be obtained from the M-convex intersection theorem (Theorem 8.17) as in Example 8.27. • It is emphasized again that the discrete duality theorems are of combinatorial nature and cannot be obtained through mere combination of the convex-extensibility theorem (Theorems 6.42 and 7.20) with the separation theorem (Theorem 3.5) or the Fenchel duality theorem (Theorem 3.6) for (ordinary) convex functions. Examples 1.5 and 1.6 should be convincing enough to demonstrate this point.

8.3

M2-Convex Functions and L2-Convex Functions

Two additional classes of discrete functions, called M2-convex functions and "L^convex functions, are considered here. An M2-convex function is a function representable as the sum of two M-convex functions, and an L^-convex function is the integer infimal convolution of two L-convex functions. These functions play crucial roles in combinatorial optimization. In Edmonds's intersection theorem (Theorem 4.18), for example, the left-hand side of the min-max relation (4.29) corresponds to M2-convexity and the right-hand side to L2-convexity. 8.3.1

M2-Convex Functions

A function / : Zv —> R U {+00} with dom/ ^ 0 is said to be M^-convex if it ca be represented as the sum of two M-convex functions, i.e., if / = /i + /2 for some /i 1/2 G .M[Z —> R]. We denote by M-^jL —> R] the set of M2-convex function and by .A/^Z —> Z] the subclass of M2-convex functions / = /i + /2 with some /i> /2 G M\L —> Z]. An J\4-convex function is denned similarly as the sum of tw M^-convex functions, which is obtained as the projection of an M2-convex function. The notations M$L —*• R] and M\[Z —* Z] are defined in an obvious way. We ha

Note that a set is M2-convex (resp., M2-convex) if and only if its indicator function is M2-convex (resp., M^-convex). The effective domain and the set of minimizers of an M2-convex function are M2-convex sets; the latter is a consequence of the M-convex intersection theorem (Theorem 8.17).

8.3.

M2-Convex Functions and L2-Convex Functions

227

Proposition 8.29. (1) For an M%-convex function f, dom/ is M^-convex. (2) For an M%-convex function f, dom/ is M^-convex. Proof. This follows from the relation dom (/i + /2) = dom /i n dom /2 and the Mor M^-convexity of dom/, (i = 1, 2) given in Proposition 6.7. D Proposition 8.30. (1) For an M%-convex function f, argmin/ is M^-convex if it is not empty. (2) For an M^ -convex function f, argmin/ is M^-convex if it is not empty. Proof. This follows from argmin(/i + /2) — argmin/i[—p*] n argmin/ 2 [+p*] in (8.29) and the M- or M11 -convexity of argmin fi[—p*] and argmin/ 2 [+£>*] given in Proposition 6.29. D M2-convexity implies integral convexity. Theorem 8.31. An M^-convex function is integrally convex. In particular, an A/2 -convex set is integrally convex. Proof. For / = /i + /2 with /i,/ 2 6 M*[Z -» R] and x G Ry, Theorem 6.44 implies where /, /i^jind /2_are the local convex extensions (3.61) of /, /i, and /^respectively, and /i and /2 are the convex closures (3.56) of /i and /2. Since /i + /2 is convex, so is /. D For the minimality of an M2-convex function we have the following criterion. Theorem 8.32 (M2-optimality criterion). For an M2- convex function f 6 -M2[Z —> R] and x €. dom /, we have

Proof. By Theorem 8.31 the optimality criterion for an integrally convex function (Theorem 3.21) applies. We may impose the condition |V| = \Z\ because x(V) is constant for any x e dom/. D The optimality criterion above is not suitable for polynomial-time verification. If the summands f\ and /2 in / = /i + /2 are known, the minimality can be verified in polynomial time by the following criterion, as will be explained in Note 9.21. We mention that the M-convex intersection theorem (Theorem 8.17) also serves as an optimality criterion for M2-convex functions when the summands are known.

228

Chapter 8. Conjugacy and Duality

Theorem 8.33 (M2-optimality criterion). For M-convex functions fi, f? € M.\L —> R] and a point x £ dom /j n dom /2, we have

if and only if

for any u1,..uk1,u1...,uk e V with u1,..uk}n{v1,...,uk} =0, where uk+1= u1 by convention.

Proof. The proof is given later in Note 9.21. A scaling version of the optimality criterion above leads to a proximity theorem for M2-convex functions. Theorem 8.34 (M2-proximity theorem). Let /i,/2 € M[Z —> R] be M-convex functions, and assume a & Z++ and n = \V\. If xa & dom/i n dom/2 satisfies

for any u1,...,uk,v1,...,uk3V with {u1,..,uk}n{v1,...,uk} =0, where uk+1= u1, then arg min(f1+f2)= 0 and there exists x* e arg min(f1+f2)with

Proof. See Murota-Tamura [162]. Straightforward calculations based on the M-convex intersection theorem yield the following two theorems. Theorem 8.35.

8.3. M2-Convex Functions and L2-Convex Functions

229

Proof. Using the M-convex intersection theorem (Theorem 8.17), we see that

For (2) and (3), note that dnfi(x) e C\[Z R] from Theorem 6.61 (2).

D

where DZ denotes the integer infimal convolution (6.43) and * is the discrete Legendre-Fenchel transformation (8.11)z- A relation conjugate to this also holds for M-convex functions as follows. Theorem 8.36. For integer-valued Afi-convex functions /i,/2 € M^[Z —> Z] with dom/i n dom/2 ^ 0, we have (/i + /2)* = /i*Qz h" and (fi + /2)** = /i + hProof. For p e dom (fi + /2)* there exists q € Zv such that

by the M-convex intersection theorem (Theorem 8.17). This shows that (/i +/ 2 )* > /i* n z /2*, whereas < is obvious from (p,o;) - /i(x) - / 2 (x) < /*(g) + /*(p -
8.3.2

L2-Convex Functions

A function g : Zv —> R U {+00} is said to be L-2-convex if it can be represented as the infimal convolution of two L-convex functions, i.e., if g = R] the set of L2-convex functi and by £2[Z —> Z] the subclass of L2-convex functions 5 = ffidz^ with some Z]. An L^-convex function is defined similarly as the integer infimal convolution of two lAconvex functions, which is obtained as the restriction of an L2-convex function. The notations C\[Z —> R] and jC^[Z —> Z] are defined in n obvious way. We have

Note that a set is L2-convex (resp., L2-convex) if and only if its indicator function is L2-convex (resp., L^-convex).

230

Chapter 8. Conjugacy and Duality

Note 8.37. Here is a technical supplement concerning the definition of an L2convex function. By definition, a function g : Zv —> R U {+00} is L2-convex if it can be represented as

for some R] such that

As an example, consider a pair of L-convex functions in two variables:

The infimal convolution g — gi^z92 is identically zero, with the infimum in (8.40) unattained. An obvious valid choice for (8.41) is 171 = g% = 0 (identically). • Note 8.38. For #1,52 G £[Z —> R], it can be shown that (51 DZfl^Xpo) = —oo (3p 0 ) =*> (Si n z02)(p) = -oo ( V p e domsri+dom^)• The effective domain and the set of minimizers of an L2-convex function are L2-convex sets. Proposition 8.39. (1) For an L-2-convex function g, doing is L^-convex. (2) For an L\-convex function g, domg is L^-convex. Proof. This follows from the relation dom (giOz g2) = domgi +dom<72 and the Ivor L^-convexity of dom <jij (i — 1, 2) given in Proposition 7.8.

Proposition 8.40. (1) For an L%-convex function g, argming is L^-convex if it is not empty. (2) For an I\-convex function g, argming is L\-convex if it is not empty. Proof. By (8.41) this follows from a general fact in Proposition 8.41 below and the L- or L^-convexity of argming, (i = 1,2) given in Proposition 7.16. D Proposition 8.41. If g\,gz '• Zv —> RU {+00} are such that

then we have

8.3.

M2-Convex Functions and l-2-Convex Functions

231

Proof. It suffices to prove

since the converse inclusion is always true, independently of (8.42). Take p* & argmin(<7iD2 9i[-x](pi) for some pi, we would have

a contradiction to the choice of p*. Hence p* € argmin p e S T l N(p)_ (see (3.71))^J3y S = £>i + D2 = TTl+T>2 we have p = p\ + p2 for some pi € D\ and p2 e D2. Put 01 = pi — [pij and 0"2 = Ip2] — P2, where 0 < a,k(v) < I for k = 1,2 and v € V. Denoting the distinct values among {ai(v),a2(v) \ v e V} by ai > a2 > • • • > am (> 0) and defining Ukt = {v e V | afc(u) > QJ} for k = 1,2 and i = 1 , . . . , m, we have

and, hence,

where OQ = 1, a m+ i = 0, and f/10 = C/2o = 0. This implies p € S(~)N(p), since ft = [PiJ + Xt/ii + [^2! - Xu2i belongs to S D JV(p) for z = 0 , 1 , . . . , m, as shown below.

232

Chapter 8. Conjugacy and Duality

[Proof of qi e S] We have [pi\ + xuu £ DI by Theorem 5.10 for pi e Dl. Since — p2 6 — D?, we similarly see [pal ~~ Xu2i € -^2- Hence q^ & DI+ D? = S. [Proof of qi € -/V(p)] We are to show

we have

Ifp(w) e Z, (8.46) shows X£/ii( w ) = Xu2i(v), which implies (8.44). Supposep(v) £ Z. We put and divide into two cases: (i) v G W and (ii) v €E V" \ W. In case (i), let i be such that v € t/ij. Then v e C/2i follows from

Therefore, -1 < Xc/ii(f) - Xu2i(v) < 0, which implies (8.45). In case (ii), let i be such that v G U^i- Then TJ € C/H follows from

Therefore, 0 < xuu(v) — Xu2i(v) < 1, which implies (8.45). For the minimality of an L2-convex function, we have the following criterion. Theorem 8.43 (L2-optimality criterion). For an L^-convex function g € £2[Z —•> R] and p & dom g, we have

Proof. By Theorem 8.42 this is obtained as a special case of the optimality criterion for an integrally convex function (Proposition 3.22). A scaling version of the optimality criterion above leads to a proximity theorem for L2-convex functions. Theorem 8.44 (L2-proximity theorem). Let g : Zv —> RU {+00} be an L2-convex function such that g(p) — g(p + 1) (Vp € Z y ), and assume a £ Z++ and n = \V\. If pa £ domg satisfies

8.3.

Mg-Convex Functions and l-2-Convex Functions

233

then argmin0 ^ 0 and there exists p* € argming with

Proof. Let g be represented as (8.41) with L-convex functions g\ and 02, where gi(p + 1) = 0j(p) (Vp) for i = 1, 2 as a consequence of g(p) = g(p + 1) (Vp). There exist p?,p2* e Zy such that 0(pa) = 0i(p") + ^2(^2) and Pa = Pi + Pa • For anY V C I/ we have

by the definition of infimal convolution, whereas 0i(p?) + 52(^2 ) = 5(pa) 5= ff(Pa + axr) by (8.49). Hence

By the L-proximity theorem (Theorem 7.18) there exist p* & argmingi and p^ & argmin<72 such that

Then p* — p* + p\ satisfies (8.50). Moreover, p* is a minimizer of g because p\ G argmingi and p% € argmin^The following two theorems are the counterparts of Theorems 8.35 and 8.36. Theorem 8.45. (1) .For0i,02 e /^[Z —*• R] withgiOzg2 > -oo and(8A2) andp e dom(0in z 0 2 ), there exist pt e dom0j (i = 1,2) such that p = p\ +P2 and

(2) For 01,02 6 ^[Z —> Z] with giOzg2 > -oo andp e dom(0inz52), i/iere exzsi ^ 6 dom 0j (i = 1,2) SMC/I tftai p = Pi + P2 «wrf

(3) -For 0 € >C^[Z —> Z] and p € dom0, <9z0(p) is an A^ -convex set. For g e £2[Z —> Z] anrfp e dom0, dzg(p) is an M^-convex set. Proof. Recall the relation x € Z] TOi/i 0idz02 > —oo, we /iaw (0iQz02)** = 0iDzfi f 2 J where * means the discrete Legendre-Fenchel transformation (8.11)z-

234

Chapter 8. Conjugacy and Duality

Proof. Applying Theorem 8.36 to /; = gf <E M^[Z -> Z] shows this. Note 8.47. An Li2-convex function g(p) = min{(?i (q) +52(^ — 9) 1 € Z^}, represented as in (8.41), can be evaluated efficiently, since gi(q) + g^P — q) is an L-convex function in q, to which the minimization algorithms in section 10.3 can be applied. •

8.3.3

Relationship

The relationship between M2- and L2-convex functions is discussed here. The first theorem shows the conjugacy relationship between M2- and La-convex functions. Theorem 8.48. The two classes of functions M.^\L —> Z] and H^\L —> Z] are in one-to-one correspondence under the discrete Legendre-Fenchel transformation (8.11)z, and similarly for M\[Z -> Z] and £ 2 [Z -> Z]. Proof. This is due to (8.38) and Theorems 8.36, 8.46, and 8.12.

D

Separable convex functions are characterized as functions possessing both M2convexity and Lj-convexity. Theorem 8.49. For a function f : Zv —> RU {+00}, we have f is A^-convex and I\-convex •<=> / is A^-convex and L^-convex •^=> / is separable convex. Proof. It suffices to show that, if / is both M^-convex and L^-convex, then it is separable convex. We may assume that dom / is bounded. Take any p & Hv. By Propositions 8.30 and 8.40, the set argmin/[-p] is both M^-convex and L^-convex, and therefore it is an integer interval. This means that / is a separable convex function. D

8.4

Lagrange Duality for Optimization

8.4.1 Outline On the basis of the conjugacy and duality theorems we can develop a Lagrange duality theory for a (nonlinear) integer program:

where c : Zv —> Z U {+00} and 0 ^ B C Zv. The canonical "convex" case consists of problems in which

8.4.

Lagrange Duality for Optimization

235

(REG) B is an M-convex set, and (OBJ) c is an M-convex function. We refer to a problem with (REG) and (OBJ) as an M-convex program. We follow Rockafellar's conjugate duality approach [177] to convex/nonconvex programs in nonlinear optimization. The whole scenario of the present section is a straightforward adaptation of it, whereas the technical development leading to a strong duality assertion for "convex" programs relies heavily on fundamental theorems of a combinatorial nature. An adaptation of the Lagrangian function in nonlinear programming affords a duality framework that covers "nonconvex" programs. We follow the notation of [177] to emphasize the parallelism. In the canonical "convex" case, the problem dual to P turns out to be a maximization of an L2-concave function, where the strong duality holds between the pair of primal/dual problems. This is a consequence of the conjugacy between M2and L2-convexity and the Fenchel-type duality theorem for M-/L-convex functions. In the literature of integer programming we can find a number of duality frameworks, such as the subadditive duality. The present approach is distinguished from those in the following ways: 1. It is primarily concerned with nonlinear objective functions. 2. The theory parallels the perturbation-based duality formalism in nonlinear programming. 3. In particular, the dual problem is derived from an embedding of the given problem in a family of perturbed problems with a certain convexity in the direction of perturbation. 4. It identifies M-convex programs as the well-behaved core structure to be compared to convex programs in nonlinear programming.

8.4.2

General Duality Framework

We describe the general framework, in which neither (REG) nor (OBJ) is assumed. First we rewrite the problem P as follows:

with where SB '• Zv —> {0, +00} is the indicator function of B. We say that the problem P is feasible if f ( x ) < +00 for some x G Zv. Next we embed the optimization problem P in a family of perturbed problems. As the perturbation of / we consider F : Zv x Zu —> Z U {+00}, with U being a finite set, such that

236

Chapter 8.

Conjugacy and Duality

Here the second condition (8.55) means that the integer biconjugate of F(x,u) as a function in u for each fixed x coincides with F(x, u) itself. Note 8.50. By Proposition 8.11, the condition (8.55) is satisfied if, for each x, either F(x, •) = +00 or the integer subdifferential of F(x,u) with respect to u is nonempty for each u € domF(:r, •). Recall that an integer-valued M^- or Lj-convex function has a nonempty integer subdifferential (Theorems 8.35 and 8.45). • The resulting family of optimization problems, parametrized by u G Zu, reads as follows:

We define the optimal value function (f>: Zu —> Z U {±00} by

and the Lagrangian function K : Zv x Zu —> Z U {±00} by

For each x 6 Zv, the function K(x,-) : y >-* K(x,y) is the concave discrete Legendre-Fenchel transform of the function —F(x, •) : u H-> —F(x,u). Our assumptions (8.54) and (8.55) on F(x,u) guarantee the following. Proposition 8.51. (1) F(x,u) = sup{K(x,y)-(u,y)\y£Zu} ( 2 ) f ( x ) = sup{K(x,y) yeZu}(xeZv). Proof. (1) Abbreviate F(x,u) and K(x,y) to F(u) and K(y), respectively. We have F'(y) = -K(-y) by (8.58), while F(u) = F"(u} by (8.55). Therefore,

(2) This follows from (1) with u = 0 and (8.54). We define the dual problem to P as follows:

where the objective function g : Zu —> Z U {±00} is defined by

We say that the problem D is feasible if g(y) > — oo for some y 6 Zu. If the problem P is feasible, we have g(y) < +00 for all y € Zu, since g(y) < K(x,y) < f(x] for all x € Zv and y e Zu.

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237

We use the following notations:

We write min(P) instead of inf (P) if the problem P is feasible and the infimum is finite, in which case the infimum is attained (i.e., opt(P) ^ 0), and similarly for max(D). Theorem 8.52 (Weak duality). inf(P) > sup(D). Proof. We have

Hence, sup(D) = supyg(y) < inf(P). Our main interest lies in the strong duality, namely, in the case where the inequality in the weak duality turns to an equality with a finite common value. Theorem 8.53. (1) g(y) = -v*(-y). (2)sup(D) = p"(0). (3) inf(P) =¥>(()). (4) inf(P) = sup(D) <=> ip(0) = dz
(2) By using (1) we have

(3) This is obvious from (8.54) and (8.57). (4) The equivalence is due to (2) and (3). (5), (6) We have the following chain of equivalence: y € — dz f ( u ) — (u,-y} (Vu e Zu) &• inf u ( ¥ >(u) + (u,y)) = y>(0) <^> g(y) = ip(0), where

Chapter 8. Conjugacy and Duality

238

mfu((p(u) + (u, y)} = g(y) is shown in the proof of (1). This implies the claims when combined with the weak duality (Theorem 8.52). Theorem 8.54 (Saddle-point theorem). Both inf(P) and sup(D) are finite and min(P) = max(D) if and only if there exist x £ Zv and y 6 Zu such that K(x,y) is finite and

If this is the case, we have x & opt(P) and y G opt(D). Proof. By Proposition 8.51 (2) we have f(x) = supyK(x,y) for any x € Zv, whereas g(y) = inf x K(x,y) for any y G Zu by the definition (8.60). In view of the weak duality (Theorem 8.52) and the relation

we see that

8.4.3

Lagrangian Function Based on M-Convexity

As the perturbation F, we choose F = Fr : Zv x Zv —> Z U {+00} defined by

where r : Zv —> Z U {+00} is an M-convex function with r(0) = 0. (We take V as the U in the general framework.) The special case with r = 0 is distinguished by the subscript 0. Namely,

We single out the case of r = 0 because the technical development in this special case can be made within the framework of M-/L-convex functions, whereas the general case involves M2-/L2-convex functions. Throughout this section we assume (REG), i.e., that B is an M-convex set. We use the subscript r to denote the quantities derived from Fr; namely,

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Lagrange Duality for Optimization

239

Our choice of the perturbation (8.61) is legitimate, meeting the requirements (8.54) and (8.55), as follows. Proposition 8.55. Assume (REG). (1) Fr(x,0) = f ( x ) (x£Zv). (2) For each x & Zv, FQ(X, ii) is M-convex in u or FQ(X, u) = +00 for all u. (3) For each x & Zv, Fr(x,u) is M^-convex in u or Fr(x,u) = +00 for all u. (4)Fr(x,-)"=Fr(x,-) (x£Zv). (5) Fr(x,u) = sup{Kr(x,y) ~(u,y)\ye Zv} (x,u& Zv}. Assume (REG) and (OBJ). (6) For each u 6 Zv, Fr(x,u) is M^-convex in x or Fr(x,u) = +00 for all x. Proof. (I) This follows from r(0) = 0. (2) We have F0(x, u) = +00 unless x e dom c. For each x, 5B(x+u) = SB-X(U) is the indicator function of B — x (translation of B by x), which is again an M-convex set. Therefore, SB (x + u) is M-convex in u. (3) We have Fr(x,u) = +00 unless x e dome. Besides SB(X + u), r(u) is M-convex by the assumption. Hence, Fr(x, •) is the sum of two M-convex functions for each x € dom c. By definition, such a function is either M2-convex or identically equal to +00. (4) This follows from (3) and Theorem 8.36. (5) This follows from (4) and Proposition 8.51 (1). (6) The proof is similar to (3) by the symmetry between c(x) and r(u). D The Lagrangian function Kr(x,y] has the following properties. It should be clear that SB' is the support function of B and 0-B^zr[y\ means the integer infimal convolution of the indicator function of — B = {x \ —x e B} and r[y\(u) = r(u) + (u,y). Proposition 8.56.

Proof. It suffices to prove (2), since (1) is its special case with r = 0. Assume x e dome. Substituting (8.61) into (8.64) we obtain

The alternative expression is easy to see.

D

Chapter 8. Conjugacy and Duality

240

Theorem 8.57. Assume (REG). (1) For each x & Zv, KQ(X, y) (2) For each x e Zv, Kr(x, y) Assume (REG) and (OBJ). (3) For each y & Zv, K$(x, y) all x. (4) For each y 6 Zv, Kr(x,y) for all x.

is L-concave in y or KQ(X, y) = +00 for all y. is L2-concave in y or Kr(x, y) = +00 for all y. is M-convex in x or KQ(X, y) € {+00, —00} for is M2-convex in x or Kr(x,y) £ {+00, — 00}

Proof. (1), (3) The expression of Ko(x,y) in Proposition 8.56 (!) shows these. (2) In the expression of Kr(x,y) in Proposition 8.56 (2) we have SB-X + f G M2[L —> Z] or = +00 (see Note 6.17). Then the conjugacy in Theorem 8.48 implies this. (4) In the expression of Kr(x,y) in Proposition 8.56 (2) the second term (S-Bazf[y])(—x) is M-convex or e {+00, — 00} since it is the integer infimal convolution of two M-convex functions (Theorem 6.13 (8)). D In the case of M-convex programs the dual objective function gr and the optimal value function ipr are well behaved, as follows. Theorem 8.58. Assume (REG) and (OBJ). (1) go is L-concave or go(y) = —oo for all y. (2) gr is L2-concave, gr(y) = —oo for all y, or gr(y) — +00 for all y. (3) o is M-convex or tf>o(u) e {+00, -00} for all u. (4) (fr is M2-convex or yv(u) G {+00, — 00} for all u. Proof. We prove (1), (3), (4), and, finally, (2). (1) Using Proposition 8.56 (1) we obtain

This shows ^o is L-concave or = — oo, since the sum of two L-concave functions is again L-concave provided the effective domains of the summands are not disjoint. (3) We have (po(u) = inf z (c(x) + SB(X + u)) = (cDz <5_ B )(—u). The assertion follows from Theorem 6.13 (8). (4) It follows from Fr(x,u) = FQ(X,U) + r(u) that <pr(u) = +00 unless u e domr and that (pr(u) = Z], then ipr G M-2\L —> Z] or ipr = +00. If Z], the conjugacy in Theorem 8.48 implies the L2-concavity of gr. If (pr = +00, then gr = +00. If (pr(u) = —oo for some u, then gr = — oo. D Strong duality holds true for M-convex programs with the Lagrangian function Kr(x,y). Theorem 8.59 (Strong duality). Assume (REG), (OBJ), and that the problem P

8.4.

Lagrange Duality for Optimization

241

is feasible and bounded from below. (1) min(P) = !f>r(0) =
from which is derived the following dual problem:

This is the polymatroidal version of the optimal common base problem explained in Example 8.27. The optimal solution y = y* to D gives the weight splitting w* = w — y* and w\ = y*. For a concrete instance, take V = {1,2},

We have B H B' = {(0,0), (1, -1)} and

8.4.4

Symmetry in Duality

So far we have derived the dual problem D from the primal P by means of a perturbation function F(x,u) such that F(x,0) = f ( x ) and F(x, •) 6 Mi\L —> Z]. Namely,

242

Chapter 8.

Conjugacy and Duality

We have seen that g is L2-concave, i.e., —g € £a[Z —> Z], in the "convex" case where (REG) and (OBJ) are satisfied. We are now interested in the reverse process, i.e., how to restore the primal problem P from the dual D in a way consistent with the general duality framework of section 8.4.1. We embed the dual problem D in a family of maximization problems defined in terms of another perturbation function G(y, v) such that G(y, 0) = g(y) and -G(y, •) e £2[Z -> Z]. Namely,

With reference to (8.60) and Proposition 8.51 we define a perturbation function G : Zu x Zv -» Z U {±00} by54

By this we intend to consider a family of maximization problems parametrized by veZv: Maximize G(y,v) subject to y G Zu. The optimal value function 7 : Zv —> Z U {±00} is accordingly defined by

It is then natural to introduce the dual Lagrangian function K(x, y) : Zv x Zu —> Z U {±00} as

The problem dual to the problem D is to minimize

As can be imagined from the corresponding constructions in convex analysis (cf. section 4 of Rockafellar [177]), K(x,y) and f ( x ) thus constructed do not necessarily coincide with the original K(x,y) and f ( x ) . We show, however, that the dual of the dual comes back to the primal in the canonical case with a bounded M-convex set B using the Lagrangian function Kr. Example 8.61. For KQ of (8.66) in Example 8.60 we can calculate

We observe that Ko(x,y) = Ko(x,y) where they take finite values. 54

Here we have v € Zv and not v dV.

8.4.

Lagrange Duality for Optimization

243

In what follows we always assume (REG), (OBJ), and that B is bounded. We consider the Lagrangian function Kr with U = V. Proposition 8.62. Assume (REG), (OBJ), and that B is bounded. Then

Proof. The definitions (8.67) and (8.69) show Gr(y, •) = ~(Kr(-, y))* and Kr(; y) = (-Gr(y,-))' for each y. Since Kr(-,y) 6 M2[Z -> Z] or = +00 by Theorem 8.57 (4) when B is bounded, we have KT(-,y) — (Kr(-,y))" for each y. Hence follows Kr = Kr. Then Proposition 8.51 (2) and (8.70) imply / = /. D Proposition 8.63. Assume (REG), (OBJ), and that B is bounded. (l}Gr(y,0)=gr(y)(y£Zv). (2) For each y 6 Zv, Gr(y,v) is L^-concave in v or Gr(y,v) = +00 for all v. (3) For each v e Zv, Gr(y, v) is L2-concave in y, Gr(y, v) = —oo for all y, or Gr(y, v) = +00 for all y. Proof. (1) This is obvious from (8.65) and (8.67). (2) The definition (8.67) shows Gr(y,-) = -(Kr(-,y))' for each y, while Kr(-,y) € M%[Z —> Z] or = +00 by Theorem 8.57 (4) when B is bounded. Hence -Gr(y, •) € £ 2 [Z -> Z] or = -oo by Theorem 8.48. (3) By (8.67), (8.64), and (8.61), we have the expression

in which c[—v] is M-convex. On the other hand, Theorem 8.58 (2) shows that

is L2-concave, gr(y) = —oo for all y, or gr(y) = +00 for all y. By replacing c with c[—v] we obtain the claim. D The optimal value function jr, defined by (8.68) with reference to Gr, enjoys the following properties. Theorem 8.64. Assume (REG), (OBJ), and that B is bounded. (1) f ( x ) = -7,°(-x). (2) 7,. is L>2-concave or jr(v) = +00 for all v. Proof. (1) Using Proposition 8.51 (2), (8.69), and Proposition 8.62 we obtain

244

Chapter 8. Conjugacy and Duality

(2) Since / 6 M.i\L —> Z] or = +00, the assertion follows from (1) and the conjugacy between £2[Z —> Z] and M.i\L —> Z]. D Theorem 8.65. Assume (REG), (OBJ), and iftai ifte problem P is feasible and B is bounded. (1) min(P) = 7r(0) = 7r°°(0) = max(D r ). (2)opt(P)=d z (-7,)(0)^0. Proof. The proof is essentially the same as that of Theorem 8.59. To be specific, we have the following chain of equivalence: x & dz(—7r)(0) •<=> Tr(^) ~ 7r(0) < (v, -x) (Vv e Zv) &-mfv((v, -z)-7 r (t>)) = 7r(0) <=> f ( x ) = jr(0). This implies the claim when combined with the weak duality (Theorem 8.52). D

Bibliographical Notes The conjugacy relationship between M-convexity and L-convexity was established first for integer-valued functions (Theorem 8.12) by Murota [140], whereas the present proof is based on Murota [147]. The conjugacy theorem for polyhedral M-/L-convex functions (Theorem 8.4) is due to Murota-Shioura [152]. The polarity between M-/L-convex cones in Theorem 8.5 is stated in [147] and Proposition 8.11 for the integer biconjugate is in [140]. Theorem 8.1 is a special case of a theorem .of Topkis [202], stated explicitly as Corollary 2.7.3 in Topkis [203]. The M-separation theorem (Theorem 8.15) is given in Murota [137], [140], [142] and the L-separation theorem (Theorem 8.16) in [140]. The Fenchel-type duality theorem for M-convex functions originated in [137] (see also [140]); the present form (Theorem 8.21) is in Murota [147]. The M-convex intersection theorem (Theorem 8.17) is in [137], [142]. The Fenchel-type duality theorem for submodular set functions described in Example 8.26 is due to Fujishige [62]. The weight-splitting theorem for weighted matroid intersection in Example 8.27 is due to Frank [54], and that for valuated matroid intersection in Example 8.28 is due to Murota [135]; see also Theorem 5.2.40 of Murota [146]. M2-convex and L2-convex functions were introduced by Murota [140], to which Theorems 8.35, 8.36, 8.45, and 8.46 and the conjugacy theorem (Theorem 8.48) ar ascribed. Theorems 8.31 and 8.42 (integral convexity) as well as Theorem 8.49 are due to Murota-Shioura [153]. Theorems 8.32 and 8.43 (M2-/L2-optimality criteria and Theorems 8.34 and 8.44 (M2-/L2-proximity theorems) are given by MurotaTamura [162]. See Tamura [198] for Notes 8.37 and 8.38. The Lagrange duality of section 8.4 is developed in Murota [140]. See NemhauserRinnooy Kan-Todd [166] and Nemhauser-Wolsey [167] for the subadditive duality.

Chapter 9

Network Flows

In Chapter 2 we had a glimpse of the intrinsic relationship between M-/L-convexity and network flows (nonlinear electrical networks). Pursuing this direction further we show the following facts in this chapter, (i) The minimum cost flow problem can be generalized to the submodular flow problem, where M-/L-convexity plays a fundamental role, (ii) The submodular flow problem with an M-convex function admits nice optimality criteria in terms of potentials and negative cycles, (iii) The optimality criterion using potentials is equivalent to the Fenchel-type duality theorem, (iv) A conjugate pair of M-convex and L-convex functions is transformed to another conjugate pair of M-convex and L-convex functions through network flows. Algorithms are treated in Chapter 10.

9.1

Minimum Cost Flow and Fenchel Duality

To single out the role of M-/L-convexity we first review standard results on the conventional minimum cost flow problem. Emphasis is placed on the equivalence of the optimality criterion in terms of potentials and the Fenchel duality theorem for convex functions.

9.1.1

Minimum Cost Flow Problem

Let G = (V, A) be a directed graph with vertex set V and arc set A. Suppose that each arc a e A is associated with an upper capacity c(a), a lower capacity c(a), and a cost 7(0) per unit flow. Furthermore, for each vertex v & V, the amount of flow supply at v is specified by x(v). The minimum cost flow problem is to find a flow £ = (£(a) a € A) that minimizes the total cost (7, £)A = SaeA 7(a)£(a) subject to the capacity constraint and the supply specification. Here the supply specification means a constraint that the boundary d£ of £, defined by

245

246

Chapter 9.

Network Flows

should be equal to the given x. The problem is described by a graph G = (V, A), an upper capacity c : A —> R U {+00}, a lower capacity c : A —> R U {—oo}, a cost vector 7 : A —> R, and a supply vector x : V —> R, where it is assumed that c(a) > c(a) for each a G A The variable to be optimized is the flow £ : A —> R.

Minimum cost flow problem MCFP0 (linear arc cost)55

The minimum cost flow problem is a typical well-behaved combinatorial problem that has nice properties, such as 1. an optimality criterion in terms of potentials (dual variables), 2. an optimality criterion in terms of negative cycles, 3. the integrality of optimal solutions, and 4. efficient algorithms. Precise statements for the first three above are given later in Theorems 9.4, 9.5, and 9.6, respectively. In particular, the integrality of optimal solutions refers to the fact that, if the capacity constraint and the supply specification are given in terms of integer-valued functions, c : A —> Z U {+00}, c : A —> Z U {—oo}, and x : V —* Z, then there exists an integer-valued optimal flow £ to the above problem. This implies that the problem MCFPo specified by such integer-valued data remains essentially the same even if the integrality condition

is additionally imposed on the flow £. We refer to the problem with (9.6) in place of (9.5) as the minimum cost integer-flow problem. To discuss the relationship to convex analysis it is convenient to consider a more general form of the minimum cost flow problem. The generalization is twofold. First, the linear arc cost /^ae.4 7( a )£( a ) is replaced with a nonlinear cost represented by a separable convex function ^2aeA / 0 (£(a)) with a family of univariate polyhedral convex functions fa S C[R —> R] indexed by a G A. Second, with a polyhedral convex function / : Rv —> R U {+00}, an additional term /(d£) for the flow boundary <9£ is introduced in the cost function as a generalization of the supply specification <9£ = x.

Minimum cost flow problem MCFPs (nonlinear cost)56

55

MCFP stands for minimum cost flow problem. We have MCFP; for i = 0, 3 and not for i = 1, 2. This is for consistency with section 9.2.

56

9.1.

Minimum Cost Flow and Fenchel Duality

247

Obviously, MCFPo is a special case of MCFPs, where

for a € A and / is the indicator function 6{xj of the singleton set {x}. Among the four nice properties of MCFP0 listed above, the optimality criterion by potentials is generalized to MCFPa, as we will see in section 9.1.3, whereas the other three fail to survive for a general /. In considering the integer-flow version of the problem it is natural to assume /„ e C[Z —> R] (or /„ e C[Z|R —> R]) for each a e A, but it is not clear what combinatorial property to impose on / to ensure the integrality of optimal solutions. M-convexity gives an answer to this, as we will see in section 9.4. Note 9.1. In MCFPs we have restricted / and fa (a € A] to be polyhedral convex functions. This is for consistency with our theoretical framework of polyhedral M-/L-convex functions. The optimality criterion by potentials (Theorem 9.4), as well as its equivalence to the Fenchel duality to be discussed in section 9.1.4, remains valid for nonpolyhedral convex functions under appropriate assumptions; see Iri [94] and Rockafellar [178]. •

9.1.2

Feasibility

For the minimum cost flow problem MCFPo, a feasible flow means a function £ : A —» R that satisfies

We say that MCFPo is feasible if it admits a feasible flow. For X C V we denote the sets of arcs leaving and entering X by

and define the cut capacity function K : 2V —> R U {+00} by

Proposition 9.2. The cut capacity function K is submodular. Proof. It is easy to verify

248

Chapter 9. Network Flows

where the summation is taken over all arcs a connecting X \ Y and Y\X.

D

If a flow £ meets the capacity constraint (9.12), its boundary x = <9£ satisfies

for all X C V and also x(V) = 0 = «(V). This means x e B(re), where B(K) is the base polyhedron (4.13) associated with n. The above argument shows that the condition x € B(K) is necessary for MCFPo to be feasible. It is also sufficient, as stated in the following theorem. Theorem 9.3 (Feasibility). For c : A ->• R U {+00}, c : A —> R U {-oo}, and x : V —> R, there exists a flow £ : A —> R satisfying (9.12) and (9.13) if and only if

That is, If c and c are integer valued, we may restrict £ to be integer flows; namely,

Proof. This follows from the max-flow min-cut theorem or a variant thereof, called Hoffman's circulation theorem (see, e.g., (2.65) of Fujishige [65] or Theorem 3.18 of Cook-Cunningham-Pulleyblank-Schrijver [26]). D

9.1.3

Optimality Criteria

The minimum cost flow problem MCFPs, which has convex boundary cost and separable convex arc cost, admits a nice optimality criterion in terms of potentials. The conventional case MCFP0 admits, in addition, an optimality criterion in terms of negative cycles and the integrality of optimal solutions. A potential means a function p : V —> R (or a vector p e Rv) on the vertex set. The coboundary of a potential p is a function 5p : A —> R defined by

The inner product (pairing) of tension 77 : A —> R and flow £ : A —+ R can be expressed as if x = 9£ and p is a potential such that

9.1. Minimum Cost Flow and Fenchel Duality

249

The identity (9.21) is a fundamental relation, frequently used in the subsequent arguments. It should be clear that

With reference to a potential p we modify the cost functions / and fa (a e A) to the reduced cost functions f[-p] and fa[op(a)} (a € A) defined by

A straightforward calculation with the use of (9.21) yields

where inf /[—p] and mi fa[6p(a)] with a € A mean the infima of the reduced cost functions. The inequality (9.25) gives a lower bound for the minimum of F3. In particular, if

for some p, then £ is an optimal flow satisfying (9.25) with equality. This statement is true for any functions / and fa (a & A). The converse is also true under a fairly general assumption that / and fa (a €. A) are convex. Theorem 9.4 (Potential criterion). In the minimum cost flow problem MCFPs with polyhedral convex f and fa (a & A), we have the following: (1) For a feasible flow £ : A —> R, the two conditions (OPT) and (POT) below are equivalent. (OPT) £ is an optimal flow. (POT) There exists a potential p : V —» R such that (i) £(o) € &Tgmmfa[Sp(a)] for every a & A, and (ii) d£ € argmin/[-p]. (2) Suppose that a potential p : V —> R satisfies (i) and (ii) above for an optimal flow £. A feasible flow £' is optimal if and only if

250

Chapter 9.

(i) £'(a) € argmin fa[5p(a)] (ii) d£' e argmin f[—p].

Network Flows

for every a G A, and

Proof. (1) (POT) =$> (OPT) is already shown. To prove (OPT) => (POT), suppose that £ is an optimal flow. Putting

we see

where inf FS is finite and x = <9£ attains the infimum of the last expression. Noting that /A is a polyhedral convex function (see Note 2.17) and dom/^ n dom/ ^ 0, we apply the Fenchel duality theorem in convex analysis (Theorem 3.6 and (3.42)) to obtain p : V —> R such that

The second equation shows (ii) in (POT). We will show that the first equation above implies (i) in (POT). It follows from (9.26) and (9.21) that

for any x' e Ry, and therefore,

On the other hand, the optimality of £ implies /yi(d£) = ^2a€A /a(£( a ))> which, in combination with (9.21), yields

Substituting (9.28) and (9.29) into the first equation in (9.27) shows

9.1.

Minimum Cost Flow and Fenchel Duality

251

Figure 9.1. Characteristic curve (kilter diagram) for linear cost.

which is equivalent to (i) in (POT). (2) This is obvious from (1) and (9.25).

D

A potential p satisfying (i) and (ii) in (POT) is called an optimal potential. Though this definition refers to a particular optimal flow £, it is, in fact, independent of the choice of £ by Theorem 9.4 (2). Condition (i) in (POT) is closely related to the characteristic curve (or kilter diagram) Fa introduced in section 2.2 with an illustration in Fig. 2.3. Since by (2.34) and (2.35), condition (i) in (POT) says that flow £(a) and tension r?(a) = —6p(a) should satisfy the constitutive equation in every arc a G A. In the case of linear arc cost, the characteristic curve Fa takes the form of Fig. 9.1, and, accordingly, condition (i) in (POT) is expressed as

in terms of the reduced cost jp : A —> R defined by In the conventional case MCFPo with linear arc cost, the optimality criterion can be reformulated in terms of negative cycles in an auxiliary network. For a feasible flow £ : A —> R, let G^ = (V, A^) be a directed graph with vertex set V and arc set A^ = A^ U jB| consisting of two disjoint parts:

252

Chapter 9. Network Flows

(a: reorientation of a), and define a function £j : A$ —> R, representing arc lengths, by

We refer to (G^l£) as the auxiliary network. We call a directed cycle of negative length a negative cycle. Theorem 9.5 (Negative-cycle criterion). For a feasible flow £, : A —> R to the minimum cost flow problem MCFPo, conditions (OPT) and (NNC) below are equivalent. (OPT) £ is an optimal flow. (NNC) There exists no negative cycle in (G^,l^) with t% o/(9.34). Proof. By (9.31), (9.32), and the definition (9.34) of £?, condition (i) of (POT) in Theorem 9.4 is equivalent to whereas condition (ii) of (POT) is void for MCFPo. On the other hand, the existence of a potential p : V —*• R satisfying (9.35) is equivalent to (NNC), as is well known in network flow theory. Hence follows the equivalence of (NNC) and (OPT) by Theorem 9.4. D The minimum cost flow problem MCFP0 is endowed with remarkable integrality properties: 1. An integer-valued optimal flow exists if the upper and lower capacities and the supply vector are integer valued (primal integrality). 2. An integer-valued optimal potential exists if the cost vector is integer valued (dual integrality). Theorem 9.6 (Integrality). Suppose that the minimum cost flow problem MCFPo has an optimal solution. (1) [Primal integrality] // c : A —> Z U {+00}, c : A —> Z U {-oo}, and x : V —> Z, then there exists an integer-valued optimal flow £ : A —> Z. (2) [Dual integrality] The set of optimal potentials II* = {p p : optimal potential} is an L-convex polyhedron. If^:A^Z, then II* is an integral L-convex polyhedron and there exists an integer-valued optimal potential p : V —> Z. Proof. (I) Let p be an optimal potential. By (9.31) and (9.32), a flow £ is optimal if and only if it is a feasible flow with respect to a more restrictive capacity constraint c*(a) <£,(a)
9.1.

Minimum Cost Flow and Fenchel Duality

253

for each a & A. Since c*(a) and c*(a) are integers for every a € A, the claim follows from (9.19) in Theorem 9.3. (2) Since condition (i) of (POT) in Theorem 9.4 is equivalent to (9.35) in the proof of Theorem 9.5, II* coincides with the polyhedron described by (9.35) with an optimal £. This implies the L-convexity of II* (see section 5.6). The integrality assertion follows from Proposition 5.1 (4). D The nice features of the minimum cost flow problem discussed so far (Theorems 9.4, 9.5, and 9.6) are derived mainly from the combinatorial structure inherent in the underlying graph, as well as the convexity of the cost functions. Further combinatorial properties stemming from the M-convexity of the cost functions will be investigated in section 9.4 and section 9.5. Note 9.7. Here is a comment on the definition of the coboundary. In this book we follow the convention of defining Sp(a) by 6p(a) = (p at the initial vertex of a) — (p at the terminal vertex of a). The boundary d£(v) is defined to be the amount of flow leaving v and the tension 77 is defined as 77 = -8p (see (9.1), (9.20), and (9.22)). Then follows the fundamental identity Another convention of defining 6p(a) by Sp(a) = (p at the terminal vertex of a) — (p at the initial vertex of a) and the tension 77 by r\ = 5p results in

The notations div and A in Rockafellar [178] are related to ours as div = d and

A = -5.

9.1.4

m

Relationship to Fenchel Duality

We discuss here the relationship between the potential criterion for optimality for the minimum cost flow problem MCFPa and the Fenchel duality in convex analysis. The potential criterion for MCFPs (Theorem 9.4 in section 9.1.3) has been derived from the Fenchel duality applied to / and -/A, where

and the evaluation of JA amounts to solving a minimum cost flow problem with nonlinear arc cost /„ but without boundary cost /. Thus, the minimum cost flow problem MGFPs with boundary cost can be understood as a composition of the

254

Chapter 9.

Network Flows

Figure 9.2. Minimum cost flow problem for Fenchel duality.

minimization/maximization problem of the Fenchel duality and the minimum cost flow problem without boundary cost. The proof of Theorem 9.4 yields, as a byproduct, a min-max identity for MCFP3:

where

with g = /* and ga = fa* for a € A. The identity (9.36) is an immediate consequence of the Fenchel duality (3.41):

in which f ' ( p ) = g(p) and

by (9.28). The left-hand side of (9.36) is MCFPs in disguise, and accordingly, we may think of the maximization problem on the right-hand side of (9.36) as an optimization problem dual to MCFPa. Although the potential criterion for MCFPa has been derived from the Fenchel duality, they are essentially equivalent, which we demonstrate here. To be specific, we derive the Fenchel duality theorem (Theorem 3.6, Case (a2)) from the optimality criterion for MCFP3 (Theorem 9.4). Given a polyhedral convex function f\ : Ry —> R U {+00} and a polyhedral concave function h^ : Rv —> R U {—00} with dom/i n dom/i2 ^ 0, we consider a minimum cost flow problem MCFPa on the bipartite graph G = (V\ U V^A) in Fig. 9.2. The vertex set of G consists of two copies of V, i.e., V\ and Vi, and the

9.2. M-Convex Submodular Flow Problem

255

arc set is A = {(^1,^2) | v € V}, with v\ € V\ and v-z € V-z denoting the copies of v € V. We define the boundary cost function / : R/1 x R72 -» R U {+00} by

and assume that the arc cost functions fa (a e A) are identically zero without capacity constraints. Note that x\ = —x2 if (xi,x2) = d£ for a flow £ in this network. Assuming inf (/i — /la) > — oo, let £ be an optimal flow, which exists since / is a polyhedral convex function. Let (piiPi} 6 R^1 x R^2 be an optimal potential satisfying (POT) in Theorem 9.4. Condition (i) of (POT) implies pi = p2- Since

condition (ii) of (POT) gives

for x = d£|vi and p = p\. This implies the Fenchel duality (3.41) for f\ and h^\ see also (3.30) and (3.42).

9.2

M-Convex Submodular Flow Problem

A series of generalizations of the minimum cost flow problem to the M-convex submodular flow problem is described. Recall the conventional minimum cost flow problem MCFPo introduced in section 9.1.1. It is described by a graph G — (V, A), an upper capacity ~c : A —» R U {+00}, a lower capacity c : A —> R U {—oo}, a cost vector 7 : A —> R, and a supply vector x : V —> R, where c(o) > c(a) for each a e A. A generalization of MCFPo is obtained by relaxing the supply specification (9£ = x to the constraint that 9£ belong to a given set B of feasible or admissible supplies: The nice properties described in section 9.1 are maintained if -B is a base polyhedron represented as B = B(p) with a submodular set function p : 2V —> RU{+oo}. Such a problem described by some p G <S[R] is called the submodular flow problem. Submodular flow problem MSFPi (linear arc cost)57

57 MSFP stands for M-convex submodular flow problem. We use the notation MSFPj with i = 1, 2, 3 to indicate the hierarchy of generality in the problems.

256

Chapter 9. Network Flows

In the integer-flow version of the problem, with £(o) e Z (a e A) instead of (9.41), we assume p & S[Z]. A further generalization of the problem is obtained by introducing a cost function for the flow boundary d£ rather than merely imposing the constraint <9£ e B. Namely, with a function / : Hv —> R U {+00} we add a new term f(d£) to the objective function, thereby imposing the constraint d£ € B — dom / implicitly. The aforementioned nice properties are maintained if / is a polyhedral M-convex function. Such a problem described by some / 6 ./Vf [R —> R] is called the M-convex submodular flow problem. M-convex submodular flow problem MSFP2 (linear arc cost)

Note that the M-convex submodular flow problem with a {0, +00}-valued / reduces to the submodular flow problem MSFPi. In the integer-flow version of the problem we assume c : A -> Z U {+00}, c : A -> Z U {-oo}, and / 6 M[Z -> R] (or

/eM[Z|R->R]).

A still further generalization is possible by replacing the linear arc cost in T% with a separable convex function. Namely, using univariate polyhedral convex functions fa e C[R -» R] (a € A), we consider Y^aeA /a(£(°0) instead of ^2aeA 7(a)£(a) to obtain MSFPa below, a special case of MCFPs with / being M-convex. M-convex submodular flow problem MSFPs (nonlinear arc cost)

In the integer-flow version of the problem we assume / e M[Z —> R] and fa e C[Z -> R] for a e A (or / e M[Z\R -^ R] and /„ e C[Z|R -^ R] for a e A). Obviously, MSFP2 is a special case of MSFPs with

The converse is also true; i.e., MSF?3 can be put into a problem of the form of MSFP2, as is explained in Note 9.8. Throughout this chapter we assume

9.2. M-Convex Submodular Flow Problem

257

since d£(V) = 0 for any flow £ and d£ & dom / = B — B(/j) is imposed. In subsequent sections we will see that the optimality criteria in terms of potentials and negative cycles, as well as efficient algorithms for the conventional minimum cost flow problem MCFPo, can be generalized for the M-convex submodular flow problem. Note 9.8. The problem MSFP3 on G = (V,A) can be written in the form of MSFP2 on a larger graph G = (V, A). We replace each arc a = (u, v) G A with a pair of arcs, a+ = (u, i>~) and a~ = (v£,v), where v+ and v~ are newly introduced vertices. Accordingly, we have A = {a + ,a~ | a G A} and V = V(J{v^,v~ \ a G A}. For each a G A we consider a function /0 : R2 —> R U {+00} given by

and define / : R^ -> R U {+00} by

where x\v denotes the restriction of x to V. For a flow £ : A —> R, we have |(o+) = f(oT) if (d£(v+),d|(u-)) G dom/0. The problem MSFP3 is thus reduced to MSFP2 with the objective function f 2(|) = /(<9£). Note that, if / e .M[R -> R] and /0 G C[R ->• R] for a e .A, then / G Ai[R -> R]. Note 9.9. The cost function FS of MSFPs consists of two terms, the separable arc cost X^aeA /o(£(a)) and the M-convex boundary cost /(<9£). Noting that the former is M^-convex, one might be tempted to consider a (nonseparable) M^-convex cost function defined on the arc set. The integer-flow version of such a problem, however, contains the Hamiltonian path problem, a well-known NP-complete problem, as a special case. Suppose that we want to check for the existence of an (s, i)-Hamiltonian path in a directed graph G = (V, A), where we may assume s ^ t G V and 5~s = 6+t = 0. We construct another directed graph G = (V, A) by replacing each arc a = (u, v) G A with three arcs connected in series:

where v£ and va are newly introduced vertices. Hence, V — V U {v£,va \ a € ^4} and A = A+ U A° U A~, with A+ = {a+ \ a e A}, A° = {a° \ a G A}, and A~ = {a~ \ a £ A}. We consider three matroids, say, M+, M"", and M° on A+, A~, and A°, respectively. M+ is a partition matroid in which B+ C A+ is a base if and only if

M is another partition matroid defined similarly (with + replaced with —), and M° is the graphic matroid in which B° C. A° is a base if and only if {a G A \ a° G B°} is

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Chapter 9. Network Flows

a tree of the original graph G. Let Q be the set of characteristic vectors of a subset B of A such that B H A+ is a base of M+, B n A~ is a base of M~, and B r\ A° is an independent set of M°. Then a {0, l}-flow £ in G with £ e Q and <9£ = Xs — corresponds to an (s, i)-Hamiltonian path in G. Since Q is an M^-convex set, the constraint | e Q can be represented by a {0, +oo}-valued M^-convex cost fun on the arc set A. •

9.3

Feasibility of Submodular Flow Problem

The feasibility of the submodular flow problem MSFPi is investigated here. Recall that we are given a graph G = (V, A), an upper capacity c : A —> Ru{+oo}, a lowe capacity c : A —> R U {—oo}, and a submodular set function p : 2V —» R U {+00 where c(o) > c(a) for a £ A and p(0) = p(V) = 0. A feasible flow means a function £ : A —> R that satisfies

The problem MSFPi is said to be feasible if it admits a feasible flow. In section 9.1.2 we considered (9.52) to obtain Theorem 9.3. We now combine (9.52) and (9.53) for the feasibility of MSFPi. Theorem 9.10 (Feasibility). A submodular flow problem MSFPi is feasible if and only if Moreover, if c, c, and p are integer valued and the problem is feasible, there exists an integer-valued feasible flow £ : A —> Z. Proof. Let K be the cut capacity function denned by (9.16). By Theorem 9.3 a feasible flow exists if and only if B(/c) n B(p) 7^ 0. The latter condition is equivalent to by Edmonds's intersection theorem (Theorem 4.18) and further to (9.54) by

In a feasible problem with integer-valued c, c, and p, both B(K) and B(p) are integral base polyhedra (integral M-convex polyhedra), and B(K) n B(p) n Zv is nonempty by (4.32). Then (9.19) in Theorem 9.3 guarantees the existence of an integer flow £ : A -+ Z with d£ G B(K) n B(p) n Zv. D Note 9.11. The necessity of (9.54) is easy to see. For any X C V, the net amount of flow entering X is equal to zero:

9.3.

Feasibility of Submodular Flow Problem

259

and the constraints (9.52) and (9.53) should be satisfied:

Combining these two yields (9.54). Theorem 9.10 claims that this "obvious" necessary condition is in fact sufficient. • Note 9.12. In a feasible submodular flow problem, the set of boundaries of feasible flows, dE = {d£ £ : feasible flow}, is an M2-convex polyhedron, and it is an integral M2-convex polyhedron if c, c, and p are integer valued. This can be seen from the proof of Theorem 9.10. • The maximum submodular flow problem is to find a feasible flow £ that maximizes £(ao) for a specified arc ao € A. Maximum submodular flow problem maxSFP

A max-flow min-cut theorem holds for this problem. Note that for any X C V with OQ G /\+X we have an "obvious" inequality:

by (9.55) and (9.56). Theorem 9.13 (Max-flow min-cut theorem). For a feasible maximum submodular flow problem maxSFP,

where this common value can be +00. If c, c, and p are integer valued and (9.61) is finite, there exists an integer-valued maximum flow £ : A —> Z. Proof. Divide the arc ao = (u, v) into two arcs in series, say, ao = (u, w) and a0 = (w,v), and denote by G = (V,A) the resulting graph, where V = V U {w} and A = A U {o0}. Define the capacities of a0 by C(OQ) = t and c(a0) = +00 with a parameter i, and let p be defined for all subsets of V by p(X U {w}) = p(X) for X C V. The maximum in (9.61) is equal to the maximum (or supremum) of t such

260

Chapter 9. Network Flows

that the submodular flow problem on G = (V, A) is feasible. With this relationship, Theorem 9.10 implies (9.61) as well as the integrality assertion. D If £ and X attain the maximum and the minimum in (9.61), respectively, and if £(ao) < c(a0), then we have

9.4

Optimality Criterion by Potentials

In section 9.1.3 we saw a potential criterion for optimality (Theorem 9.4) for the minimum cost flow problem MCFPa- Since the M-convex submodular flow problem MSFPa is a special case of MCFPa, the following optimality criterion for MSFPs is immediate from Theorem 9.4. Theorem 9.14 (Potential criterion). In the M-convex submodular flow problem MSFP3 with fa e C[R -> R] (a € A) and f g M[H —> R], we have the following. (1) For a feasible flow £ : A —> R, the two conditions (OPT) and (POT) below are equivalent. (OPT) £ is an optimal flow. (POT) There exists a potential p : V —> R such that (i) £(a) e argmin/ a [<5p(a)] for every a e A, and (ii) <9£e arg min/[-p]. (2) Suppose that a potential p : V —* R satisfies (i) and (ii) above for an optimal flow £. A feasible flow £' is optimal if and only if (i) £'(a) € arg min fa [Sp(a)] for every a 6 A, and (ii) d£' £ arg min f[—p]. A comment is in order on the role of the M-convexity of /. Since fa is a univariate convex function for every a G A, condition (i) in (POT) can be expressed in terms of directional derivatives as:

If / is M-convex, condition (ii) in (POT) can also be expressed in terms of directional derivatives as:

by the M-optimality criterion in Theorem 6.52 (1). These expressions show how the conditions in (POT) can be verified efficiently for a given p. It is also mentioned that these expressions lead to another optimality criterion in terms of negative cycles, to be established in section 9.5, and furthermore to the cycle-canceling algorithm for the M-convex submodular flow problem, to be explained in section 10.4.3.

9.4.

Optimality Criterion by Potentials

261

An alternative representation of condition (ii) in (POT) is obtained from the M-convexity of /. The function conjugate to /, say, g, is a polyhedral L-convex function with g(p+ 1) = g(p) for all p (by Theorem 8.4 and (9.51)). It follows from (3.30) and the L-optimality criterion (Theorem 7.33 (1)) that

This shows that argmin/[—p] coincides with the base polyhedron B(gp) associated with the set function gp denned by

which is submodular by Theorem 7.43 (1). Hence,

This expression is used in the primal-dual algorithm for the M-convex submodular flow problem, to be explained in section 10.4.4. We go on to discuss integrality properties of the M-convex submodular flow problem. This generalizes the well-known facts (Theorem 9.6) for the minimum cost flow problem MCFP0. Recall the notation M[Z\R -> R] and M[R -»• R|Z] for the sets of integral and dual-integral polyhedral M-convex functions, respectively. Theorem 9.15. Suppose that an optimal solution exists in the M-convex submodular flow problem MSFP3 with fa <E C[R -> R] (a e A) and f e X[R -> R]. (1) The set of the boundaries of optimal flows,

is an M2-convex polyhedron, and the set of optimal potentials,

is an L-convex polyhedron. (2) [Primal integrality] If fa £ C[Z|R -> R] (a e A) and f e X[Z|R ->• R], then 95* is an integral M^-convex polyhedron, and there exists an integer-valued optimal flow £ : A —> Z. (3) [Dual integrality] ///„ e C[R -> R|Z] (a G A) and f 6 M[R ->• R Z], then II* is an integral L-convex polyhedron, and there exists an integer-valued optimal potential p : V —> Z. Proof. (1) Let p be an optimal potential. Since argmin/ a [5p(a)] forms an interval, say, [c*(a),c*(a)]R, condition (i) in (POT) of Theorem 9.14 can be expressed as c*(a) < £(a) < c*(o) (a 6 A). Just as in (9.16) and (9.18), the set of <9£ for such £ coincides with the base polyhedron B(K*) for K* defined by

262

Chapter 9. Network Flows

Combining this with (9.65) we obtain 9H* = B(K*) nB(<7 p ), which is an M2-convex polyhedron. Now let £ be an optimal flow. Potentials p satisfying (i) in (POT) form an L-convex polyhedron, say, DI, by (9.63), whereas those satisfying (ii) in (POT) form another L-convex polyhedron D% by (9.64). Therefore, II* = DI n D% is an L-convex polyhedron. (2) Both B(K*) and B(0P) are integral M-convex polyhedra. The integrality of dE* = B(/c*) n B(gp) follows from (4.32). (3) Both DI and D% are integral L-convex polyhedra by Theorem 6.61 (1). The integrality of II* = DI n D2 follows from Theorem 5.7. D For linear arc cost, with fa given by (9.50), the integrality conditions are simplified as follows:

Finally, we state the optimality criterion for the integer-flow version of the M-convex submodular flow problem MSFPa. This is a corollary of Theorems 9.14 and 9.15. Theorem 9.16 (Potential criterion). Consider the M-convex submodular integerflow problem MSFP3 with fa e C[Z -» R] (a e A) and f e M[Z -> R]. (1) For a feasible integer flow £ : A —> Z, i/ie two conditions (OPT) and (POT) below are equivalent. (OPT) £ is an optimal integer flow. (POT) There exists a potential p : V —> R such that (i) £(a) e argmin/a[(5j3(a)] /or eaerj/ a £ A, and (ii) d£ e argmin/[-p]. (2) Suppose that a potential p : V —> R satisfies (i) and (ii) above for an optimal integer flow £. A feasible integer flow £' is optimal if and only if

(i) £'(a) e argmin/ a [£p(a)] for every a e A, and (ii) <9£' e argmin/[-pj.

(3) T/ie set of the boundaries of optimal integer flows, d'E* = {<9£ £ : optimal integer flow}, is an M-2-convex set. (4) If the cost functions are integer valued, i.e., if fa e C[Z —> Z] (a e A) and f £ M[Z —> Z], then there exists an integer-valued potential p :V—>Zin (POT). Moreover, the set of integer-valued optimal potentials, II* = {p p : integer-valued optimal potential}, is an L-convex set. In connection to (i) and (ii) in (POT) in Theorem 9.16, note the equivalences

9.5. Optimality Criterion by Negative Cycles

263

These are the discrete counterparts of (9.63) and (9.64). Note 9.17. The Fenchel-type duality theorem for M-convex functions (Theorem 8.21) is essentially equivalent to the optimality criterion for the M-convex submodular integer-flow problem (Theorem 9.16). See section 9.1.4 and note that, for an M-convex function /i and an M-concave function /i 2 > f ( x i , X 2 ) = /i(a^i) — /i2(—#2) is an M-convex function.

9.5

Optimality Criterion by Negative Cycles

The optimality of an M-convex submodular flow can also be characterized by the nonexistence of negative cycles in an auxiliary network. This fact leads to the cycle-canceling algorithm to be described in section 10.4.3.

9.5.1

Negative-Cycle Criterion

We consider the M-convex submodular flow problem MSFP2 with M-convex boundary cost and linear arc cost. This is not restrictive, since MSFPa, having nonlinear convex arc cost, can be put in the form of MSFP2, as explained in Note 9.8. We consider real-valued flows and then integer-valued flows. We assume / e .M[R —> R] in considering real-valued flows. For a feasible flow £ : A —> R, we define an auxiliary network as follows. Let Gj = (V,A^) be a directed graph with vertex set V and arc set A^ = A*^ U _B| U Cj consisting of three disjoint parts:

We define a function i% : A% —> R, representing arc lengths, by

We refer to (Gj,^) as the auxiliary network. We call a directed cycle of negative length a negative cycle. The following theorem gives an optimality criterion in terms of negative cycles. Theorem 9.18 (Negative-cycle criterion). For a feasible flow £ : A —> R to the M-convex submodular flow problem MSFP2 with f e M[R —> R], the conditions (OPT) and (NNC) below are equivalent.

264

Chapter 9. Network Flows

(OPT) £ is an optimal flow. (NNC) There exists no negative cycle in (Gf,l^) with I^ o/(9.71). Proof. As is well known in network flow theory, (NNC) is equivalent to the existence of a potential p : V —> R such that

By (9.31), (9.32), (9.64), and the definition (9.71) of i^ this condition is equivalent to conditions (i) and (ii) of (POT) in Theorem 9.14. Hence follows the equivalence of (NNC) and (OPT) by Theorem 9.14. D Note 9.19. In a problem with dual integrality the arc length ^ is integer valued. The integrality of ^(a) for a e A* U B* is due to (9.67) and that for a e Q is by Theorem 6.61 (1). For integer-valued £^, we can take an integer-valued p in the proof of Theorem 9.18. • Next we consider the integer-flow problem under the assumptions

For a feasible integer flow £ : A —> Z, we define an auxiliary network (Gj,^) in a similar manner, while modifying the definitions of C^ and (.^ to

Theorem 9.20 (Negative-cycle criterion). For a feasible integer flow £ : A —> Z to the M-convex submodular integer flow problem MSFP2 with (9.72), the conditions (OPT) and (NNC) below are equivalent. (OPT) £ is an optimal flow. (NNC) There exists no negative cycle in (G^,l^) with I £ o/(9.74). Proof. This is similar to the proof of Theorem 9.18. Note, however, that Theorem 9.16 is used here in place of Theorem 9.14. D Note 9.21. The M-convex intersection problem introduced in section 8.2.1 can be formulated as an M-convex submodular flow problem. Given two M-convex functions /i,/2 : Zv —> R U {+00}, we consider an M-convex submodular flow problem on the bipartite graph G = (Vi U V2,A) in Fig. 9.3, where V\ and Vi are copies of V and A = {(v\, v%) v e V} with vi £ V\ and v% € Vi denoting the copies of v e V. The boundary cost function / : ZVl x ZV2 -> R U {+00} is defined by /(zi,X2) = f i ( x i ) + f2(^X2) for Xi G ZVl and X2 6 Z^ 2 , whereas the arc costs are

9.5. Optimality Criterion by Negative Cycles

265

Figure 9.3. Submodular flow problem for M-convex intersection problem.

identically zero without capacity constraints. Since x\ = —x% if (xi^x^) = <9£ f a flow £ in this network, the M-convex submodular flow problem is equivalent to minimizing f i ( x ) + /2(x). The negative-cycle optimality criterion (Theorem 9.20) for this M-convex submodular flow problem yields the M2-optimality criterion in Theorem 8.33. This argument shows also that the M2-optimality criterion can be verified in polynomial time. 9.5.2

Cycle Cancellation

The negative-cycle optimality criterion states that the existence of a negative cycle implies the nonoptimality of a feasible flow. This suggests the possibility of improving a nonoptimal feasible flow by the cancellation of a suitably chosen negative cycle. Let us consider the integer-flow problem with (9.72). Suppose that negative cycles exist in the auxiliary network (G^,l^) for a feasible integer flow £, where the arc length l^ is defined by (9.74). Choose a negative cycle with the smallest number of arcs and let Q (C A^) be the set of its arcs. Modifying the flow £ along Q we obtain a new integer flow £ defined by

The following theorem shows that £ is a feasible flow with an improvement in the objective function

This gives an alternative proof for "(OPT) =>• (NNC)," which has already been established in Theorem 9.20 with the aid of Theorem 9.16. Theorem 9.22. For a feasible integer flow £ to the M-convex submodular integerflow problem MSFP2 with (9.72), let Q be a negative cycle with the smallest number of arcs in (Gj,^). Then £ in (9.75) is a feasible integer flow and

266

Chapter 9. Network Flows

The rest of this section is devoted to the proof of Theorem 9.22. The key ingredient of the proof is the unique-rain condition, denned as follows. For a pair ( x , y ) of integer vectors satisfying x G doniz/ and ||x — y\\<x> = 1, we consider a bipartite graph G(x, y) = (V+, V~\E) with vertex sets V+ = supp+(:r y) and V~ = supp^ (x - y) and arc set

and associate c(u,v) = & f ( x ; v , u ) with arc ( u , v ) G E as its weight. We say that (x, y) satisfies the unique-min condition if there exists in G(x, y) exactly one minimum-weight perfect matching with respect to c. Denote by f ( x , y) the minimum weight of a perfect matching in G(x, y), where f ( x , y ) = +00 if no perfect matching exists. Proposition 6.25 shows f(y) — f ( x ) > f ( x , y ) for any x G doniz/ and y G Zv. The unique-min condition is a sufficient condition for this inequality to be an equality. Proposition 9.23. Let f G A4[Z —> R] be an M-convex function, and assume x G domz/, y G Zv, and \\x-y\\oo — 1. I f ( x , y ) satisfies the unique-min condition, then y G domz/ and

Proof. The set function u> defined by w(X] — —f(xf\y+xx) (X C F) is a valuated matroid; see (2.77). The present claim is a reformulation of the unique-max lemma for valuated matroids (see Theorem 5.2.35 in Murota [146]). D The following proposition gives a necessary and sufficient condition for a bipartite graph to have a unique minimum-weight perfect matching. It also shows that the unique-min condition for a pair of integer vectors can be checked by an efficient algorithm. Proposition 9.24. Let G — (V+,V~;E) be a bipartite graph with \V+\ = \V~ \ (= m) and c : V+ x V —> RU{+oo} be a weight function such that c(u, v) < +00 ^=^ (u,v) G E. There exists a unique minimum-weight perfect matching if and only if there exist a potential p : V+UV~~ —> R and orderings of vertices V+ = {HI, ... ,um} and V~ = {vi,..., vm} such that

Proof. This follows from the complementarity (Theorem 3.10 (3)) in the linear program formulation in the proof of Proposition 3.14. D The following is the key fact. Note that <9£ G domz/ and \\d£ — <9£||oo — 1-

9.5. Optimality Criterion by Negative Cycles

267

Proposition 9.25. (3£,<9£) satisfies the unique-min condition. Proof. Consider the bipartite graph G(d£,d£) = (V+,V~;E), where V+ = supp+(<9£ - d£), V' = supp-(<9£ - <9£), and

We have \V+\ = \V \ — m for m = ||<9£ — d£||i/2 and the weight of arc (u,v) equal to A/(9£; v, u). We may think of G(d£, d£) as a subgraph of the graph Gj of section 9.5.1 by regarding E as a subset of C^ in (9.73). Then Q n C% determines a perfect matching in G(d£,d£). Let M — {(ui,Vi) | i = 1,...,TO}be a minimum-weight perfect matching in G(d£,d£) and p be an optimal potential in Proposition 3.14. Note that M is a subset of

Regarding M as a subset of Cj, we define Q' = (Q\C^)UM. Since M is a minimumweight perfect matching, Qr\C^ is a perfect matching, and ^(a) = Af(d£;v,u) for a = (u, v) € C%, we have ^g(M) < t$(Q n C^), from which follows

Since Q' is a union of disjoint cycles with \Q'\ = \Q\ and Q is a negative cycle with the smallest number of arcs, (9.78) implies that Q' is also a negative cycle with the smallest number of arcs. To prove by contradiction, suppose that (d£, <9£) does not satisfy the uniquemin condition. Since (KJ, vt) e (7| for i = 1,..., m, it follows from Proposition 9.24 that there exist distinct indices i*. (fc = 1,..., q; q > 2) such that (uik, vik+1) e C£ for fc = 1 , . . . , g , where i g+ i = i\. That is,

On the other hand, we have

It then follows that

i.e.,

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Chapter 9. Network Flows

For k — ! , . . . , < / , let P'(vik+1,Uik) denote the path on Q' from Vik+1 to Uik, and let Q'k be the directed cycle consisting of arc (u$ fc , Vik+1) and path P'(vik+1, itj fc ). Obviously,

where the union here (and also below) means the multiset union, counting the number of occurrences of elements. A simple but crucial observation is that

for some integer q' with 1 < q' < q. Hence,

where (9.79) and (9.78) are used. This implies that ^(<2'fc) < 0 for some k, which, however, is a contradiction, since Q'k has a smaller number of arcs than Q'. This completes the proof. D Proof of Theorem 9.22: It follows from Propositions 9.25 and 9.23 as well as the definition of / that

whereas

Adding these two results in F 2 (£) < T 2 (£) + ^(Q).

9.6

Network Duality

Transformation by a network is one of the most important operations for M-convex and L-convex functions. A given pair of M-convex and L-convex functions defined on entrance vertices of a network is transformed through the network to another pair of M-convex and L-convex functions on exit vertices. Moreover, if the functions in the given pair are conjugate to each other, the resulting pair is also conjugate. This fact reveals a deeper intrinsic relationship of M-/L-convexity to network flow, partly discussed in section 2.2. The theorems as well as their implications are stated in section 9.6.1, and the proofs are given in section 9.6.2.

9.6.

Network Duality

269

Figure 9.4. Transformation by a network. 9.6.1

Transformation by Networks

We first deal with functions of the Z —» Z type, integer-valued functions defined on integer points, and then functions of other types, Z —> R and R —> R. Let G = (V, A; S, T) be a directed graph with vertex set V, arc set A, entrance set S, and exit set T, where S and T are disjoint subsets of V; see Fig. 9.4 for an illustration. For each a & A, the costs of integer-valued flow and tension are represented, respectively, by functions /„ : Z —*• Zu{+oo} and ga : Z —> ZU{+oo}. Given functions /, 5 : Zs —> Z U {+00} associated with the entrance set S of the network, we define functions /, g : ZT —> Z U {±00} on the exit set T by

We may think of f(y) as the minimum cost to meet a demand specification y at the exit, where the cost consists of two parts, the cost f(x) of supply or production of x at the entrance and the cost 5^aeA/a(^( a )) of transportation through arcs; the sum of these is to be minimized over varying supply x and flow £ subject to the flow conservation constraint <9£ = (x, — y, 0). A similar interpretation is possible for g(q). We regard / and g as the results of transformations of / and g by the network; (9.81) and (9.82) are called transformations of flow type and of potential type, respectively.

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Chapter 9. Network Flows

The following theorem reveals the harmonious relationship between network flow and M-/L-convexity by which a conjugate pair of M-convex and L-convex functions is transformed to another conjugate pair of M-convex and L-convex functions. Note that C[Z —> Z] denotes the set of univariate integer-valued discrete convex functions, and * means the discrete Legendre-Fenchel transformation (8.11)zTheorem 9.26. Assume fa,ga <E C[Z -> Z] for each a <E A. For f,g : Zs Z U {+00}, let /, g : ZT -> Z U {±00} be the functions induced by G = (V, A- S, T) according to (9.81) and (9.82), where it is assumed that f > —oo, f ^ +00, g > —oo, and g ^ +00.

We explain the implications of this theorem by considering three special cases. The first special case is a well-known construction in matroid theory, induction of a matroid through a graph. Given a graph G = (V, A; S, T) and a matroid (S, B) on S with base family B, let B be the family of subsets of T that can be linked with some base of (S, B) by a vertex-disjoint linking in G. Then B forms the base family of a matroid on T, which is referred to as the matroid induced from (S, B) through G. To formulate this as a special case of Theorem 9.26 (1), we split each vertex v e V into two copies, v' and v", to consider a graph G = (V U V", A; S', T") with

where S' = {v1 v& S} and T" = {v" v e T}. Let / : Zs/ -> Z U {+00} be the indicator function of the set of the characteristic vectors of bases, i.e., / = SB for B = {xxr | X e B}, and, for each arc a 6 A, define fa to be the indicator function of {0,1}. Then the induced function / : ZT —> Z U {+00} represents the family B in the sense that / = 6^ for B = {XY" \Y&B}. Then the M-convexity of / stated in Theorem 9.26 (1) shows that (T,B) is a matroid. The second case is where (5, T) = (V, 0). Then the induced functions / and g are constants, having no arguments, and the conjugacy asserted in Theorem 9.26 (3) amounts to a min-max relation

for

9.6. Network Duality

271

This is the discrete counterpart of (9.36), showing the duality nature of the assertion of Theorem 9.26 (3). The third case is where 5 = 0. Then the induced functions are

which are identical to (2.42) and (2.43), respectively, and the claims in Theorem 9.26 reduce to the facts observed in section 2.2.2. Whereas Theorem 9.26 deals with integer-valued functions defined on integer points, similar statements are true for functions of type Z —> R and R —> R. Note that the conjugacy assertion is missing in the case of Z —> R. Theorem 9.27. Assume fa,ga 6 C[Z -> R] for each a e A. For f,g : Zs -> R U {+00}, let /, g : ZT -> R U {±00} be the functions induced by G = (V, A; 5, T) according to (9.81) and (9.82), where it is assumed that f > — oo, / ^ +00, g > —oo, and g ^ +00.

Theorem 9.28. Assume fa,ga £ C[R —> R] for each a e A. For f,g : Rs —> RU{+oo}, let f,g : RT -» RU{±oo} be the functions induced by G = (V,A;S,T) according to

where it is assumed that f > -oo, / ^ +00, g > -oo, and g ^ +00.

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Chapter 9. Network Flows

Figure 9.5. Bipartite graphs for aggregation and convolution operations.

where * means the Legendre-Fenchel transformation (3.26). A number of fundamental operations on M-convex and L-convex functions can be formulated as the transformation by networks, as is partly demonstrated below.

Note 9.29. The M-convexity of the aggregation fu* of an M-convex function / (Theorem 6.13 (7)) is proved here as an application of Theorem 9.27. Let V' be a copy of V and consider a bipartite graph G — (S U T, A;S, T) with S = V, T = U U {MO}, and A = {(v', v) \ v e U} U {(v', u0) \ v
9.6.

Network Duality

273

Figure 9.6. Rooted directed tree for a laminar family.

Note 9.31. An alternative proof of the M^-convexity of a laminar convex function (6.34) is given here as an application of Theorem 9.27. Let T be a laminar family of subsets of V, where we may assume that 0 ^ T, V e T, and every singleton set belongs to T. We represent T by a, directed tree G = (U, A; S, T) with root MO, where U = {ux \ X € T} U {u0}, A = {ax X e T}, S = {u0}, T = {u{vj v e V}, and d~ax = ux and d+ax = ux for X £ T, where X denotes the smallest member of T that properly contains X (and V = 0 by convention). As an example, the rooted directed tree (arborescence) for V = {1,2,3,4,5} and T = {{l},{2},{3},{4},{5},{2,3},{l,2,3},{4,5},y} is depicted in^Fig. 9.6. We associate the given function fx with arc ax for X e T. The function / on T induced from / = 0 on S by this network coincides with the laminar convex function (6.34), and its M^-convexity follows from Theorem 9.27. •

9.6.2

Technical Supplements

Proof of Theorem 9.26 It suffices to consider the case of / € A4[Z —> Z] and g G £[Z —> Z]. (1) To prove (M-EXC[Zj) for /, we fix 2/1,2/2 € dom/ and u e supp+(yi -2/2) and look for v e supp~(j/i — 2/2) such that58

by a refinement of the augmenting path argument used in Note 2.19 for the special case of S = 0. We take & e ZA and z; e Zs for i = 1,2 such that

We search for a kind of augmenting path with respect to the pair (£1, £2) that yields the desired inequality (9.84). If we are not successful in finding such a path, we modify the flow pair to a new pair with smaller ^-distance ||£i — &\\i, so that we can eventually find an appropriate augmenting path. 58

For w G T, xfu is the characteristic vector of {w} in ZT.

274

Chapter 9. Network Flows

Before giving a formal proof we explain the idea of the proof in a typical situation. Consider the difference in the flows, £2 — Ci 6 Z'4, for which we have 9(& ~ £1) = (xy ~ xi,yi - 2/2,0). Since u € supp+(<9(£2 - £1)), there exists a simple path, say, PI, compatible with £2 ~~ £1 that connects u to some vertex v\ in supp~(i9(^2 — £1)) = supp + (xi - £2) U supp~(j/i - yi). This is an augmenting path with respect to the pair of flows £1 and £2- Suppose that we have the case of vi E supp + (xi — £ 2 ). By (M-EXC[Zj) for / we obtain HI <E supp"(xi — £2) such that59 Since u\ E supp+(9(^2 — £i))> there exists a simple path, say, P2, compatible with £2 ~ £1 that connects HI to some v% € supp~(<9(£2 ~ Ci)) = supp + (xi — ^2) U supp~(yi — 7/2)- Suppose further that v? E supp~(yi — 2/2) and P% is vertex disjoint from PI. Putting v — V2 we represent the path PI U P% by TT : A —> {0, ±1} such that supp+(7r) C supp+(^2 — 6): supp'(?r) C supp^(^ 2 - Ci)> and dir = (Xui - X?i > Xu - X« > 0). For the augmented flows £{ = ^ + TT and ^2 = 6 - TT a the new bases x{ = £1 - x^ + xf j and x'2 = x2 + X^ - xf1, we have

and

since

By (9.85), (9.86), (9.88), and (9.87), we obtain

which shows the inequality (9.84). Having presented the rough idea, we are now in a position to start the proof that works in general. We shall construct a pair of flows £,( and £2 that satisfy (9.87) for some v E supp~(yi — 2/2) and £^,£2 6 Zs, and also

where

59

For w 6 S, Xw is the characteristic vector of {w} in Zs.

9.6. Network Duality

275

with b\s denoting the restriction of b to S. To obtain such (££,£2) we generate a sequence of tuples (<^i, 1^2,61,62, w) with i,ip2 £ Z"4, 61, 62 € Zy, and w e V such that

where x™ £ Zy in (9.93). We call (ipi,y>2i 61, b?,w) a flow-boundary tuple and w the frontier vertex. Note that (61,62) is determined uniquely from (ip\,(p2,w] by (9.93). We start with (i = (6i|s, -(yi - X« + XD, 0) and d 2 ,61,6 2 , w) is updated to another flow-boundary tuple ( \\ 2 ( a *) = ¥2(a*) ~ 1, an(i tne frontier vertex is changed to w' = d~~a*, the terminal vertex of a*. Symmetrically, if w = d~a* and ^(a*), the flows and the frontier vertex are updated as ^(a*) = i(a*) - 1, ¥> 2 ( a *) = ^2(0*) + 1, w' — d+a*. The boundaries remain unchanged; 6j — b\ and 62 = 62. See (9.88) for the condition (9.94). A basis exchange applies when w € S and 61 (w) > 6 2 (w). By (M-EXC[Zj) for / there exists w' G 5 such that bi(w') < 6 2 (u/) and

The frontier vertex is updated to this w' and the boundaries to b( =b\~Xw+ Xw1, 62 = 62 + Xw - Xw'- The flows remain the same: ip{ = ipi and ip'2 = y? 2 A crossover updates ((/PI, tp2,61, 62, w) with reference to another flow-boundary tuple (
according to whether

or not. The condition (9.94) is maintained, since

276

Chapter 9. Network Flows

Under the additional conditions that we have We use crossovers to avoid cycling in generating the flow-boundary tuples. The generation of flow-boundary tuples consists of stages. Each stage consists of repeated applications of flow push and base exchange, possibly followed by an application of crossover. Denote by (ip\ , y>2 \b\ , 62 j w^) the flow-boundary tuple at the beginning of a stage and by ( b[ ,b% ,w^) (j = 2 , . . . , fc) the flowboundary tuples generated so far in the stage. We end the stage if either (a) w^ G supp~(yi — 7/2) or (b) u/ fc ) = u 7^ w/ 1 ); we are done in case (a), whereas in case (b) we go on to the next stage. If w^k> — w^ for some j with 1 < j < k — 1, we apply crossover to (^\^\bf\b(^\w^) with reference to ( b% = d(p2 — Xu, and w = u for ( < P i , y 2 , & i , & 2 , w ) = (< / 5f ) ,^ f c ) ,&f ) ,6^ / c ) ,w;( f e ) ),justaswehadfor(y)i,^2,^i,^2,w) = (£i) ^2> d£i + Xu, d^2 ~Xu,u) at the beginning of the generation process. In case (c) we obtain a closer pair of flows, with which the next stage starts. The above generation process terminates with a finite number of flow-boundary tuples. This is because (i) the frontier vertices in one stage are distinct and hence the number of flow-boundary tuples generated in one stage is bounded by \V\, (ii) \\ipi —
9.6.

Network Duality

277

which shows (TRF[Zj) for g. Suppose that g(q) is finite for q = qi,q2- There exist (77i,Pi,7*i) and (772,252,7*2) such that

Here we have by the submodularity (SBF[Zj) of g and with by the convexity of ga (see Note 2.20). The submodularity (SBF[Z]) of g then follows because

(cf. (9.21)), whereas the assumed conjugacy implies

Therefore, we have

from which follows the weak duality Fix y with f ( y ) finite. By Theorem 9.16 (4) there exists (p*,q*,r*) such that

with 77* = — S(p*,q*,r*). This implies

Combining this with (9.97) shows g = /*. The proof of Theorem 9.26 is completed. It is mentioned that (1) follows from (2) and (3) with the aid of the conjugacy theorem (Theorem 8.12).

278

Chapter 9. Network Flows

Proof of Theorem 9.27 Because the functions are real valued, the infima in (9.81) and (9.82) may not be attained. The proof for Theorem 9.26 can be adapted to this case by introducing £ > 0 and letting s -> 0 as in Notes 2.19 and 2.20. Proof of Theorem 9.28 The augmenting flow argument for (1) suffers from a technical difficulty that the amount of augmenting flow may possibly converge to zero. To circumvent this difficulty we first prove (2) and (3) and then use the conjugacy (Theorem 8.4) to show (1). See Murota-Shioura [152] for details.

Bibliographical Notes The minimum cost flow problem treated in section 9.1 is one of the most fundamental problems in combinatorial optimization. For network flows, Ford-Fulkerson [53] is the classic, whereas Ahuja-Magnanti-Orlin [1] describes recent algorithmic developments; see also Cook-Cunningham-Pulleyblank-Schrijver [26], Du-Pardalos [43], Korte-Vygen [115], Lawler [119], and Nemhauser-Wolsey [167]. Thorough treatments of the network flow problem on the basis of convex analysis can be found in Iri [94] and Rockafellar [178]. The submodular flow problem was introduced by Edmonds-Giles [46] using crossing-submodular functions. The present form avoids crossing-submodular functions on the basis of the fact, due to Fujishige [61], that the base polyhedron defined by a crossing-submodular function can also be described by a submodular function. See Fujishige [65] for other equivalent neoflow problems, such as the independent flow (Fujishige [59]) and the polymatroidal flow (Hassin [87], Lawler-Martel [120], [121]). The M-convex submodular flow problem was introduced by Murota [142]. Section 9.3 is a collection of standard results on the submodular flow problem. Theorems 9.10 and 9.13 are taken from Fujishige [65] (Theorems 5.1 and 5.11, respectively), where the former is ascribed to Frank [56]. The optimality criterion by potentials for the M-convex submodular flow problem was established by Murota [142] for the integer-flow version (Theorem 9.16) and adapted to the real-valued case (Theorem 9.14) with the integrality assertion (Theorem 9.15) in Iwata-Shigeno [105] and Murota [147]. The optimality criterion by negative cycles (Theorem 9.20) was established by Murota [140], [142] for the integer-flow version and adapted to the real-valued case (Theorem 9.18) in Murota-Shioura [152]. Theorem 9.22 is in [140], [142]. Proposition 9.23 is a reformulation of the unique-max lemma due to Murota [135]. Proposition 9.25 is also from [135]; the proof technique using (9.80) originated in Fujishige [59]. Transformation by networks was found first for M-convex functions / s M [Z —> R] by Murota [137]; the proof given in section 9.6.2 is due to Shioura [188], [189]. Transformation of L-convex functions is stated explicitly in Murota [145]. The extension to polyhedral M-convex and L-convex functions is made in Murota-Shioura

9.6.

Network Duality

279

[152]. Theorems 9.26, 9.27, and 9.28 are explicit in Murota [147]. Induction of matroids through graphs is due to Perfect [171] (for bipartite graphs) and Brualdi [21]; see also Schrijver [183], Welsh [211], and White [213]. The alternative proof of the M^-convexity of a laminar convex function described in Note 9.31 is communicated by A. Shioura.

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Chapter 10

Algorithms

Algorithmic aspects of M-convex and L-convex functions are discussed in this chapter. Three fundamental optimization problems tractable by efficient algorithms are (i) M-convex function minimization, which is a nonlinear extension of the minimumweight base problem for matroids; (ii) L-convex function minimization, which includes submodular set function minimization as a special case; and (iii) minimization/maximization in the Fenchel-type min-max duality, which is equivalent to the M-convex submodular flow problem.

10.1

Minimization of M-Convex Functions

Four kinds of algorithms for M-convex function minimization are described: the steepest descent algorithm, the steepest descent scaling algorithm, the domain reduction algorithm, and the domain reduction scaling algorithm. Throughout this section we assume that / : Zv —> RU {+00} is an M-convex function, n = \V\, and F is an upper bound on the time to evaluate /.

10.1.1

Steepest Descent Algorithm

The local characterization of global minimality for M-convex functions (Theorem 6.26) immediately suggests the following algorithm of steepest descent type. Steepest descent algorithm for an M-convex function / € Ai[Z —> R] SO: Find a vector x & dom /. SI: Find u, v G V (u ^ v) that minimize f(x — Xu + Xv)S2: If f ( x ) < f(x — Xu + Xv), then stop (x is a minimizer of /). S3: Set x := x — Xu + Xv and go to SI. Step SI can be done with n2 evaluations of function /. At the termination of the algorithm in step S2, x is a global minimizer by Theorem 6.26 (M-optimality criterion). The function value / decreases monotonically with iterations. This 281

282

Chapter 10.

Algorithms

property alone does not ensure finite termination in general, although it does if / is integer valued and bounded from below. Let us derive an upper bound on the number of iterations by considering the distance to the optimal solution rather than the function value. Proposition 10.1. If f has a unique minimizer, say, x*, the number of iterations in the steepest descent algorithm is bounded by \\x° — x*\\\/1, where x° denotes the initial vector found in step SO. Proof. Put x' = x — Xu + Xv m step S2. By Theorem 6.28 (M-minimizer cut), we have x*(u) < x(u) — 1 = x'(u) and x*(v) > x(v) + I = x'(v), which implies \\x' — x*||i — \\x — x* \\ — 2. Note that ||x° — x*\\i is an even integer. D When given an M-convex function /, which may have multiple minimizers, we consider a perturbation of / so that we can use Proposition 10.1. Assume now that the effective domain is bounded and denote its ^i-size by We arbitrarily fix a bijection (p : V —> {1,2,... ,n} to represent an ordering of the elements of V, put Vi = tp~l(i) for i = 1,... ,n, and define a vector p € Ry by p(vi) = el for i = 1,... ,n, where e > 0. The function /£ = f\p] is M-convex by Theorem 6.13 (3) and, for a sufficiently small e, it has a unique minimizer that is also a minimizer of /. Suppose that the steepest descent algorithm is applied to the perturbed function fe. Since fs(x - Xu + Xv) = f(x - Xu + Xv) + E"=i ^x(vi) e v(«) -)_ £v(v) ^ this amounts to employing a tie-breaking rule: where

in the case of multiple candidates in step SI of the steepest descent algorithm applied to /. With this tie-breaking rule we have the following complexity bound. Proposition 10.2. For an M-convex function f with finite K\ in (10.1), the number of iterations in the steepest descent algorithm with tie-breaking rule (10.2) is bounded by K\/2. Hence, if a vector in dom/ is given, the algorithm finds a minimizer of f in O(F • n2Ki) time. By Theorem 6.76 (quasi M-optimality criterion) and Theorem 6.77 (quasi Mminimizer cut), the steepest descent algorithm can also be used for minimizing quasi M-convex functions satisfying (SSQM^). Note 10.3. For integrally convex functions we have the local optimality criterion for global optimality (Theorem 3.21). This naturally suggests the following.

10.1.

Minimization of M-Convex Functions

283

Steepest descent algorithm for an integrally convex function / SO: Find a vector x € dom/. SI: Find disjoint Y,ZCV that minimize f(x — XY + Xz)S2: If f ( x ) < f(x — XY + X z ) , then stop (x is a minimizer of /). S3: Set x := x — XY + Xz and go to SI. The steepest descent algorithm for M-convex functions is a special case of this. It is emphasized that no efficient algorithm for step SI is available for general integrally convex functions. •

10.1.2

Steepest Descent Scaling Algorithm

Scaling is one of the fundamental general techniques in designing efficient algorithms. The proximity theorem for M-convex functions leads us to the following steepest descent scaling algorithm for M-convex function minimization. We assume that the effective domain is bounded and denote its £00-size by

Steepest descent scaling algorithm for an M-convex function / 6 M\L -f R]

SO: Find a vector x e dom/, and set a ~ 2^82(^00^n)^ B .= dom /. SI: Find an integer vector y that locally minimizes f(,,\ _ / f(x +ay) (x + ay&B), J(y> ~ { +00 (x + ayiB) in the sense of f ( y ) < f(y — Xu + Xv) (Vu, v & V) by the steepest descent algorithm of section 10.1.1 with initial vector 0 and set x := x + ay. S2: If a = 1, then stop (x is a minimizer of /).

S3: Set B := B n (y e Zv to SI.

\\y - x^

< (n - 1)(a - 1)} and a := a/2 and go

By the M-proximity theorem (Theorem 6.37 (1)), the set B always contains a global minimizer of / and, at the termination of the algorithm in step S2, x is a global minimizer by the M-optimality criterion (Theorem 6.26 (1)). The number of iterations is bounded by [Iog2(-ft'oo/4n.)] • Some remarks are in order concerning step SI. If the function / is such that the scaled function / remains M-convex, step SI can be done in O(F • n4) time by the steepest descent algorithm with tie-breaking rule (10.2). This time bound follows from Proposition 10.2 with K\ < 4ra2. For a general /, however, / is not necessarily M-convex (see Note 6.18) and no polynomial bound for step SI is guaranteed, although we have an obvious exponential time bound 0(F-(4n)"n 2 ). On the basis of Theorem 6.76 (quasi M-optimality criterion) and Theorem 6.78 (quasi M-proximity theorem), the steepest descent scaling algorithm can be adapted to the minimization of quasi M-convex functions satisfying (SSQM^).

284

10.1.3

Chapter 10. Algorithms

Domain Reduction Algorithm

The domain reduction algorithm is a kind of bisection method that searches for the minimum of an M-convex function by generating a sequence of nested subsets of the domain on the basis of the M-minimizer cut theorem (Theorem 6.28). For an M-convex function with bounded effective domain, the algorithm finds a minimizer in time polynomial in n and Iog2 KOC, where K^ is defined by (10.3). We introduce the following notations for a bounded nonempty set B C Zv:

The set B° is intended to represent the central part of B, i.e., the set of vectors of B lying away from the boundary. The set B° is nonempty if B is M-convex; see Proposition 10.6 below. Domain reduction algorithm for an M-convex function / S Ai[Z —*• R] SO: Set B :=dom/. SI: Find a vector x <E B°. S2: Find u, v € V (u ^ v) that minimize f(x — Xu + Xv)S3: If f ( x ) < f(x — Xu + Xv), then stop (x is a minimizer of /). S4: Set B := B D {y € Zv \ y(u) < x(u) - 1, y ( v ) > x(v) + 1} and go to SI. The vector x G B° in step SI can be found with O(n2log2K00) evaluations of / by the procedure to be described below. The set B forms a decreasing sequence of M-convex sets, which contain a minimizer of / because of the M-minimizer cut theorem (Theorem 6.28). Since x is taken from the central part of B, UB(W) — IB(W) for w e {u, v} decreases with a factor of (1 — ~) and hence the number of iterations is bounded by O(n2 Iog2 -ftToo)- The above algorithm, therefore, finds a minimizer of / with O(n 4 (log 2 Koo)2) evaluations of / provided that it is given a vector in dom /. Proposition 10.4. If a vector in dom / is given, the domain reduction algorithm finds a minimizer of an M-convex function f in O(F • n4(log2 Koc)2) time. It remains to show how to find x e B° in step SI when given a vector of B. For a vector y of an M-convex set B and two distinct elements u,v of V, the exchange capacity is defined by

which is a nonnegative integer representing the distance from y to the boundary of B in the direction of Xv - Xu- In the domain reduction algorithm, B is always an M-convex set and the exchange capacity can be computed by a binary search with [Iog2 KOO] evaluations of /. For x e B we define

10.1.

Minimization of M-Convex Functions

285

with an obvious observation that V° (x) = V if and only if x 6 B°. If V°(x) i- V, we can modify x to x' e B with the property \V°(x')\ > \V°(x)\ + 1 as follows. Take any u e V \V°(x) and assume x(u) > u°B(u); the other case with x(u) < i°B(u) can be treated in a similar manner. Putting {vi,V2, • • • ,vn-i} = V\ {u} and x0 = x, we define a sequence r c i , x 2 , . . . ,xn-i by Xi = Xi-i + oti(xVi - Xu) with

for i = 1 , 2 , . . . , n — 1 and put x' = xn-iProposition 10.5. V°(x') D V°(x) U {u}. Proof. The inclusion V°(x'} 2 V°(x) is obvious. To prove x'(u) = u°B(u) by contradiction, suppose x'(u) > u°B(u) and take x* & B°, where B° ^ 0 by Proposition 10.6 below. Since u & supp+(o;' — #*), it follows from (M-EXC[Z]) that x" = x' + Xvi — Xu € B and x'(vi) < x*(vi) < u°B(vi) for some Vi. Since Vi 6 supp+(x" — Xi) and supp~(o;" — Xi) = {u}, (M-EXC[Zj) implies Xi + \Vi — \u 6 B. But this contradicts the definition of x,. D The modification of x to x' described above can be done with n evaluations of the exchange capacity. Repeating such modifications at most n times we arrive at x with V°(x) = V. Thus, given a vector in B, we can find x e B° with at most n2 evaluations of the exchange capacity. We finally prove the nonemptiness of B°. Proposition 10.6. B° ^ 0 if B is an M-convex set. Proof. Let p 6 <S[Z] be the submodular function satisfying B = B(/o) n Zv and p be its Lovasz extension. Then we have IB(V) — p(V) — p(V\{v}} and UB(V) — p(v). We can assert the nonemptiness of B° by establishing

(see Theorem 3.8 in Pujishige [65]). We prove (i) here; a similar argument works for (ii). Fix X C V and put p = xx, Pv = 1 — Xv (v € X), and k — \X\. It follows from

that kp(V) = /3(fcl) = p(p + Y^vexPv) and

286

Chapter 10. Algorithms

With these identities we see

The last inequality holds true since

by the positive homogeneity and convexity of p. Hence follows i°B(X} < p ( X ) .

D

By Theorem 6.77 (quasi M-minimizer cut), the domain reduction algorithm can also be used for minimizing quasi M-convex functions satisfying (SSQM5^) provided that the effective domain is a bounded M-convex set.

10.1.4

Domain Reduction Scaling Algorithm

We present here the domain reduction scaling algorithm for M-convex function minimization, a combination of the idea of the domain reduction algorithm of section 10.1.3 with a scaling technique based on the theorem of M-minimizer cut with scaling (Theorem 6.39). The algorithm works with a pair (x, (.) of integer vectors, where x is the current solution and £ is a lower bound for an optimal solution. Specifically, two conditions

are maintained, where The algorithm consists of scaling phases parametrized (or labeled) by a nonnegative integer a, called the scaling factor, which is initially set to be sufficiently large and is decreased until it reaches unity. In each scaling phase with a fixed a, the pair (x, (.} is modified so that it satisfies an additional condition

At the end of the algorithm we have a = 1 and hence x = i by (10.6). This means S(l) n dom/ = {x}, since x(V) = y(V) for any y G dom/ by Proposition 6.1 and since x(V) < y(V) for any y e S(£) distinct from x. Furthermore, we see from the second condition in (10.5) that a; is a minimizer of /, since {x} — S(K) n dom/ D S(f)nargmin/^0. The outline of the algorithm reads as follows, where step SI for the a-scaling phase is described later and K^ is defined in (10.3).

10.1.

Minimization of M-Convex Functionsiions

287

Domain reduction scaling algorithm for an M-convex function

/ e M[Z -> R]

SO: SI: S2: S3:

Find a vector x € dom/ and set t := x - K^l, a := ^lo^K^/2n^. Modify (x,£) to meet (10.6) (a-scaling phase). If a = 1, then stop (x is a minimizer of /). Set a := a/2 and go to SI.

The a-scaling phase is now described. In view of (10.6) we employ a subset V of V such that

Initially, V* is set to V and then decreases monotonically to the empty set. ct-scaling phase for (x, (., a) SO: Set V := V. SI: If V — 0, then output (x,l) and stop. S2: Take any ueV*. S3: Find v e V that minimizes f(x + a(\v - x«))S4: If v = u or x(u) - a < l(u), then set l(u) := max[l(u), x(u) - (n - l)(a - 1)] and V := V \ {u} and go to SI. S5: Otherwise, set £(v) := max[l(v),x(v) + a — (n — l)(a — \)}, x := x + a(xv ~Xu) and V := V \ {v} and go to SI. As is easily seen, the first condition in (10.5) is maintained in steps S4 and S5. The second condition in (10.5) is also maintained by virtue of Theorem 6.39 (Mminimizer cut with scaling). The subset V is nonincreasing, although it may be that v $. V* in step S5, and then the operation V := V \ {v} is void and V* remains unchanged. Denote the initial value of (x,l) by (x°,l°). For each w € V the value of x(w) is decreased at most (x°(w) — l°(w))/a times before it is deleted from V*. Hence, the number of iterations in the a-scaling phase is bounded by \\x° — i°\\\/a. In particular, the a-scaling phase terminates with V = 0. The time complexity of the domain reduction scaling algorithm is given as follows, where it is assumed for simplicity that K^ is known. Proposition 10.7. // a vector in dom/ is given, the domain reduction scaling algorithm finds a minimizer of an M-convex function j in 0(F • n3log2(K00/n)) time. Proof. At the beginning of the a-scaling phase we have \\x°— (°\\\ < n(n— l)(2a—1) by (10.6). Since step S3 in the a-scaling phase can be done with n evaluations of /, the a-scaling phase terminates in O(F • n3) time. The number of scaling phases is equal to \\og2(Koc/2n)~\. D On the basis of Theorem 6.79 (quasi M-minimizer cut with scaling), the domain reduction scaling algorithm can be adapted to the minimization of quasi M-convex functions satisfying (SSQM^) provided that the effective domain is a bounded M-convex set.

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10.2

Minimization of Submodular Set Functions

The minimization of submodular set functions is one of the most fundamental problems in combinatorial optimization. In this section we deal with algorithms for minimizing submodular set functions, which we will use as an essential component in algorithms for L-convex functions.

10.2.1

Basic Framework

Let p : 2 —> R be a submodular set function,60 where p(0) = 0 and n = \V\. In discussing the efficiency or complexity of algorithms it is customary to categorize them into finite, pseudopolynomial, weakly polynomial and strongly polynomial algorithms. For our problem of minimizing p, a finite algorithm is trivial; we may evaluate p(X) for all subsets X to find the minimum. This takes O(F • 2") time, where F is an upper bound on the time to evaluate p. The complexity of an algorithm may depend on the complexity or size of p; if p is integer valued, V

often serves as a measure of the size of p. An algorithm for minimizing p is said to be (i) pseudopolynomial, (ii) weakly polynomial, or (iii) strongly polynomial, according as the total number of evaluations of p as well as other arithmetic operations involved is bounded by a polynomial in (i) n and M, (ii) n and Iog2 M, or (iii) n alone. Our objective in this section is to describe two strongly polynomial algorithms for minimizing p. Let

be the base polyhedron associated with p. Recall that a point in B(/o) is called a base and an extreme point of B(p) is an extreme base. For any base x and any subset X we obviously have

where x~ is the vector in R^ defined by x~(v) = min(0,x(v)) for v e V. The inequalities are tight for some x and X, as follows. Proposition 10.8. For a submodular set function p : 2V —-> R, we have

If p is integer valued, the maximizer x can be chosen to be an integer vector. Proof. Although this is an easy consequence of Edmonds's intersection theorem (Theorem 4.18) with pi = p and p2 = 0, a direct proof is given here. Let x be a 60

Note that we assume p to be finite valued for all subsets. An adaptation to the general case p : 2V —> RU {+00} is explained in Note 10.14.

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maximizer on the left-hand side. For any u € supp~(z) and v €. supp + (x), there exists a subset Xuv such that u 6 Xuv C V \ {v} and x(Xuv) — p(Xuv). Put X = U u6 sup P -(x) a6,Upp+(a,) Xuv. We have x(X) = p(X) by (4.23) and x'(V) = x(X) since supp~(x) C X and supp+(a;) C V \ X.. The integrality assertion can be established by the same argument starting with an integral base x that maximizes x~(V) over all integral bases. D The min-max relation (10.11) shows that we can demonstrate the optimality of a subset X by finding a base x with x~(V) — p(X). But how can we verify that a vector x belongs to B(p)? By definition, x e B(p) if and only if mmx(p(X) — x(X)) = p(V) - x(V) = 0. Thus, testing for membership in B(p) for an arbitrary x seems to need a submodular function minimization procedure. To circumvent this difficulty, we recall from Note 4.10 that we can generate an extreme base by the greedy algorithm. Let L = (vi,v<2, • • •, vn) be a linear ordering of V and define L(VJ) = {i>i, t>2, • • •, vj} for j — 1,...,n. Then the extreme base y associated with L is given by Any base x can be represented as a convex combination of a number of extreme bases, say, {yi \ i £ I}, as

where we may assume |J| < n by the Caratheodory theorem. Combining (10.12) and (10.13) shows that any base can be represented by a list of linear orderings {Li | i € /} (that generate {j/j | i 6 /}) and coefficients of the convex combination {A; | i e /}. With this representation of x we can be sure that x is a member of B(p). For u, v € V and i e I we use the notation

where w ^ v means w -<, v or w = v. The following proposition gives a sufficient condition for optimality in terms of the linear orderings {Li | i € /}. Proposition 10.9. Let x be a base represented as (10.13) with {(Lj,Aj) i e 1} and W be a subset of V. (1) //supp-(z) C W and supp+(x) CV\W, then x~(V) = x(W). (2) Ifu^v for every u e W, v e V \ W, and i G /, then x(W) = p(W). (3) If the conditions in (1) and (2) are satisfied, x and W are optimal in (10.11). Proof. (1) is obvious. By (10.12) the condition in (2) implies p(W) = yi(W) for every i € /. Then x(W) = Eie/^W = P(W)- (3) follows from (1), (2), and (10.10). D

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We are thus led to a basic algorithmic framework: 1. We maintain (and update) a number of linear orderings {Li i € /}, together with the associated extreme bases {yi \ i 6 /} and the coefficients of convex combination {A^ i e /}, to represent a base x. 2. We terminate when the conditions in Proposition 10.9 are satisfied. Two strongly polynomial algorithms using this framework are described in subsequent sections. Note 10.10. Here is a brief historical account of submodular function minimization. Its importance seems to have been recognized around 1970 by J. Edmonds [44] and others. The first polynomial algorithm was given by M. Grotschel, L. Lovasz, and A. Schrijver—weakly polynomial in 1981 [82], and strongly polynomial in 1988 [83]. These algorithms, however, are based on the ellipsoid method and, as such, are not so much combinatorial as geometric. Efforts for a combinatorial polynomial algorithm have been continued with major contributions made by W. H. Cunningham and others [15], [28], [29], who showed the basic framework above as well as a combinatorial pseudopolynomial algorithm for submodular function minimization. In 1994, extending the min-cut algorithm of Nagamochi-Ibaraki [163], M. Queyranne [172] came up with a combinatorial algorithm for symmetric submodular function minimization, which is to minimize over nonempty proper subsets a submodular function p such that p(X) = p(V \ X) for all X C V. Combinatorial strongly polynomial algorithms for general submodular functions were found, independently, in the summer of 1999 by two groups, S. Iwata, L. Fleischer, and S. Fujishige [101], [102] and A. Schrijver [182]. Both of these follow Cunningham's framework, but they are significantly distinct in technical aspects. Subsequently, a new problem was recognized by Schrijver. These two algorithms are certainly combinatorial, but they rely on arithmetic operations (division, in particular) in computing the coefficients of convex combination in (10.13). The question posed by Schrijver is as follows: Is it possible to design a fully combinatorial strongly polynomial algorithm that is free from division and relies only on addition, subtraction, and comparison? This was answered in the affirmative by Iwata [99] in the fall of 2000. • Note 10.11. Because the minimizers form a ring family (see Note 4.8), there exists a unique minimal minimizer as well as a unique maximal minimizer of a submodular set function p : 2V —> R. Given an optimal base x with the representation x = Sie/^il/i m (10.13), we can compute the minimal and maximal minimizers in strongly polynomial time as follows. With the notation T>(x) for the family of tight sets at x, introduced in (4.22) of Note 4.9, we have a representation

for the family of the minimizers of p. Noting that T>(x) is a ring family, let Gx = (V,AX) be the directed graph associated with T>(x), as defined by (4.20) in Note

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4.7. This is equivalent to saying that (u, v) € Ax if and only if v G dep(x, u), where dep(x,w) means the smallest tight set at x that contains u. Then the minimal minimizer can be identified as the set of vertices reachable from supp~ (x) in Gx and the maximal minimizer as the complement of the set of vertices reachable to supp+ (x) in Gx. Moreover, the graph Gx enables us to enumerate all the minimizers. By (10.13) we have

Since each yi is an extreme base, we can easily compute dep(yi,u). For each u e V, we start with D := V and update D to D \ {v} as long as D \ {v} € T^(yi) for some v £ D \ {u}; we obtain D = dep(yi,u) at the termination. We can thus compute dep(yi, u) with O(n 2 ) evaluations of p. The number of function evaluations can be reduced to O(n) if a linear ordering Lj = (vi,vz,... ,vn) generating yi is also available; assuming u = Vk, start with D = {vi,f2> • • • ,Vk-i} and, for j = k - 1, k - 2 , . . . , 1, update D to D \ {vj} if D \ {vj} e £>(&). Therefore, the graph Gx can be constructed with O(n 3 |/|) or 0(n 2 |/|) evaluations of function p, where it is reasonable to assume |/| < n. • Note 10.12. The minimal minimizer of a submodular set function p : 1V —> R can also be computed using any submodular function minimization algorithm n + I times. Let X be a minimizer of p. For each v e X, compute a minimizer Yv of a submodular set function pv : 2X\^ —> R, the restriction of p to X \ {v}, defined by pv(Y) = p(Y) for Y C X \ {v}. Then the minimal minimizer of p is given by {v G X | p(Yv) > p ( X ) } , since p(Yv) > p(X) if v is contained in the minimal minimizer and p(Yv) = p(X) if not. The maximal minimizer can be computed similarly. • Note 10.13. An alternative way to find the minimal minimizer of a submodular set function p : 2V —> R is to introduce a penalty term to represent the size of a subset and to minimize a modified submodular set function p(X) = p(X) + s\X\ with a sufficiently small positive parameter e. If p is integer valued, e = l/(n + 1) is a valid choice. The maximal minimizer can be computed similarly. • Note 10.14. It has been assumed that the submodular function p is finite valued for all subsets. This assumption is not restrictive but for convenience of description. Let p 6 <S[R] be a submodular set function on V taking values in RU{+oo}; T> = domp is a ring family with {0, V} C T>. For v € V we denote by Mv the smallest member of T> containing v and by Nv the largest member of T> not containing v. For X C V we denote by X the smallest member of T> including X. We assume that we can compute Mv for each v & V efficiently, say, in time polynomial in n. Then we can compute X and Nv by

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Let us assume, without loss of generality, that the length of a maximal chain of V is equal to n = \V\. Consider now a set function ~p : 2V —> R denned by

with c e R^ given by

As is shown below, (i) p is a finite-valued submodular function and (ii) X £ arg min p implies X e arg min p. Thus we can minimize p via the minimization of p. Proof of (i): p is obviously finite valued. If X e V, v £ X, and Y = X U {v} e T>, we have YL)NV = NVU {v} and Y D Nv = X and, therefore,

This means that n(X) = p(X}-\-c(X) is nondecreasing on T>. Putting Ji(X) — n(X) for X C V and noting X U F = XUY and X n F D X n F, we see

which shows the submodularity of /I and hence that of p. Proof of (ii): For any Y e D we have p(T) < p(T) + c(X) - c(X) = p(X) < p(Y) = p(Y). m Note 10.15. The method in Note 10.14 can be used to minimize a submodular function defined on a (finite) distributive lattice. By a lattice we mean a tuple (S, V, A) of a nonempty set S and two binary operations V and A on S1 such that

for any a, b, c €. S. A lattice (S, V , A ) is a distributive lattice if, in addition, the distributive law

holds true. A ring family T> is a typical distributive lattice, where (S, V, A) = (Z>, U , n ) . The converse is essentially true: any distributive lattice (S, V , A ) can be represented in the form of a ring family (Birkhoff's representation theorem). The size of the underlying set of the ring family is equal to the length of a maximal chain of S. A function p : S —> R defined on a distributive lattice (S, V, A) is said to be submodular if it satisfies

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Thus, with an appropriate representation of (S, V, A), a submodular function p on S can be minimized by using the method in Note 10.14. • Note 10.16. Maximizing a submodular set function is a difficult task in general. It is known that no polynomial algorithm exists for it (and this statement is independent of the P ^ NP conjecture); see Jensen-Korte [106] and Lovasz [122], [123]. In this context, M^-concave functions on {0, l}-vectors form a tractable subclass of submodular set functions. Recall that an M^-concave function is submodular (Theorem 6.19) and that it can be maximized efficiently by algorithms in section 10.1. •

10.2.2

Schrijver's Algorithm

We explain here Schrijver's strongly polynomial algorithm for submodular function minimization. This algorithm achieves strong polynomiality using a distance labeling with an ingenious lexicographic rule. Following the basic framework introduced in section 10.2.1, the algorithm employs the representation of a base in terms of a convex combination of extreme bases associated with linear orderings. Given {(I/i,Ai) | i € /}, the algorithm constructs a directed graph G = (V, A) with arc set

(see (10.14) for the notation -
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Index the elements of V so that Lk = (vi, • - •, vn) and assume vp = s. Then we have vp+a = t and ( s , t } ^ k = {vp+i,...,vp+a}. For j = l , . . . , a , consider a linear ordering

which is obtained from Lk by moving vp+j to the position just before vp = s, and let Zj be the extreme base associated with L*. Proposition 10.17. For some S > 0, j/fc + 6(xt ~ Xs) can be represented as a convex combination of {zj \ j = 1 , . . . , a}. Proof. Put Vh = Lk(vh) = {vi,..., vh} for h = 1,..., n and VQ = 0. By (10.12) we have

and, therefore,

where the inequalities follow from submodularity; for instance, for h = p+j, we have

in which (Vp-i U {vp+j}) U Vp+j-i = Vp+j and (Vp-i U {vp+j}) n Vp+j-! = Vp-!. The sign pattern of (10.18), as well as that of (xt~Xs)(vh), for h with p < h < p+j looks like:

Note also that each row sum is equal to zero since Zj(V) = yk(V). If all the diagonal entries marked by @ are strictly positive, we can represent xt ~ Xs as a nonnegative combination of {zj —y^ \ j = 1 , . . . , a} with a positive coefficient for j — a; namely,

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X t ~ X s = E°=i »j(zi ~ yk) with //j > 0 (j = 1,..., a - 1) and p,a > 0. Then the claim is true for 5 = 1/($3?=1 Mj)- ^ a diagonal entry, say, in the j'oth row, vanishes, then Zj0 = yk and the claim is true for 6 = 0. D Define x = x + \k$(Xt — Xa)- This vector can be represented as a convex combination of Y" = {y, | i € / \ {fc}} U {zj \ j = 1,..., a} by Proposition 10.17 and (10.13). Let x' be the point on the line segment connecting x and x that is closest to x with the t-component x'(t) < 0. This means

Note that x' can be represented as a convex combination of Y U {yk} and, moreover, {yk} can be dispensed with if x'(t) < 0. By a variant of Gaussian elimination we can obtain a convex combination representation of x' using at most n vectors from Y U {yk}- We update {(Li, A,) | i e /} according to this representation. Since \Y\ + 1 < 2n, step S4 can be done with O(n3) arithmetic operations. Proposition 10.18. The number of iterations in Schrijver's algorithm is bounded by O(n 6 ). Hence, Schrijver's algorithm finds a minimizer of a submodular set function p : 2V —> R with O(n8) function evaluations and 0(n 9 ) arithmetic operations, where n = \V\. Proof. Denote by 0 the number of indices i £ I such that |(s,t]^J = a. Let x',d',A',P',N',t',s',a',/3' be the objects x,d,A,P,N,t,s,a,/3 in the next iteration. We first observe that a new arc appears only if it connects two vertices lying between s and t with respect to -<&: (a) For each arc (v, w) e A' \ A we have s - d(v) for all v € V and, (c) if d'(v) = d(v) for all v 6 V, then (d'(t'),t',s',a',/3') smaller than (d(t),t,s,a,(3).

is lexicographically

Proof of (b): Note that P' C P. If (b) fails, there exists an arc (v, w)eA'\A with d(w) > d(v) + 2. By (a) we have s -
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j — 1,..., a, (s, £]_<• is a proper subset of (s, t]->k. This implies a' < a. If a' = a, then /3' < (3, since x'(t) = x'(t') < 0 and L^ disappears in the update. Hence (c). It follows from (c) that d(v) increases for some v € V in O(n 4 ) iterations because each of d(t), t, s, a, (3 is bounded by n and there are at most n pairs (d(t),t) if d does not change. For each v € V, d(v) can increase at most n times. Therefore, the total number of iterations is bounded by O(n 6 ). D Note 10.19. A more detailed analysis of Vygen [208] yields an improved bound of O(n 5 ) on the number of iterations in Schrijver's algorithm. •

10.2.3

Iwata-Fleischer-Fujishige's Algorithm

We explain here Iwata-Fleischer-Fujishige's strongly polynomial algorithm for submodular function minimization. While sharing the basic framework of section 10.2.1, this algorithm differs substantially from Schrijver's in that it is based on a scaling technique rather than distance labeling. Weakly Polynomial Scaling Algorithm

We start with a scaling algorithm for minimizing a submodular set function p : 2V —> R. The algorithm is weakly polynomial for integer-valued p. It is emphasized that the value of M in (10.8) need not be computed. Recall from Proposition 10.8 that the problem dual to minimizing p is to maximize x~(V) over x e B(/o). To add flexibility in solving this maximization problem we introduce a scaling parameter 6 > 0 to relax (or enlarge) the feasible region B(p) to B(p + Kg), where

The function Kg is submodular and therefore B(p + K,j) — B(p) + B(K^) by Theorem 4.23 (1). For a concrete representation of B(K^) we observe that Kg is the cut capacity function associated with a complete directed graph G = (V, A), where

and the arc capacities are all equal to 6; indeed, Kg coincides with K in (9.16) with c(a) = 8 and c(a) = 0 for every arc a € A. By a S-feasible flow we mean a function if : A —> R such that for each (u,v) € A we have (i) 0 <
(10.21)

The algorithm consists of scaling phases, each of which corresponds to a fixed parameter value 5. We start with an arbitrary linear ordering, the extreme base x associated with it, zero flow (p = 0, and

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where x+ is the vector in Rv defined by x+(v) = m&x(Q,x(v)) for v € V. In each scaling phase, we construct an approximate solution to (10.21) from a given pair of a base x and a ^-feasible flow

Proof. Since S C W C V \ T, we have z(v) < 6 for every v £ W and z(v) > —5 for every v € V \ W. Therefore, Since x(W) = p(W] by the nonexistence of active triples and Proposition 10.9 (2) and d 0 by the nonexistence of arcs leaving W in Gv, we have z~(V) > p(W) — n5. Since dtp(v) < (n — 1)5 for every v 6 V, we have

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With 5 < A/n 2 we have x~(V) > p(W) - n25 > p(W) - A, whereas x~(V) < p(Y) for all Y C V by (10.10). Hence W is a minimizer of p. D On the basis of Proposition 10.20 above we terminate the scaling phase if neither augmenting path nor active triple exists. Otherwise, while keeping z invariant, we aim at "improving the situation" by either 1. eliminating an active triple or 2. enlarging the reachable set W. With a view to eliminating an active triple (i, u, v) we modify Li by swapping u and v in Li; denote the old pair (L/i,yi) by (Lk,yk) with a new index fc. The extreme base associated with the updated Li is given by yi — yk + /3(Xu — Xv) with

(see (10.12)). Defining a = mm((f>(u,v),Xi(3)

let us modify x and ip by setting

Then z = x + dip is invariant and ip remains ^-feasible. The updated x is equal to

which is a convex combination of {yj j € / U {k}}. In the saturating case where a = \i/3, the old extreme base yk disappears from (10.25). Since u precedes v in the new L^, the active triple (i,u,v) is successfully eliminated, whereas the size of the index set / remains the same. In the nonsaturating case where a < Aj/3, the old extreme base y^ remains and the size of I increases by one. Nevertheless, the situation is somewhat improved; namely, the reachable set W is enlarged to contain v as a result of (p(u, v) = 0 for the updated flow (p. The above task is done by the procedure Double-Exchange(z,M, v) below. Procedure Double-Exchange(i,«, v) SI: Set /3 := p(Li(u) \ {v}} - p(Lt(u)) + yi(v), a := mm(^(u, v), A;/3). S2: If a < \i/3, then let k be a new index and set / := 7 U {k}, Afc := A* - a//?, A* := a/j3, yk := j/», Lk := Lt. S3: Set ^ := yi + j3(\u - Xv), x := x + a(xu ~ Xv),
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IFF scaling algorithm for submodular function minimization

SO: Take any linear ordering L\ and let y\ be the associated extreme base. If 3/i~(V) = 0, then output V as a minimizer and stop. If l / i ( V ) = 0, then output 0 as a minimizer and stop. Set / := {!}, A! := 1, x := j/i, tp := 0, S := mm{x+(V), \x~ (V)\} / H* . SI: Let W be the set of vertices reachable from S in Gv. S2: If W n T ^ 0, then let P be a ^-augmenting path, apply Augment(<£>, P) and Reduce(x,I), and go to SI. S3: If there exists an active triple, then apply Double-Exchange to an active triple (i,u,v) and go to SI. S4: If S < A/n 2 , then output W as a minimizer and stop. S5: Apply Reduce(o;,/), set 5 := 5/2, and

Z with O(n5 Iog2 M) function evaluations and arithmetic operations, where n = \V\ and M = max.{\p(X}\ X C V}. Proof. This can be derived from the properties listed in Proposition 10.22.

D

Proposition 10.22.

(1) The number of scaling phases is O(log 2 (M/A)). (2) The first scaling phase calls Augment O(n 2 ) times. (3) A subsequent scaling phase calls Augment O(n 2 ) times. (4) Between calls to Augment, there are at most n — 1 calls to nonsaturating Double-Exchange. (5) Between calls to Augment, there are at most 2n3 calls to saturating DoubleExchange. (6) We always have \I\ < In. (7) Reduce(a;, /) with \I\ < 2n can be done in O(n 3 ) arithmetic operations. Proof. (1) We have 6 < M/n2 in step SO, since x+(V) = x(X) < p(X) < M for X = supp+(:c). The number of scaling phases is bounded by log 2 ((M/n 2 )/(A/n 2 )) = log2(M/A). (2) Let x denote the initial base in step SO. Then z (V) = x (V) at the beginning of the scaling phase. Throughout the scaling phase we have z~(V) < z(V) = x(V) as well as z~(V) < 0. Since Augment increases z~(V) by <5, the number of calls to Augment is bounded by

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(3) At the beginning of a subsequent scaling phase, we have z~(V) > p(W) —nS by Proposition 10.20, where z = x + dtp with (p = 2ip and 6 — 26. Since

this implies that z~(V) > p(W) — 2n8 — «2<5/4 at the beginning of the scaling phase. Throughout the scaling phase we have

Therefore, the number of calls to Augment is bounded by (2n5 + n25/2)/S = 2n + n2/2. (4) Each nonsaturating Double-Exchange adds a new element to W. (5), (6) A call to Reduce results in |/| < n. A new index is added to / only in a nonsaturating Double-Exchange. By (4), |/| grows to at most 2n — 1. Hence, the number of triples (i,u,v) is bounded by 2n3. (7) Reduce can be performed by a variant of Gaussian elimination. D The following is a key property of the scaling algorithm that we make use of in designing a strongly polynomial algorithm. Recall that a scaling phase ends when the algorithm reaches step S4. Proposition 10.23. At the end of a scaling phase with parameter S, the following hold true. (1) If x(w) < —n28, then w is contained in every minimizer of p. (2) // x(w) > n2S, then w is not contained in any minimizer of p. Proof. We have x~(V) > p(W) — n2S by Proposition 10.20, whereas, for any minimizer X of p, we have p(W) > p(X) > x(X) > x ~ ( X ) . Hence, x~(V) > x ~ ( X ) - n25. Therefore, if x(w) < —n2S, then w e X. On the other hand,

Therefore, if x(w] > n2S, then w £ X.

D

Strongly Polynomial Fixing Algorithm

Using the scaling algorithm as a subroutine, we can devise a strongly polynomial algorithm for submodular function minimization. The strongly polynomial algorithm, which we call the IFF fixing algorithm, exploits two fundamental facts: • The minimizers of p form a ring family (see Note 4.8).

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• A ring family can be represented as the set of ideals of a directed graph (see Note 4.7). Let D° be the directed graph representing the family T>° of minimizers of p. This means that W C V is a minimizer of p if and only if it is an ideal of D° such that min£>° C W C max 2}°. The algorithm aims at constructing the graph by identifying arcs of D° or elements of (mini?0) U (V \ max£>°) one by one with the aid of the scaling algorithm. To be more specific, we maintain an acyclic graph D = (U, F) and two disjoint subsets Z and H of V such that • Z is included in every minimizer of p, i.e., Z C min£>°; • H is disjoint from any minimizer of p, i.e., H C V \ maxX>°; • each u € U corresponds to a nonempty subset, say, F(u), of V, and (F(u) | u e U} is a partition of V \ (Z U H); • for each u & U and any minimizer W of p, either F(u) C W or F(w) f~l W = 0; • an arc (u, w) € F implies that every minimizer of p including T(u) includes F(w;) Using the notation T(Y) = Uusr ^(u) f°r ^ — U, we define a function p : 2U —»• Rby It is easy to verify that • p is submodular; • a subset W of V is a minimizer of p if and only if W = T(Y) U Z for a minimizer V C U of /5; • an arc (u, w) G F implies that every minimizer of p containing u contains w; i.e., a minimizer of p is an ideal of D. The algorithm consists of iterations. Initially we set U := V, Z := 0, H := 0, and F :— 0. At the beginning of each iteration, we compute

where R(u) denotes the set of vertices reachable from u £ U in D. If rj < 0, we are done by Proposition 10.24 below. Otherwise, we either enlarge Z Li H or add an arc to D, where directed cycles that may possibly arise in this modification are contracted to a single vertex; the partition F of V \ (Z U H) is modified accordingly. Proposition 10.24. // 77 < 0, then V \ H is the maximal minimizer of p. Proof. Let Y be the unique maximal minimizer of p. If Y ^ U, there is an element u £ U \Y such that Y U {u} is an ideal of D. By Y U {u} 2 R(u) and the submodularity of p, we have

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Chapter 10. Algorithms

which contradicts the definition of Y. Thus, U is the maximal minimizer of p and hence F(t/) U Z = V \ H is the maximal minimizer of p. D Suppose that 77 > 0 and let u G U be the vertex that attains the maximum in (10.26). Then and we have at least one of the following three cases: (i) p(U) > 77/8, (ii) p(R(u) \ {u}) < -77/3, or (iii) p(R(u)) - p(U) > 77/8. Case (i): If p(U) > 77/8, we invoke a procedure Fix + (p, D, 77), described below, to find an element w 6 U that is not contained in any minimizer of p. Since T(w) cannot be included in any minimizer of p, we add T(w) to H and delete w from D. Case (ii): If p(R(u) \ {u}) < — 77/8, we invoke another procedure Fix~(/5, D, 77), described below, to find an element w € U that is contained in every minimizer of p. Since T(w) must be included in every minimizer of p, we add T(w) to Z and delete w from D. Case (iii): If p(R(u)) — p(U) > 77/8, we consider the contraction of p by R(u), which is a submodular function p* on U \ R(u) defined by

and find an element w G U \ R(u) that is contained in every minimizer of p*. As explained below, we can do this by applying Fix" to (p*,D*,/7), where D* means the subgraph of D induced on the vertex set U \ R(u). A subset X C. U \ R(u) is a minimizer of p* if and only if X U R(u) minimizes p over subsets of U containing R(u). Therefore, if a minimizer of p containing u exists, then it must contain w. Equivalently, if a minimizer of p including T(u) exists, then it must include F(u>). Accordingly, we add a new arc (u, w) to F, where the arc (u, w) is new because w ^ R(u). If the added arc yields directed cycles, we contract the cycles to a single vertex, with corresponding modifications of U and F. Thus, in each iteration with 77 > 0, we either enlarge Z U H or add a new arc to D. Therefore, after at most r? iterations, we can terminate the algorithm with 77 < 0 when we have a minimizer of p by Proposition 10.24. The procedures Fix" (p, D, 77) and Fix + (p, D,rj) are as follows. Given a submodular function p : 2U —> R, an acyclic graph D — (U,F), and a positive real number 77 such that

the procedure F\x~(p,D,r]) finds an element w & U that is contained in every minimizer of p. Similarly, Fix+(/5, D, 77) finds an element w G U that is not contained in any minimizer of p when The procedures F\x.~(p,D,r]) and Fix + (/5, D,rj) are the same as the IFF scaling algorithm except that they start with 6 = 77 and a linear extension of the partial order represented by D and return w in step S4. We put h = \U\.

10.2.

Minimization of Submodular Set Functions

303

Procedure Fix~(/5, D, 77) SO: Take any linear extension LI of the partial order represented by D and let yi be the associated extreme base. Set / := {!}, A! := 1, x :— yi,

/2, and go to SI. Procedure F\x+(p,D,rj) is identical to Fix" (p,D,n) except that step S4 is replaced with S4: If there exists w G U with x(w) > n2S, then return such a w. The correctness of the above procedures at the termination in step S4 is guaranteed by Proposition 10.23. As to the complexity we have the following as well as Proposition 10.22 (3)-(7). Proposition 10.25. The following statements hold true for Fix ± (p, D,rj). (1) The number of scaling phases is O(log2 n), where n = \U . (2)//

then the first scaling phase calls Augment O(n) times. Proof. (1) Assume 8 < n/(3n3). By (10.28) we have x(U) = p(U) > ry/3 > hz5 and hence x(w) > n26 for some w & U. Therefore, the number of scaling phases in Fix+ is bounded by [Iog2(3n3)] = O(log 2 n). For Y in (10.27), we have x(Y) < p(Y) < —rj/3 < -n38 and hence x(w) < -n28 for some w G Y. Therefore, the number of scaling phases in Fix" is bounded by O(log2 n). (2) Let x denote the initial base in step SO. By the proof of Proposition 10.22 (2), the number of calls to Augment is bounded by x+(U)/S, whereas x+(U) < nrj by (10.29). Since S — 77, the number of calls to Augment is bounded by n. D The applications of Fix* in the IFF fixing algorithm are legitimate. Proposition 10.26. Let r? be defined by (10.26). (1) Conditions (10.28) and (10.29) are satisfied by (p,D,r)) in Case (i). (2) Conditions (10.27) and (10.29) are satisfied by (p,D,n) in Case (ii). (3) Conditions (10.27) and (10.29) are satisfied by (p*,D#,rj) in Case (in). Proof. In Case (i), (10.28) is obviously satisfied. Condition (10.27) is satisfied with Y = R(u) \ {w} in Case (ii) and Y = U \ R(u) in Case (iii). To show (10.29) for

304

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(p,D,rj), let y be an extreme base generated by a linear extension of the partial order of D. For each u € U we have y(u) = p(Y) — p(Y \ {u}) for some Y D R(u), whereas

by the submodularity of p and the definition of n. This proves (10.29) for (p, D, 77). Finally, for each u e U \ R(u), put J?*(u) = R(u) \ R(u) and observe

This shows (10.29) for (p,,D*,ri).

D

We are now in the position to assert the correctness and the strong polynomiality of the IFF fixing algorithm. Proposition 10.27. The IFF fixing algorithm finds a minimizer of a submodular set function p : 2V —+ R with O(n 7 log 2 n) function evaluations and arithmetic operations, where n = |V|. Proof. This follows from Proposition 10.25 and Proposition 10.22 (3)-(7).

D

Finally, we note that the maximal minimizer is found. Proposition 10.28. The IFF fixing algorithm finds the maximal minimizer of p. Proof. This follows from Proposition 10.24.

D

Note 10.29. For minimization of a submodular set function p : 2V —> R U {+00} defined effectively on a general ring family, the IFF scaling/fixing algorithm can be applied to the associated finite-valued submodular function ~p in Note 10.14. Alternatively, the IFF scaling/fixing algorithm can be tailored to this general case if the ring family is represented as the set of ideals of a directed graph (V, E). The min-max relation (10.11) in Proposition 10.8 holds true in this general case. The representation (10.13) of a base x as a convex combination of extreme bases y, (i € /) should be augmented by an additional term 9£ as

where 9£ is the boundary of a nonnegative flow £ : E —> R+. Then we have

10.3.

Minimization of L-Convex Functions

305

where tp is a flow in the complete graph on V representing the superposition of £ and (p. It is possible to design an algorithm that finds the minimum of p by maintaining the extreme bases and the flow tp. See Iwata [99] for more details. •

10.3

Minimization of L-Convex Functions

Three kinds of algorithms for L-convex function minimization are described: the steepest descent algorithm, the steepest descent scaling algorithm, and the reduction to submodular function minimization on a distributive lattice. All of them depend heavily on the algorithms for submodular function minimization in section 10.2. Throughout this section g : Zv —> R U {+00} denotes an L- or L''-convex function with \V\ = n. For an L-convex function g, it is assumed that

since otherwise g does not have a minimum.

10.3.1

Steepest Descent Algorithm

The local characterization of global minimality for L-convex functions (Theorem 7.14) naturally leads to the following steepest descent algorithm. Steepest descent algorithm for an L-convex function g € £[Z —» R] SO: Find a vector p e domg. Si: Find X C V that minimizes g(p + xx)S2: If g(p) < g(p + X x ) , then stop (p is a minimizer of g). S3: Set p := p + xx and go to SI. Step SI amounts to minimizing a set function

over all subsets X of V. As a consequence of the submodularity of g, pp is submodular and can be minimized in strongly polynomial time by the algorithms in section 10.2. At the termination in step S2, p is & global minimizer by Theorem 7.14 (1) (L-optimality criterion). The function value g decreases monotonically with iterations. This property alone does not ensure finite termination in general, but it does if g is integer valued and bounded from below. We introduce a tie-breaking rule in step SI:

Thus, we can guarantee an upper bound on the number of iterations. Let p° be the initial vector found in step SO. If g has a minimizer at all, it has a minimizer p* satisfying p° < p* by (10.31). Let p* denote the smallest of such minimizers, which exists since p* A q* € argming for p*, q* € argmin
306

Chapter 10. Algorithms

Proposition 10.30. In step SI, p < p* implies p + xx < P*• Hence the number of iterations is bounded by \\p° - p*\\iProof. Put Y = {v e V | p(v) — P*(v)} and p' = p + xx- By submodularity we have whereas g(p*) < g(p*Vp') since p* is aminimizer of g. Hence g(p') > g(p*/\p'). Here we have p' = p + Xx and p* A p' — p + Xx\Y, whereas X is the minimal minimizer by the tie-breaking rule (10.33). This means that X \ Y = X; i.e., X n Y = 0. Therefore, p' = p + xx < P*- D It is easy to find the minimal minimizer of pp using the existing algorithms for submodular set function minimization (see Notes 10.11, 10.12, and 10.13 and Proposition 10.28). Assuming that the minimal minimizer of a submodular set function can be computed with O(
where it is noted that domg itself is unbounded by (10.31). Proposition 10.31. For an L-convex function g with finite K\, the number of iterations in the steepest descent algorithm with tie-breaking rule (10.33) is bounded by KI . Hence, if a vector in dom g is given, the algorithm finds a minimizer of g in O((a(n)F + r(n))Ki) time. Proof. We have \\p° — p*||i < KI since p°(v) = p*(v) for some v e V. Then the claim follows from Proposition 10.30. D The steepest descent algorithm can be adapted to L''-convex functions. Let g be an L''-convex function and recall from (7.2) that it is associated with an L-convex function g as The steepest descent algorithm above applied to this L-convex function g yields the following algorithm for the L''-convex function g. Steepest descent algorithm for an L^-convex function g G £&[Z —>• R] SO: Find a vector p e doing. SI: Find e € {1, —1} and X C V that minimize g(p + £Xx)S2: If g(p) < g(p + exx), then stop (p is a minimizer of g). S3: Set p := p + sxx and go to SI. Step SI amounts to minimizing a pair of submodular set functions

10.3.

Minimization of L-Convex Functions

307

Let X+ be the minimal minimizer of p+ and X~ be the maximal minimizer of p~. The tie-breaking rule in step SI above reads

This is a translation of the tie-breaking rule (10.33) for g in (10.35) through the correspondence

where p = (0,p) & Z x Zv. Since (l,Xv\x-) cannot be minimal in the presence of (0, Xx+); we choose (1, X+) in the case of min/9+ = min/9~. At the termination in step S2, p is a global minimizer by Theorem 7.14 (2) (L-optimality criterion). In view of the complexity bound given in Proposition 10.31 we will derive a bound on the size of dom g in terms of the size of dom g. Let K\ (g) be denned by (10.34) for g. The li-size and ^-size of doing are denoted, respectively, by

Proposition 10.32. K^(g) < KI + nKx < min[(n + 1)^,

2nK00\.

Proof. Take p = {po,p} and q = (qo,q) in dom g such that Ki(g) = \po — qo\ + \\p - q\\i and either (i) po = qQ or (ii) p(v) = q(v) for some v € V. We may assume Po > Qo and p > q since p V q,p A q e domg and \\(p V q) - (p A q)\\i = \\p - q\\iThe vectors p' = p — pol and q' = q — qol belong to domg. In case (i), we have Ki(g] = \\p-q\\\ = \\p'~
Note finally that KI < nKx and Kx < KI.

D

The steepest descent algorithm could be used for minimizing quasi L-convex functions satisfying (SSQSBW) because of Theorem 7.53 (quasi L-optimality criterion). Note, however, that the set function pp of (10.32), to be minimized in step SI, is not necessarily submodular and hence no efficient procedure is available for step SI.

308

10.3.2

Chapter 10. Algorithms

Steepest Descent Scaling Algorithm

The steepest descent algorithm for L-convex function minimization can be made more efficient with the aid of a scaling technique. The efficiency of the resulting steepest descent scaling algorithm is guaranteed by the complexity analysis in section 10.3.1 combined with the proximity theorem for L-convex functions. The algorithm for an L-convex function g with (10.31) reads as follows, where

Steepest descent scaling algorithm for an L-convex function g G JC.[Z —*• R] SO: Find a vector p € domg and set a := 2^l°S2(Kaa/'2n)] SI: Find an integer vector q that locally minimizes g(q) = g(p + aq) in the sense of g(q) < g(q + Xx) (VX C V) by the steepest descent algorithm of section 10.3.1 with initial vector 0 and set p := p + aq. S2: If a = 1, then stop (p is a minimizer of g). S3: Set a := a/2 and go to SI. Note first that the function g(q) = g(p + aq) is an L-convex function. By the L-proximity theorem (Theorem 7.18 (1)), there exists a minimizer q of g satisfying 0 < q < (n - 1)1. Then, by Propositions 10.30 and 10.31, the steepest descent algorithm with tie-breaking rule (10.33) finds the minimizer in step SI in O((cr(n)F + r(n))n 2 ) time, where
10.3.3

Reduction to Submodular Function Minimization

The effective domain of an L''-convex function g is a distributive lattice (a sublattice of Zv) on which g is submodular. Hence we can make use of submodular function minimization algorithms as adapted to functions on distributive lattices (see Note 10.15). If the £i-size of domg is given by KI, domg is isomorphic to a sublattice of the Boolean lattice 2V for a set V of cardinality KI . Hence, the complexity of this algorithm is polynomial in n and K\. It may be noted, however, that, being dependent only on the submodularity of g, this approach does not fully exploit L^-convexity.

10.4

Algorithms for M-Convex Submodular Flows

Five algorithms for the M-convex submodular flow problem are described: the twostage algorithm, the successive shortest path algorithm, the cycle-canceling algo-

10.4. Algorithms for M-Convex Submodular Flows

309

rithm, the primal-dual algorithm, and the conjugate scaling algorithm. Because the optimality criterion for the M-convex submodular flow problem is essentially equivalent to the duality theorems for M-/L-convex functions, these algorithms can be used for finding a separating afnne function in the separation theorem and the optimal solutions in the minimization/maximization problems in the Fenchel-type duality.

10.4.1

Two-Stage Algorithm

This section is intended to provide a general structural view on the duality nature of the M-convex submodular flow problem. It is based on the recognition of the M-convex submodular flow problem as a composition of the Fenchel-type duality and the minimum cost flow problem that does not involve an M-convex function. The algorithm presented in this section, called the two-stage algorithm, computes an optimal potential by solving an L-convex minimization problem in the dual problem and constructs an optimal flow as a feasible flow to another submodular flow problem. As an adaptation of our discussion in section 9.1.4, the relationship between the M-convex submodular flow problem MSFPs and the Fenchel-type duality may be summarized as follows. To be specific, we consider the integer-flow version of MSFP3 on the graph G = (V, A) with / <E M[Z -> Z] and /„ e C[Z -> Z] for a e A. M-convex submodular flow problem MSFP3 (integer flow)

We assume the existence of an optimal solution. First, we identify the problem dual to MSFPs and indicate how to compute an optimal potential. With the introduction of a function

we obtain

where / e M[Z -> Z] and fA € M(Z -> Z]. Putting g = /*, ga = /0* for a € A, and

310

Chapter 10. Algorithms

we have gA = fA* (see (9.28)) and also g e £[Z -> Z], #0 e C[Z -> Z] for a € v4, and gA e £[Z -> Z]. The Fenchel-type duality (Theorem 8.21 (3)) gives

which is equivalent to61

The function

to be minimized on the right-hand side of (10.44) is an L-convex function and a minimizer of g is an optimal potential for MSFPa, and vice versa, in the sense of Theorem 9.16. Next we discuss how to construct an optimal flow. Let p* be a minimizer of g and define c* : A -> Z U {-oo}, c* : A -> Z U {+00}, and B* C Zv by

where B* is an M-convex set by Proposition 6.29. Since p* is an optimal potential, a flow £* is optimal if and only if

The two-stage algorithm is described as follows. Two-stage algorithm for MSFP3 (integer flow) SI: Find a minimizer p* of g in (10.45). S2: Find a flow £* satisfying (10.48). The feasibility of this approach is guaranteed by the following facts if the given functions, / and fa for a € A, can be evaluated. 1. We can evaluate g by applying an M-convex function minimization algorithm to /. Similarly, we can evaluate ga for a e A. 2. We can find a minimizer p* in step SI by applying an L-convex function minimization algorithm to g. 3. We can find a member of B* by applying an M-convex function minimization algorithm to /[—p*}. 4. We can find a flow £* in step S2 as a feasible flow to the submodular flow problem defined by c*, c*, and B*. This can be done, e.g., by the successive shortest path algorithm described in section 10.4.2. 61

We have seen (10.44) in (9.83) as a special case of Theorem 9.26 (3).

10.4.

Algorithms for M-Convex Submodular Flows

10.4.2

311

Successive Shortest Path Algorithm

We present the successive shortest path algorithm to find a feasible integer flow in an integral submodular flow problem. We adopt the most primitive form of the algorithm to better explain the basic idea without being bothered by technicalities. Given a graph G = (V,A), an upper capacity c : A —> Z U {+00}, a lower capacity c : A —> Zu{—oo}, and an M-convex set B C Zv, we are to find an integer flow £ : A —> Z satisfying

It is assumed that c(a) < c(a) for each a e A. The algorithm maintains a pair (£, z) e ZA x Zv of integer flow £ satisfying (10.49) and base x e B and repeats modifying (£, z) to resolve the discrepancy between d£ and x. For such (£,x) let G^,x = (V,j4j >x ) be a directed graph with vertex set V and arc set A^tX = A*c U B£ U Cx consisting of three disjoint parts:

and define

In order to reduce the discrepancy ||z - d£\\i = X^ev \x(v) ~ <9£(v)|, tne algorithm augments a unit flow along a shortest path P from S+ to S~ (shortest with respect to the number of arcs) and modifies £ to £ given by

Obviously, £ satisfies the capacity constraint (10.49). The algorithm also updates the base x to

which remains a base belonging to B; see Note 10.33. For the initial vertex s G S+ of the path P, either d£(s) increases or x(s) decreases by one and hence |z(s) — <9£(s)| reduces by one. A similar result is true for the terminal vertex of P in S~, whereas x(v) = d£(v) is kept invariant at every inner vertex v of P. Therefore, each augmentation along a shortest path decreases ||z —
312

Chapter 10. Algorithms

Successive shortest path algorithm for finding a feasible integer flow SO: Find £ e ZA satisfying (10.49) and x <E B. SI: If S+ is empty, then stop (£ is a feasible flow). S2: If there is no path from S+ to S~, then stop (no feasible flow exists). S3: Let P be a shortest path from S+ to S~ and. for each arc a £ P. set

and go to SI.

Note 10.33. The updated vector x in (10.52) remains a base, i.e., x e B, by Proposition 9.23 with / = SB (indicator function of B). Let G(x, x) be the bipartite graph, as defined in section 9.5.2. All the arcs have zero weight. Hence, (x, x) meets the unique-min condition if and only if G(x, x) has a unique perfect matching. The latter condition holds in the algorithm because P is chosen to be a shortest path, whereas Proposition 9.23 says that the unique-min condition implies x £ B. • Note 10.34. Instead of augmenting a unit flow along P it is more efficient to augment as much as possible. The maximum admissible amount is given by 6 = min{c(a) | a € P} with

where cg(-, •, •) means the exchange capacity defined in (10.4). It can be shown that stays in B; see Lemma 4.5 of Fujishige [65]. The successive shortest path algorithm can be adapted to finding a real-valued feasible flow to a nonintegral submodular flow problem. For a polynomial complexity bound of the algorithm, it is important to choose, in the case of multiple candidates, an appropriate shortest path with reference to some lexicographic ordering. The successive shortest path algorithm can be generalized for optimal flow problems involving cost functions; see [65], Fujishige-Iwata [66], and Iwata [98] for such algorithms for MSFPi (without Mconvex cost), whereas such an algorithm for the integer-flow version of MSFP2 (with M-convex cost) is given in Moriguchi-Murota [132]. • Note 10.35. A number of algorithms are available for finding a feasible submodular flow; e.g., Fujishige [59], Frank [56], Tardos-Tovey-Trick [199], and Fujishige-Zhang [70]. The reader is referred to Fujishige [65], Fujishige-Iwata [66], and Iwata [98] for expositions. • 10.4.3

Cycle-Canceling Algorithm

For the M-convex submodular integer-flow problem with linear arc cost, we have seen in section 9.5 that a feasible flow is optimal if and only if there exists no negative

10.4. Algorithms for M-Convex Submodular Flows

313

cycle in an auxiliary network (Theorem 9.20) and that a nonoptimal flow can be improved by augmenting a flow along a suitably chosen negative cycle (Theorem 9.22). These two facts suggest the following cycle-canceling algorithm, which works on the auxiliary network (G^,^) introduced in section 9.5. Cycle-canceling algorithm for MSFP2 (integer flow) SO: Find a feasible integer flow £. SI: If (G^,^) has no negative cycle, then stop (£ is an optimal flow). S2: Let Q be a negative cycle with the smallest number of arcs. S3: Modify £ to £ of (9.75) and go to SI. The objective function I^ decreases monotonically by Theorem 9.22. This property alone does not ensure finite termination in general, but it does if F^ is integer valued and bounded from below. In the special case with dom/ C {0,1}^, which corresponds to the valuated matroid intersection problem (Example 8.28), some variants of the cycle-canceling algorithm are known to be strongly polynomial (see Murota [136]). Cycle-canceling algorithms can also be designed for problems with real-valued flow on the basis of Theorem 9.18. Note 10.36. For MSFPi (submodular flow problem with linear arc cost and without M-convex cost), a number of cycle-canceling algorithms are proposed, including Fujishige [59], Cui-Fujishige [27], Zimmermann [222], Wallacher-Zimmermann [210], and Iwata-McCormick-Shigeno [103]. •

10.4.4

Primal-Dual Algorithm

A primal-dual algorithm for the M-convex submodular flow problem is described. The algorithm maintains a pair of flow and potential and modifies them to optimality. We deal with the case of linear arc cost with dual integrality. M-convex submodular flow problem MSFP2

Here, c : A -^ R U {+00}, c : A -> R U {-oo}, / € M[TL -> R Z], and 7 : A -> Z (see (9.67)). The feasibility of the problem is also assumed. By Theorems 9.14 and 9.15 as well as (9.31), (9.32), and (9.65), a feasible flow £ : A —> R is optimal if and only if there exists an integer-valued potential p : V —> Z such that

314

Chapter 10. Algorithms

where 7P : A —> Z is the (integer-valued) reduced cost defined by

gp : 2V —> R U {+00} is the submodular set function derived from g = f* e £[Z|R -> R] by

with gp(V) = 0 by (9.51), and B(gp) is the base polyhedron (M-convex polyhedron) associated with gp. The algorithm maintains a pair (£,p) 6 R"4 x Zv of a feasible flow £ and an integer-valued potential p that satisfies (10.59) and repeats modifying (£,p) to increase the set of arcs satisfying (10.57) and (10.58). We say an arc is in kilter with respect to (£,p) if it satisfies (10.57) and (10.58) and out of kilter otherwise. Note that exactly one of (10.57) and (10.58) fails for an out-of-kilter arc. An initial (£,p) satisfying (10.59) can be found as follows. For any feasible flow £ we consider a graph Gf — (V, Q) with vertex set V and arc set

and define the length of arc ( u , v ) as /'(<9£; — Xu + Xv), which is an integer by the assumed dual integrality (Note 9.19). By solving a shortest path problem we can find an integral vector p such that

which implies d£ 6 argmin/[-p] = B(gp) by (9.64) and (9.65). To classify out-of-kilter arcs we define

where the dependence on p is implicit in the notation. Note that D^(v) is the set of arcs leaving v for which (10.58) fails, D7 (v) is the set of arcs entering v for which (10.57) fails, and {D^(v) \ v £ V} gives a partition of the set of out-of-kilter arcs. If D^(v) = 0 for all v € V, condition (i) of (POT) is satisfied and the current (£,£>) is optimal. Otherwise, the algorithm picks62 any v* € V with nonempty D^(v*) and tries to meet the conditions (10.57) for a 6 D7(v*) and (10.58) for a e Dt(v*) by changing the flow to £'. The flow £' is determined by solving the following maximum submodular flow problem on G = (V, A) with a more restrictive capacity constraint c*(a) < £'(a) < c*(a) with 62

The original primal-dual algorithm' picks an out-of-kilter arc and tries to meet (10.57) and (10.58) for that arc. The present strategy to pick a vertex is to improve the worst-case complexity bound.

10.4.

Algorithms for M-Convex Submodular Flows

315

for each a 6 A. Maximum submodular flow problem maxSFP

where £' : A —> R is the variable to be optimized. Since the capacity interval [c* (a) ,c* (a)] is included in the original capacity interval [c(a),c(a)], any feasible flow in maxSFP is feasible in MSFP2. Note also that maxSFP has a feasible flow £' = £. In maximizing the objective function

it is intended to~ meet the conditions (10.58) and (10.57) by increasing the flow in a G D£(V*') to the upper capacity and decreasing the flow in a G D^(v*) to the lower capacity. The maximum submodular flow problem is a special case of the feasibility problem for the submodular flow problem and a number of efficient algorithms are available for it (see section 10.4.2). Let £' be an optimal solution to maxSFP above. If D^/(v*) is empty, the algorithm updates £ to £' without changing p. The condition (10.59) is maintained because of (10.65). If D^(v*) is nonempty, the algorithm finds a minimum cut W C V (explained later) and updates p to

as well as £ to £'. The condition (10.59) is maintained, as is shown in Proposition 10.38 below. The primal-dual algorithm for the M-convex submodular flow problem with a linear arc cost (MSFP2) with dual integrality is summarized as follows. Primal-dual algorithm for MSFPz with dual integrality SO: Find (£,p) e R/4 x Zv satisfying (10.54) and (10.59). SI: If D$(v) = 0 for all v 6 V, then stop ((£ ,p) is optimal). S2: Take any v* e V with D^(v*) ^ 0 and solve maxSFP to obtain £'. S3: If D£I(V*) ^ 0, then find a minimum cut W and set p := p + xwS4: Set £ := £' and go to SI. It remains to explain the minimum cut for maxSFP. For W C V containing v*, we define the cut capacity i/(W) by

316

Chapter 10.

Algorithms

Figure 10.1. Structure of G and G at v*.

where A+W and A W mean the sets of arcs leaving W and entering W, respectively, as in (9.14) and (9.15). The following proposition states that the flow value £'(D£(v*))-£'(Dt(v*)) is bounded by v(W] for any W C V with v* e W and that this bound is tight for some W, which is referred to as a minimum cut in the above. A minimum cut can be found with the aid of an appropriately defined auxiliary network. Proposition 10.37. In the maximum submodular flow problem maxSFP, we have

For a maximum flow £' and a minimum cut W, we have

Proof. We prove this by applying Theorem 9.13 (max-flow min-cut theorem) to a maximum submodular flow problem on a graph G = (V, A), which is obtained from G = (V,A) by a local modification at v* illustrated in Fig. 10.1. The vertex v* in G is split into two vertices, v* and v*, and a new arc OQ = (v*,i)*) is introduced; V — V U {{>*} and A = A U {a0}. The initial vertex of a € D£(V*) is changed to v* and the terminal vertex of a & D7(v*) is changed to v*. For W C V we denote by A + jy and A~W the sets of arcs leaving and entering W, respectively, in G. The problem maxSFP is equivalent to maximizing the flow £'(OQ) in OQ in G = (V,A), where the conservation of flow at v* (i.e., d£'(v*) = 0) is assumed and no

10.4. Algorithms for M-Convex Submodular Flows

317

capacity constraint is imposed on UQ. Note that £'(ao) = £'(Dt(v*))—t;'(D7(v*)) as a consequence of the flow conservation at v*. As for cuts, we note the correspondence between W C V with OQ e A+W and W C V with v* G W and observe the identities

for such a W. Then we obtain (10.68) from (9.61) in Theorem 9.13. Note that d(p + Xw) — g(p) = 9p(W) in v(W) corresponds to p in (9.61). Finally, (10.69), (10.70), and (10.71) are shown in (9.62). D The condition (10.59) is maintained when (£,p) is modified to (£',?/)• Proposition 10.38. #£' G B(<jy) for an optimum flow £' mmaxSFP and potential p' = p + xw with a minimum cut W. Proof. It follows from (10.65), (10.69), and discrete midpoint convexity (7.7) that

This shows #£' &B(gp>).

D

The following proposition shows the key properties for the correctness and complexity of the primal-dual algorithm. Proposition 10.39.

(1) The set of out-of-kilter arcs is nonincreasing. (2) For each arc a, |7P(a)| is nonincreasing while the arc stays out of kilter. (3) The potential is changed in at most \V\ iterations and, each time the potential is changed, the value of max.a€rj^(v*) |7p(a)| decreases at least by one. (4) Each time (£,p) is changed, the value of

decreases at least by one. Therefore, the primal-dual algorithm terminates in at most NO iterations, where NO denotes the value of N at step SO. Proof. The reader is referred to the kilter diagram in Fig. 9.1. (1) We show that an in-kilter arc a with respect to (£,p) remains in kilter in updating (£,p). It follows from £'(a) e [c*(a),c*(a)j and (10.62) that a remains in kilter with respect to (£',p). Suppose that p is updated to p' = p + \w- Since

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we may assume that a is in kilter with respect to (£',p) and (i) a G A+VP7 \ Dt(v*) or (ii) a e A~VF \ D^(v*). In case (i) we have £'(a) = c*(a) from (10.70), whereas

In case (ii) we have £'(a) — ^*(a) from (10.71), whereas

Thus the conditions (10.57) and (10.58) are preserved in either case. (2) By (10.73), it suffices to show that, if (i) 7p(a) > 0, a e A+VF, or (ii) 7P(a) < 0, a e A~W, then a is in kilter with respect to (£',p')- In case 0)i we have a e <\+W\D+(v*), from which follows £'(a) = c*(a) = c(a) by (10.70). Since 7P'(a) > 0, a is in kilter with respect to (£',p')> and similarly for case (ii). (3) If the potential does not change, we have D^i(v*) = 0 as well as D^(v) C D^(v) for all w 6 V^ by (1). Therefore, the potential must be updated in |V| iterations. Suppose now that p is changed to p'. An arc a & Df(v*).\ A+W is in kilter with respect to (£',£>'), since 7 P /(a) < 7P(a) < 0 and £'(a) = c*(a) = c(a) by (10.71). Similarly, a e D7(v*) \A~W is in kilter with respect to (£',p') because of (10.70). Fora e D+(w*)nA+VF, we have |7j/(a)| = |7P(a)|-l froni7 p /(a) =7 p (a) + 1 and 7P(a) < 0. Similarly, for a e D~^(v*) n A^H 7 , we have |7P'(a)| = 7 P (a)| - 1 from 7P'(a) = 7P(a) — 1 and 7P(a) > 0. Therefore, maxo6£){(«*) |7P(«)| decreases at least by one. (4) When the potential changes, TV decreases because of (3). When the potential remains invariant, N decreases because of Z?£/(u*) =0. D Primal-dual algorithms can also be designed for problems without dual integrality by using real-valued potential functions. Note 10.40. The framework of the primal-dual algorithm for MSFPi (submodular flow problem without M-convex cost) was established in Frank [55] and CunninghamFrank [30]. See Fujishige [65], Fujishige-Iwata [66], and Iwata [98] for expositions. •

10.4.5

Conjugate Scaling Algorithm

With the use of conjugate scaling of the M-convex cost function, the primal-dual algorithm is enhanced to a polynomial-time algorithm. We continue to deal with the M-convex submodular flow problem MSF?2 with dual integrality; i.e., we assume 7 : A -> Z and / e M[K. -> R Z]. First we explain the intuition behind the cost-scaling algorithm for the submodular flow problem MSFPi without M-convex function (see section 9.2 for MSFPi). As Proposition 10.39 (4) shows, the time complexity of the primal-dual algorithm depends essentially on

10.4.

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319

Motivated by this fact, we consider MSFPi with a new objective function

where a is a positive integer representing cost scaling and [ • ] means rounding up to the nearest integer. It is expected (or hoped) that such a scaling will result in smaller values of (10.74) and hence in an improvement in the computation time of the algorithm. On the other hand, the scaled problem with (10.75) is fairly close to the original problem, since a|~7(a)/a] ~ 7(0), and, therefore, the solution to the scaled problem is likely to be a good approximation that can be used as an initial solution in solving the original problem by the primal-dual algorithm. The scaling algorithm embodies the above idea by starting with a large a and successively halving a until a = 1. When an M-convex function / is involved, as in MSFP2, it is natural to try a scaling of the form [/(•)/<*]. This approach, however, does not seem to work in general, since \f(-)/oi\ is not necessarily M-convex for an M-convex function /. Conjugate scaling is a kind of scaling operation compatible with M-convexity. Let / : Rv —> Ru{+oo} be a polyhedral convex function with dual integrality in the sense that the conjugate function g = /* has integrality (6.75). Then we have where the supremum is taken over integer points. Replacing g(p) with ga(p) = g(ap)/a (p € Zv) in this expression we define

which we call the conjugate scaling of / with scaling factor a £ Z++. Note that /^ is again a dual-integral polyhedral convex function. It is easy to see that

and that doniR,/^ = doniR,/ provided /^ > —oo. Figure 10.2 illustrates the conjugate scaling of a univariate function / with a = 2. Proposition 10.41. For a dual-integral polyhedral M-convex function f £ M [R —> R|Z], we have /< a > e M[R -» R|Z] provided /<"> > -oo. Proof. We have /* = g & £[Z|R -> R] and hence ga e £[Z -» R] by Theorem 7.10 (2). Therefore, /< Q > = (ga)' e M\R, -> R|Z] by (8.10). D We are now in the position to present the conjugate scaling algorithm. Initially the algorithm finds a feasible flow £o and an integer-valued potential po satisfying (10.59) and applies the conjugate scaling to the objective function rewritten as

Chapter 10. Algorithms

320

Figure 10.2. Conjugate scaling f^

and scaling ga for a = 2.

where 7 = 7po and / = /[—p 0 ]. We denote the conjugate scaling of / by /^ and put g = /*. Note that / e M[H -> R|Z] and

where we regard g as a member of £[Z —> R]. Recall the notation n = \V\.

Conjugate scaling algorithm for MSFP2 with dual integrality

SO: Find a feasible flow £o G R"4 and an integer-valued potential po 6 Zv satisfying (10.59) and define 7, /, and g accordingly. Set p* := 0, K :== max a€A |7(a)|, a := 2^2 K] _ SI: If a < 1, then stop ((£,Po + p*) is optimal). S2: Find an integer vector p € Zv that minimizes ga(p) — (p,d£) subject to Ip* < p < 2p* + (n- 1)1. S3: Solve MSFP2 for (7,/) = (py/a], / < Q > ) by the primal-dual algorithm starting with (£,p) to obtain an optimal (£,*,P*) € R"4 x Zv. S4: Set £ := £* and a := a/2 and go to SI.

The correctness of the algorithm is ensured by Theorem 7.18 (L-proximity theorem), which implies that the minimizer p found in step S2 under the restriction 2p* < p < 2p* + (n — 1)1 is in fact a global minimizer of ga(p) — (p, d£). Hence, the condition <9£ € B(gap) = argmin/< a >[-p] for (10.59) is maintained. In step S2, the minimizer p can be found by the L-convex function minimization algorithms of section 10.3, where the number of evaluations of ga is bounded by a polynomial in n. Given /, an evaluation of ga amounts to minimizing a polyhedral M-convex function, since g(p) = sup{(p, x) — f ( x ) \ x (E R^}. If / has primal

10.4.

Algorithms for M-Convex Submodular Flows

321

integrality, i.e., if / G A4[Z|R —> R], theng(p) = swp{(p,x)—f(x) \ x e Zv}, which can be computed by the algorithms in section 10.1. In step S3, the number of iterations (updates of (£,p)) within the primal-dual algorithm is bounded by n2. Denote by pa the value of p at the beginning of step S3 and put p2a = p*, 7™ = |~7/of|, and -J2a = py/(2a:)]. Then we have

from which follows

This means that at the beginning of step S3 we have

for every out-of-kilter arc a and therefore N in (10.72) is bounded by n2. Obviously, steps SI through S4 are repeated flog2 K~\ times. For the value of K we have if the initial potential po in step SO is computed from a shortest path on the graph Gj = (V^Cf), as explained in section 10.4.4, where in the second term on the right-hand side we consider only those x, u, v for which f ' ( x ; —\u + Xv) is finite. If / 6 .M[Z|R —> R], the second term can be bounded as

which implies

Bibliographical Notes The tie-breaking rule (10.2) for M-convex function minimization, as well as Proposition 10.2, is due to Murota [148]. Variants of steepest descent algorithms are reported in Moriguchi-Murota-Shioura [133] with some computational results. Scaling algorithms for M-convex function minimization, including the one described in section 10.1.2, were considered first in [133], although the proposed algorithms run in polynomial time only for a subclass of M-convex functions that are closed under scaling. A polynomial-time scaling algorithm for general M-convex functions is given by Tamura [197]. The domain reduction algorithm in section 10.1.3 is due to Shioura [190] and its extension to quasi M-convex functions is observed in MurotaShioura [154]. The domain reduction scaling algorithm in section 10.1.4, with its extension to quasi M-convex functions, is due to Shioura [192]. Minimizing an Mconvex function on {0, l}-vectors is equivalent to maximizing a matroid valuation, for which a greedy algorithm of Dress-Wenzel [41] works; see also section 5.2.4 of Murota [146].

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The literature of submodular function minimization was described in Note 10.10. The algorithmic framework expounded in section 10.2.1 is due to Cunningham [28], [29] as well as Bixby-Cunningham-Topkis [15]. The algorithm in section 10.2.2 is by Schrijver [182], whereas an earlier version of this algorithm based on partial orders associated with extreme bases (presented at the Workshop on Polyhedral and Semidefinite Programming Methods in Combinatorial Optimization, Fields Institute, November 1-6, 1999) is described in Murota [147]. The algorithm in section 10.2.3 is due to Iwata-Fleischer-Fujishige [102]. Improvements on those algorithms in terms of time complexity were made by Fleischer-Iwata [50] and Iwata [100]. See McCormick [127] for a detailed survey on submodular function minimization. Note 10.12 was communicated by K. Nagano, Note 10.14 is based on [182], and Proposition 10.28 was communicated by S. Iwata. Favati-Tardella [49] proposes a weakly polynomial algorithm for submodular integrally convex function minimization. This is the first polynomial algorithm for L-convex function minimization, when translated through the equivalence between submodular integrally convex functions and lAconvex functions. The steepest descent algorithm for L-convex function minimization in section 10.3.1 is given in Murota [145]. The tie-breaking rule (10.33), as well as Proposition 10.31, is due to Murota [148]. The steepest descent scaling algorithm in section 10.3.2 is due to S. Iwata (presented at Workshop on Matroids, Matching, and Extensions, University of Waterloo, December 6-11, 1999), where step SI is performed not by the steepest descent algorithm but by the algorithm in section 10.3.3. The framework of M-convex submodular flow problems is advanced by Murota [142]. The successive shortest path algorithm for a feasible flow described in section 10.4.2 originates in Fujishige [59] and the present form is due to Frank [56]. The cycle-canceling algorithm of section 10.4.3 is devised in [142] as a proof of the negative-cycle criterion for optimality (Theorem 9.20). In the special case of valuated matroid intersection, the algorithm can be polished to a strongly polynomial algorithm (Murota [136]); see also Note 10.36. The primal-dual algorithm of section 10.4.4 is due to Iwata-Shigeno [105]; see also Note 10.40. A strongly polynomial primal-dual algorithm for the valuated matroid intersection problem is given in [136]. The conjugate scaling algorithm of section 10.4.5 is due to [105]. A scaling algorithm for a subclass of the M-convex submodular flow problem is given by Moriguchi-Murota [132]. Capacity scaling algorithms for submodular flow problems (without M-convex costs) are given in Iwata [97] and Fleischer-Iwata-McCormick [51]. For other algorithms for submodular flow problems (without M-convex costs), see the book of Fujishige [65] and surveys of Fujishige-Iwata [66] and Iwata [98].

Chapter 11

Application to Mathematical Economics

This chapter presents an application of discrete convex analysis to a subject in mathematical economics: competitive equilibria in economies with indivisible (or discrete) commodities. For economies consisting of continuous commodities, represented by real-valued vectors, a rigorous mathematical framework was established around 1960 on the basis of convexity, compactness, and fixed-point theorems. For indivisible commodities, however, no general mathematical framework seems to have been established. Such a framework, if any, should embrace both convexity and discreteness; the present theory of discrete convex analysis appears to be a promising candidate for it. It is shown that, in an Arrow-Debreu type model of competitive economies with indivisible commodities, an equilibrium exists under the assumption of the M^-concavity of consumers' utility functions and the M'-'-convexity of producers' cost functions. Moreover, the equilibrium prices form an L''-convex polyhedron, and, therefore, they have maximum and minimum elements. The conjugacy between M-convexity and L-convexity corresponds to the relationship between commodities and prices.

11.1

Economic Model with Indivisible Commodities

As an application of discrete convex analysis we deal with competitive equilibria in economies with a number of indivisible commodities and money. Indivisible commodities mean commodities (goods) whose quantities are represented by integers, such as houses, cars, and aircraft, whereas money is a real number representing the aggregation of the markets of other commodities. We consider an economy (of Arrow-Debreu type) with a finite set L of producers, a finite set H of consumers, a finite set K of indivisible commodities, and a perfectly divisible commodity called money. Productions of producers and consumptions of consumers are integer-valued vectors in ZK representing the numbers of indivisible commodities that they produce or consume. Here producers' outputs are represented by positive numbers, while negative numbers are interpreted as inputs to them, and consumers' inputs are represented by positive numbers, 323

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Chapter 11. Application to Mathematical Economics

while negative numbers are interpreted as outputs from them. Given a price vector p = (p(k) : k € K) 6 R^ of commodities, each producer (assumed to be male) independently schedules a production in order to maximize his profit, each consumer (assumed to be female) independently schedules a consumption to maximize her utility under the budget constraint, and all agents exchange commodities by buying or selling them through money. An important feature of this model is that the independent agents take the price as granted; i.e., they assume that their individual behaviors do not affect the price. Such an economy is called a competitive economy. We assume that a producer I e L is described by his cost function GI : ZK —> RU {+00}, whose value is expressed in units of money. He wishes to maximize the profit (p,y) — Ci(y) in determining his production y = yi e ZK. This means that yi is chosen from the supply set

where the function Si : RK —> 2Z is called the supply correspondence. Accordingly, the profit function TTJ : R^ —> R is defined by

To avoid possible technical complications irrelevant to discreteness issues, we assume that dom Ci is a bounded subset of 7iK for each I 6 L. This guarantees, for instance, that Si (p) is nonempty for any p. Each consumer h € H has an initial endowment of indivisible commodities and money, represented by a vector (x^,m^) € ZK x R+, where x^(fc) denotes the number of the commodity k e K and m°h the amount of money in her initial endowment. Consumers share in the profits of the producers. We denote by dih the share of the profit of producer I owned by consumer h, where

Thus, consumer h gains an income

where /3h '• RK —> R, and accordingly her schedule (x, m) = (x/j, mh) should belong to her budget set

We assume that a consumer h is associated with a utility function Uh '• "ZK x R —> RU {—00} that is quasi linear in money; namely,

11.1.

Economic Model with Indivisible Commodities

325

Figure 11.1. Consumer's behavior.

with a function63 [//, : Zx —> Ru{—oo}. Consumer h maximizes Uh under the budget constraint; that is, (x,m) — (xh,TTih) ls a solution to the following optimization problem:

(see Fig. 11.1). Under the assumption that domUh is bounded0 and mh is sufficiently large, we can take

to reduce the above problem to an unconstrained optimization problem:

This means that Xh is chosen from the demand set

The function Dh '• RK —> 2Z is called the demand correspondence. A tuple ((xh | h 6 H), (yi \ I e L),p), where xh e ZK, j/j e ZK, and p 6 RK, is called an equilibrium or a competitive equilibrium if 63 In economic terminology, U^ is called the reservation value function, although we refer to it as the utility function in this book. 64 The boundedness of dom I//, is a natural assumption because no one can consume an infinite number of indivisible commodities. This assumption is also convenient for concentrating on discreteness issues in our discussion.

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That is, each agent achieves what he or she wishes to achieve, the balance of supply and demand holds, and an equilibrium price vector is nonnegative. Denoting the total initial endowment of indivisible commodities by

we can rewrite the supply-demand balance (11.11) as

On eliminating Xh and yi using (11.9) and (11.10), we see that p e R^ is an equilibrium price if and only if

where the right-hand side is a Minkowski sum in ZK. It is noted that money balance

is implied by (11.11) with (11.3), (11.4), (11.7), and 7rj(p) = (p ) W ) - d(yi). We are concerned with mathematical properties of equilibria, rather than their economic-theoretical significance. A most fundamental question would be as follows: When does an equilibrium exist? Namely, the first problem we should address is this: Problem 1: Give a (sufficient) condition for the existence of an equilibrium in terms of utility functions Uh and cost functions C\. The conditions (11.9) and (11.10) for an equilibrium are given in terms of demand correspondences Dh and supply correspondences 5; without explicit reference to utility functions Uh and cost functions C\. This motivates the following: Problem 2: Give a (sufficient) condition for the existence of an equilibrium in terms of demand correspondences Dh and supply correspondences Si. When an equilibrium exists, we may be interested in its structure: Problem 3: Investigate the structure of the set of equilibria. A more specific problem in this category is as follows: Do the maximum and minimum exist among equilibrium price vectors? We shall answer the above problems with the use of concepts and results in discrete convex analysis. Our answers are the following.

11.2.

Difficulty with Indivisibility

327

(1) An equilibrium exists if Uh (h e H) are M^-concave functions and Ci (I € I/) are M^-convex functions (Theorems 11.13 and 11.14). (2) An equilibrium exists if Dh(p) (h € H) and Si(p) (I € L) are M^-convex sets for each p (Theorem 11.15). (3) The set P* of the equilibrium prices is an L''-convex polyhedron (Theorem 11.16). This means, in particular, that p V q,p A q € P* for any p, q e P* and that there exist a maximum and a minimum among equilibrium prices. As a preliminary consideration, the difficulty arising from indivisible commodities is demonstrated in section 11.2 by a simple example. In section 11.3 we discuss the relevance of M^-concavity as an essential property of utility functions. The results mentioned above are proved in section 11.4. Finally, in section 11.5, we show that an equilibrium can be computed by solving an M-convex submodular flow problem. Note 11.1. A special case of our economic model with L = 0, where no producers are involved, is called the exchange economy. A difficulty of indivisible commodities already arises in this case, as we will see in section 11.2. • Note 11.2. Commodities that can be represented by real-valued vectors are called divisible commodities. A framework for the rigorous mathematical treatment of equilibria in economies of divisible commodities was established around 1960 usin convexity, compactness, and fixed-point theorems as major mathematical tools. See Debreu [37], [38], Nikaido [168], Arrow-Hahn [4], and McKenzie [128]. Note 11.3. A considerable literature already exists on equilibria in economies with indivisible commodities. We name a few: Henry [88], 1970; Shapley-Scarf [187], 1974; Kaneko [107], 1982; Kelso-Crawford [111], 1982; Gale [72], 1984; Quinz [173], 1984; Svensson [196], 1984; Wako [209], 1984; Kaneko-Yamamoto [108], 19 Van der Laan-Talman-Yang [204], 1997; Bikhchandani-Mamer [13], 1997; Danilov Koshevoy-Murota [34], 1998 (also [35], 2001); Bevia-Quinzii-Silva [12], 1999; Gu Stacchetti [84], 1999; and Yang [219], 2000.

11.2

Difficulty with Indivisibility

The difficulty in the mathematical treatment of indivisible commodities is illustrated by a simple example. We consider an exchange economy consisting of two agents (H = {1,2}, L = 0) dealing in two indivisible commodities (K = {1,2}). Putting S = {(0,0), (0,1), (1,0), (1,1)} we define the utility functions Uh for h = 1,2 in (11.6) by

where domC/i = domC/2 = S (see Fig. 11.2). The demand correspondences D\ and _D2, calculated according to (11.8), are also given in Fig. 11.2. For instance, for

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Figure 11.2. Exchange economy with no equilibrium for x° = (1,1).

p = (p(l),p(2)) with 0 < p ( l ) < 1 and 0 < p(2) < 1, we have Di(p) = {(1,1)}; for p = (1,1), we have D^p) = {(1,1), (0,1), (1,0)} and £>2(p) = {(1,1)}. Given a total initial endowment x°, an equilibrium is a tuple (xi,x 2 ,p) e Z2 x Z2 x R2. such that

For x° = (1, 2), for example, the tuple of x\ = (0,1), x? = (1,1), p = (2,1) satisfies the above conditions and hence is an equilibrium. Another case, x° = (1,1), is problematic. As we have seen in (11.15), a nonnegative vector p is an equilibrium price if and only if x° € Di(p) + D^p). Superposition of the diagrams for Di(p) and £>2(p) in Fig. 11.2 yields a similar diagram for the Minkowski sum Di(p) + Dz(p), shown in Fig. 11.3. We see from this diagram that no p satisfies (1,1) 6 Di(p) + D^p) and hence no equilibrium exists for x° = (1,1). The diagram consists of eight regions, corresponding to the eight points in [0, 2]z x [0, 2]z except {(1,1)}. Hence an equilibrium exists for every

11.2. Difficulty with Indivisibility

329

Figure 11.3. Minkowski sum Di(p) + D2(p).

x° £ ([0,2] z x [0,2] z ) \ {(1,1)} and not for x° = (1,1). Let us have a closer look at the problematic case to better understand the discreteness inherent in the problem and to identify the source of the difficulty. In view of the established mathematical framework for divisible commodities, we consider an embedding of our discrete problem via a concave extension of the utility functions. Denote by Ui and U2 the concave extensions of U\ and U%, respectively. Obviously, doniRt/i = dom.Rt/2 = S, where S = [0, I]R x [0, I]R, and

The demand correspondences are defined by

for h = 1,2 and an equilibrium is a tuple ( x i , X 2 , p ) 6 R2 x R2 x R+ such that

In our case of x° — (1,1), the tuple of xi = x2 = (1/2,1/2) and p = (3/2,3/2) is an equilibrium in this sense, but it is not qualified as an equilibrium in the original problem of indivisible commodities, in which xi and x2 must be integer vectors. Thus, there is an essential discrepancy between the original discrete problem and the derived continuous problem. We can identify the reason for this discrepancy as the lack of convexity in Minkowski sum discussed in section 3.3. Since Dh(p) coincides with the convex hull Dh(p) of Dh(p) and £>i(p) + D2(p) = -Di(p) + D2(p) holds by Proposition 3.17 (4), the derived continuous problem has an equilibrium if and only if x° e

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DI(P) + D2(p). On the other hand, the original discrete problem has an equilibrium if and only if x° e Di(p) + D2(p), as noted already. For p = (3/2,3/2), we have I>i(p) = {(0,1), (1,0)}, D2(p) = {(0, 0), (1,1)}, and

which has a hole at (1,1) (see Example 3.15 and Fig. 3.4). This hole is the very reason for the nonexistence of an equilibrium for indivisible commodities.

11.3

M^-Concave Utility Functions

We demonstrate the relevance of M^-concavity to utility functions by indicating its relationship with fundamental properties such as submodularity, the gross substitutes property, and the single improvement property, discussed in the literature of mathematical economics. First, recall from Theorem 6.2 that we can define an M^-concave function as a function U : ZK —* RU {—00} with domU ^ 0 satisfying the following exchange property:

where Xi is the ith unit vector and a maximum taken over an empty set is denned to be — co. A more compact expression of this exchange property is

where xo is the zero vector, AC/(x;j, i) — U(x — Xi + Xj) - U(x) as in (6.2), &U(x;0,i) = U(x-Xi)'-U(x),aa.d&U(y,i,0) = U(y + Xi)-U(y). All the results established in the previous chapters for M^-convex functions can obviously be rephrased for M^-concave functions. In particular, an M^-concave function U has the following properties (reformulations of Theorems 6.42, 6.19, 6.26, and 6.24, Propositions 6.33 and 6.35, and Theorem 6.30). • Concave extensibility: The concave closure U of U satisfies

• Submodularity:

Utility functions are usually assumed to have decreasing marginal returns, a property that corresponds to submodularity in the discrete case.

11.3.

M^-Concave Utility Functions

331

• Local characterization of global maximality: For x € dom U,

This says that special ascent directions work for increasing a utility function. • (—M^-GSfZ]): Ifx £ argmax£/[-p+p 0 l], P < x(i) for every i £ K with p(i) = q(i) and (ii) y(K)<x(K) if p0 = qoThis is a version of the gross substitutes property. Recall the brief discussion on the gross substitutes property at the beginning of section 6.8. • (—M^-SWCSfZ]): For x € argmint/[-p], p <E RK, and i € K, at least one of (i) and (ii) holds true: (i) x e argmaxC/[—p — a\i] for any a > 0, (ii) there exist a > 0 and y e argmaxt/[—p - a\i] such that y(i) = x(i) — 1 and y(j) > x(j) for all j € K \ {i}. This is another version of the gross substitutes property, called the stepwise gross substitutes property. • M^-convexity of maximizers: For every p € R^, argmax[/[—p] is an M13convex set. The properties above are essential features of M^-concave functions and, in fact, the last four characterize M^-concavity, as follows (reformulations of Theorems 6.24, 6.34, 6.36, and 6.30). Note that submodularity and concave extensibility (even together) do not imply M^-concavity. Theorem 11.4. For a function U : ZK —> R U {—00} with a nonempty domain,

effective

Theorem 11.5. For a concave-extensible function U : ZK —> R U {—00} with a bounded nonempty effective domain,

Theorem 11.6. For a concave-extensible function U : ZK —> R U {—00} with a nonempty effective domain,

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Chapter 11. Application to Mathematical Economics

Theorem 11.7. For a function U : ZK —> R U {—00} with a bounded nonempty effective domain, U is M^-concave ^=> argmaxf/[—p] is an ^-convex set for each p 6 RK. Let us consider the special case of set functions to discuss the relationship of the above theorems to the results of Kelso-Crawford [111] and Gul-Stacchetti [84]. The gross substitutes property was denned originally for a set function U : 2K —> R in [111]. It reads as follows: (GS) If X e argmaxt/"[-p] and p < q, there exists Y e argmaxC/[—q] such that {i 6 X \ p ( i ) = q(i)} C Y, where

Two novel properties introduce are introduced in [84]; i.e., the single improvement property:

and the no complementarities property:

It is pointed out in [84] that these three properties are equivalent:65

for a set function U. It is also observed that (NC) is implied by the strong no complementarities property:

To derive these results from our previous theorems, we identify a set function U : 2K -> R with a function U : ZK -> R U {-oo}, where dom U = {0,1} K , by

When translated for U, the properties (GS) and (SI) for U may be written respectively as follows: 65 To be precise, this equivalence is shown in [84] for a monotone nondecreasing U, where p and q can be restricted to nonnegative vectors.

11.3. M^-Concave Utility Functions

333

(—Mb-GS w [Z]) If x e argmaxC/[—p] and p < q, there exists y e argmaxt/[—q] such that y(i) > x(i) for every i € K with p(z) = q(i). (-M^-SIwfZj) For p € R^ and x € domf/ \ argmax[/[-p],

Obviously, we have (-M*-GS[Z]) => (-M»-GS W [Z]) and (-M*-SI[Z]) =» (-M"SIW[Z]) and it can be shown (see Murota-Tamura [160]) that the converses are also true if domC/ = {0,1}^. It follows from this and Theorems 11.4 and 11.5 that

On the other hand, we see

from the multiple exchange axiom for a matroid (see Theorem 4.3.1 in Kung [117]) and Theorem 11.7. Thus, the three conditions (GS), (SI), and (NC) for U are all equivalent to the M^-concavity of U. Obviously, the special case of (SNC) with |/| = 1 coincides with the exchange axiom (—M I '-EXC[Z]) for U and, therefore,

Finally, we mention that the submodularity (11.19) of U is equivalent to the submodularity of U:

Note 11.8. Utility functions are to be maximized by consumers. As is discussed in Note 10.16, however, maximizing a general submodular set function is computationally intractable. M^-concave utility functions form a subclass of submodular functions that can be maximized efficiently. • Note 11.9. We mention here some examples of M^-concave utility functions treated explicitly or implicitly in the literature of indivisible commodities. See also the examples of M^-convex functions shown in section 6.3. (1) A separable concave function

denned in terms of a family of univariate discrete concave functions (ui \ i € K) (i.e., — Ui 6 C[Z —> R]) is an M^-concave function (see (6.31)).

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Chapter 11. Application to Mathematical Economics

(2) A quasi-separable concave function

defined in terms of a family of univariate discrete concave functions (ui i G /•CU{0}) (i.e., -Ui G C[Z —> R]) is an M^-concave function (see (6.32)). (3) Given a vector (0, | i & K) G RK, we denote by U(X) the maximum value of di with index i belonging to X C K. More formally, choosing a* G R U { —00} with a* < muiigx a^, we define a set function E7 : 2K —> R U {—00} by

Identifying X with its characteristic vector Xx and imposing monotonicity, we obtain a function U : ZK —>• R U {—00} given by

This is an M^-concave function. The function U represents a unit demand preference. •

11.4

Existence of Equilibria

We prove the existence of equilibria by establishing two statements: (i) an equilibrium price can be characterized as a nonnegative subgradient of the aggregate cost function and (ii) under the assumption of M^-convexity/concavity of individual cost functions and utility functions, the aggregate cost function is an M^-convex function for which the subdifferential is nonempty.

11.4.1

General Case

The conditions Xh G Dh(p) and yi G Si(p) in (11.9) and (11.10) can be rewritten by (3.30) as

using the notations for subdifferentials defined in (8.19) and (6.86). Hence, p G R+ is an equilibrium price if and only if it satisfies (11.22) for some x^ G domC/h, (h G H) and yi G domC1; (I G L) meeting the supply-demand balance (11.14) for the given total initial endowment x°. Furthermore, if ((xh h G H), (yi I G L),p) is an equilibrium, the set P*(x°) of all equilibrium prices for x° is expressed as

11.4.

Existence of Equilibria

335

In particular, the right-hand side of this expression is independent of the choice of an equilibrium allocation ((xh h 6 H), (yi \ I & L)). We define the aggregate cost function ^> : ZK —> R U {+00} by

This is the integer infimal convolution (6.43) of producers' cost functions and the negatives of consumers' utility functions. Owing to our assumptions that • dom Uh is bounded for each h £ H, and • domC; is bounded for each / € L, the infimum in (11.24) is attained for each z in

where the right-hand side is a Minkowski sum in ZK (see (6.44)). Proposition 11.10. For a total initial endowment x°, the following statements hold true under the boundedness assumption on dom Uh for h e H and dom C\ for l&L. (1) There exists an equilibrium if and only if x° G dom* and (—dR^(x°)) D R*^0.

(2)p'(x°) = (-dR<Sf(x°»nnx.

(3) I f ( ( x h h e H), (yi | I e L)) satisfies (11.9), (11.10), and (11.14) for some p (not necessarily nonnegative), then

and ((XH | h e H), (yi \ I e L),p') is an equilibrium for any p' e (—0R\l/(a;°)) nR^. Proof. Definition (11.24) can be rewritten as

which shows

Therefore, we have

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Chapter 11. Application to Mathematical Economics

Figure 11.4. Aggregate cost function 'f and its convex closure fy for an exchange economy with no equilibrium.

If ((xh h G H ) , ( y i | I 6 L),p) is an equilibrium, we have x^ € argmaxt/ft,[— p] (h 6 H), yi e argminC;[—p] (I e L), and (11.14), from which follows

Since p e R£, we have p e (-<9R#(:r0)) n R£. Conversely, for p e (-^R*(X°)) n R£, we have *[p](x°) = inf*[p]. Let (xfc | /i e jf?) and (?/; | I G L) be vectors that attain the minimum on the right-hand side of (11.28) for z = x°. Then we have (11.30) and hence Xh e arg max Uh [—p] (h e H) and yi e argminC t [-p] (I € L) by (11.29). This means that ((xh h £ H), (yi I € L),p) is an equilibrium. The expression (11.26) follows from the above argument. D Proposition 11.10 suggests that if a difficulty in the discreteness of indivisible commodities exists, it should be reflected in the emptiness of the subdifferential dR^(x°). Example 11.11. Let us investigate the previous example (section 11.2) of an exchange economy with no equilibrium. The aggregate cost function ^ takes the values shown in Fig. 11.4 (left), where dom * = [0,2]z x [0,2]z- The convex closure * of # is given on the right of Fig. 11.4. Since #(1,1) ^ *(!,!), ^,#(1,1) is empty and, by Proposition 11.10, no equilibrium exists for x° = (1,1). For x° = (1,2), on the other hand, p = (2,1) is an equilibrium price since (2,1) e ( — d n ^ ( x ° ) ) n R^_. •

11.4.

Existence of Equilibria

11.4.2

337

M"-Convex Case

The M^-concavity of utility functions and the M^-convexity of cost functions are key to the existence of equilibria, as well as to a desirable structure of equilibrium prices. In Proposition 11.10 we saw that the existence of an equilibrium is almost equivalent to the existence of a subgradient of \I>. The latter is guaranteed under M^-concavity/convexity as follows. Proposition 11.12. LetUh (h £ H) be A^-concave functions withdomUh bounded and Ci (I £ L) be Aft-convex functions with domCi bounded. Then $> is an Aflconvex function and ^^(x0) ^ 0 for any x° £ dom^. Proof. By the assumption, ^ is an infimal convolution of M^-convex functions, which is M^-convex by Theorem 6.15. The second claim follows from Theorem 6.61 (2). D We shall present three theorems on the existence of equilibria. The first is for an exchange economy (with L = 0). Theorem 11.13. Consider an exchange economy with agents indexed by H and suppose that Uh (h £ H) are nondecreasing A/*1 -concave functions with domUh bounded. Then there exists an equilibrium ((x^ \ h £ H ) , p ) for every x° £ Eft 6 H d o m ^Proof. We use Proposition 11.10. Since &R^!(X°) ^ 0 by Proposition 11.12, it suffices to show the nonnegativity of (any) p 6 — d^(x°}. The function U(z) = —*&(z) is nondecreasing and Putting z = x° + axi and letting a —> +00, we see that p(i) > 0. We next consider the existence of equilibria in the general case with consumers and producers. To separate discreteness issues from topological ones, we consider an embedding of our discrete model in a continuous space, just as we have done for the simple exchange economy in section 11.2. Utility functions Uh, being M^-concave, are extensible to concave functions Uh '• Rx -> RU{-oo}. Similarly, cost functions Ci, being M^-convex, are extensible to convex functions Ci : RK —» RU {+00}. Then demand correspondences Dh for h £ H and supply correspondences Si for I £ L are defined, respectively, as

Note that Dh(p) and §i(p) coincide respectively with the convex hulls Dh(p) and Si(p) of Dh(p) and Si(p). We define an equilibrium in the continuous model as a

338 tuple ((xh\heH),(yi

Chapter 11. Application to Mathematical Economics I e L),p) of xh & RK, yt e RK, and p 6 R£ satisfying

as well as the supply-demand balance (11.14) and the price nonnegativity (11.12). The following theorems state that the existence of an equilibrium in the continuous model implies that of an equilibrium for indivisible commodities under our assumption of M^-concavity/convexity. We emphasize here that this is by no means a general phenomenon, as we have seen in section 11.2, but is peculiar to M^concavity/convexity. For topological issues, the reader is referred to the literature in economics indicated in Note 11.2. Theorem 11.14. Suppose that utility functions UH (h G H) are Afl -concave and cost functions Ci (I € L) are Afl-convex. If the derived continuous economy has an equilibrium satisfying (11.33), (11.34), (11.14), and (11.12) for a total initial endowment x° e Z^, there exists an equilibrium of indivisible commodities for x° satisfying (11.9), (11.10), (11.14), and (11.12). Proof. By Theorem 11.7, Dh(p) and Si(p) are M^-convex sets if they are not empty. Hence, the claim follows from Theorem 11.15 below. D Theorem 11.15. Suppose that, for eachp € R+, demand sets DH(P) (h £ H) and supply sets Si(p) (I € L) are Afl-convex if they are not empty. If the derived continuous economy has an equilibrium satisfying (11.33), (11.34), (11.14), and (11.12) for a total initial endowment x° e Z+, there exists an equilibrium of indivisible commodities for x° satisfying (11.9), (11.10), (11.14), and (11.12). Proof. For an equilibrium price p e R+ in the continuous model, we have

On noting Dh(p) = Dh(p) and §i(p) = Si(p), as well as Theorem 4.12 and Proposition 3.17 (4), we see

By Theorem 4.23 (3), on the other hand, J2h€H ^h(p)-^2ieL &i(p) *san M^-convex set, which is hole free by Theorem 4.12. Hence follows

which shows (11.15).

11.4.

Existence of Equilibria

339

Finally we consider the structure of equilibrium prices. Theorem 11.16. Suppose that utility functions Uh (h € H) are Afl-concave and cost functions Ci (I 6 L) are Afi -convex and that there exists an equilibrium for a total initial endowment x°. Then the set P*(x°) of all the equilibrium price vectors is an L^-convex polyhedron. This means, in particular, that p, q € P*(x°) => p V q, pf\q&P*(x°), which implies the existence of the smallest and the largest equilibrium price vectors. Proof. The subdifferentials d^Uh(xh) (h e H) and &R.Ci(yi) (I e L) in (11.23) are lAconvex polyhedra by Theorem 6.61 (2). Since R+ is L^-convex and the intersection of L''-convex polyhedra is L''-convex, P*(x°) is an L''-convex polyhedron. As we have seen in the above, we have the correspondence

Note 11.17. Inspection of the proof of Theorem 11.15 shows that it relies solely on the following two properties of M^-convex sets: (i) M^-convex sets are hole free and (ii) Minkowski sums of M^-convex sets are M^-convex sets. This suggests a generalization of Theorem 11.15 for a class J- of sets of integer points such that (i) 5 e T, x e ZK =>• S = 5 n ZK, x - S e T, and (ii) Si, 52 E f => Si + S2 e T (cf. Proposition 3.16). Namely, if Dh(p) e T (h e H) and Si(p) e F (I £ L) for each p e R^f, where we assume 0 € F, and if the derived continuous economy has an equilibrium, then there exists an equilibrium for indivisible commodities. Note 11.18. In an exchange economy with two agents, an equilibrium exists under the L11-concavity of utility functions or the L1'-convexity of demand sets. (1) If the utility functions C/i and C/2 are nondecreasing L^-concave functions, an equilibrium exists for any x° & domC/i + dom[/2 (cf. Theorem 11.13). (2) Suppose that the utility functions C/i and f/2 are L^-concave. If the derived continuous economy has an equilibrium for x° 6 Z^f, there exists an equilibrium for indivisible commodities (cf. Theorem 11.14). (3) Suppose that, for each p e R+, the demand sets -Di(p) and D2(p) are iJconvex if they are not empty. If the derived continuous economy has an equilibrium for x° & Z+, there exists an equilibrium for indivisible commodities (cf. Theorem 11.15). Proof: (1) In the proof of Theorem 11.13, * is an L2-convex function and, therefore, 5a*(x°) ^ 0 by Theorem 8.45. (2) In the proof of Theorem 11.14, D\(p) and D2(p) are lAconvex or empty. Hence the claim follows from (3) by Proposition 7.16. (3) This follows from the proof of Theorem 11.15 by virtue of the convexity in Minkowski sum for L^-convex sets (Theorem 5.8). •

Chapter 11. Application to Mathematical Economics

340

11.5

Computation of Equilibria

In this section we show an algorithmic procedure to compute an equilibrium in the case where C\ are M^-convex and Uh are M^-concave. We use the framework of the M-convex submodular flow problem MSFP2, introduced in section 9.2. The solution to this problem yields consumptions and productions satisfying (11.9), (11.10), and (11.14), as well as a price vector, which, however, may not be nonnegative. This constitutes the first phase of our algorithm. The second phase finds an equilibrium price vector by solving a shortest path problem. We can modify the second phase so that we can find the smallest or the largest equilibrium price vector. For an M^-convex cost function Ci : ZK —•> R U {+00}, we define the corresponding M-convex function Cj : Z^UK —> RU {+00} by

compatibly with (6.4). We also define the M-concave function R U {-00} associated with Uh : ZK -> R U {-00} by

In accordance with (11.24), we consider the aggregate cost function

Obviously, we have

for the aggregate cost function ^ defined in (11.24). For the total initial endowment z° e ZK, we put x° = (-x°(K),x°) € Z<°> UK . The function \I>, being an integer infimal convolution of M-convex functions, is also M-convex and can be evaluated by solving an MSFP2 (see Note 9.30). A concrete description is given below. The instance of MSFPa for the evaluation of ^(x°) is defined on a directed bipartite graph G = (V+,V~,A) with vertex partition (V+,V~) and arc set A given by

11.5.

341

Computation of Equilibria

Figure 11.5. Graph for computing a competitive equilibrium. Note that V+, Vt+, and VJj" are copies of {0}(JK. Figure 11.5 illustrates the graph G for H = {a, /?}, L = {A, B}, and K = {1,2}. For each arc a € A, we put c(a) = —oo, c(a) = +00, and 7(0) = 0. Using the indicator function 5^°} '• %Ve —> {0, +00} of {x°}, we define a function / : Zv+uv~ -> R U {+00} by

where w G Z1^ , yi G Z17; for / G L, and i^ G Zvh for h £ H. The function / is M-convex, since C1/ (I G L) are M-convex and Uh (h G #) are M-concave. This is the instance of the MSFP2 that we use for the computation of ^>(x°). The optimal flow and the optimal potential for the MSFP2 above give the allocation and the price vector in the equilibrium, as is stated later in Theorem 11.20. The following proposition is a lemma for it. Proposition 11.19. Let £ G ZA be an optimal flow andp G R y + u y potential for the above instance of MSFP2. Define66

be an optimal

and regard x*h, y*, w*, and p as vectors on {0} U K. (1) w* = x°, x*h(0) = -x*h(K) for h&H, and ft (0) = - y f ( K ) for I G L.

(2)£° + E J6i i/r = E h€ fl^-

(3) p(fc-) = p(fc+) = p(k+) for k G {0} U K, h G H, and I G L. (4) x*h G arg max Uh [—P] for h £ H and y* G arg min Ci [— p] for I & L. Proof. (1) is obvious from the definitions of Ci, Uh, and 6^°}- By the structure of G, for each k G {0} U K, we have

66

The notation —d£\v- designates the restriction of —d£ to Vh . Hence x*h = — d(,\v- means

*J(fc) = -ae(fc)forfcev- h -.

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Chapter 11. Application to Mathematical Economics

This means (2), since w* = x° by (1). Since 7(0) = 0, c(a) = —oo, and c(a) = +00 for any a e A, condition (i) of (POT) of Theorem 9.16 implies p(d+a) — p(d~a) = 0 for all a G A. This implies (3). It follows from (3) that

We also have

Then (4) follows from (ii) of (POT) in Theorem 9.16, (11.35), and (11.37). The following theorem, an immediate consequence of Proposition 11.19, shows that consumptions and productions satisfying (11.9), (11.10), and (11.14) can be obtained from a solution to the above instance of MSFP2 and that, if the optimal potential is nonnegative, then it serves as an equilibrium price vector. Theorem 11.20. Let £ e ZA be an optimal flow and p e Ry uv potential with p(Q+) = 0 for the above instance O/MSFP2. Define

be an optimal

and regard x*h, y*, and p as vectors on K. Then we have

Therefore, ((x*h h & H),(yf \ I & L},p) is an equilibrium if p > 0. Moreover, if there exists an equilibrium at all, then ((x*h h e H), (yf \ I 6 L)) is the consumptions and productions of some equilibrium. Any algorithm for the MSFP2 will find a tuple ((x*h h e H), (yf I € L),p), which gives an equilibrium for the total initial endowment x° if the optimal potential p happens to be nonnegative. We go on to consider the set of all nonnegative optimal potentials or, equivalently, the set of all equilibrium price vectors. We already know from Theorem 11.16 that the set is an L^-convex polyhedron, and our objective here is to derive a concrete description in terms of a linear inequality system. With this knowledge, the existence of an equilibrium price vector can be checked by solving

11.5.

Computation of Equilibria

343

a linear programming problem, which can be reduced to the dual of a single-source shortest path problem. Recall from (11.23) that the set P*(x°) of all equilibrium price vectors for x° is expressed as

in which we have

by (3.30). By Theorem 6.26 (2) (M-optimality criterion), we have y^ e arg min Ci [—p] if and only if

and x*h e argmax[/h[—p] if and only if

By defining

we obtain a concrete representation of P*(x°) as follows. Note that l(j) < +00, u(j) > — oo, and u(i,j] > —oo for any i,jEK(i^ j). Theorem 11.21. The set P*(x°) of all equilibrium price vectors is an ifi-convex polyhedron described as

with l ( j ) , u ( j ) , and u(i,j) defined in (11.40), (11.41), and (11.42). By Theorem 11.21, the nonemptiness of P*(x°) can be checked by linear programming. In particular, the largest equilibrium price vector, if any, can be

344

Chapter 11. Application to Mathematical Economics

found by solving a linear programming problem:

Similarly, the smallest equilibrium price vector can be found by solving another linear programming problem:

Both (11.44) and (11.45) can be easily reduced to the dual of a single-source shortest path problem. Theorem 11.22. There exists an equilibrium price vector if and only if the problem (11.44) is feasible. The smallest and the largest equilibrium price vectors, if any, can be found by solving the shortest path problem. Thus, the existence of a competitive equilibrium in our economic model with M^-convex cost functions of producers and M^-concave utility functions of consumers can be checked in polynomial time by the following algorithm. Algorithm for computing an equilibrium SO: Construct the instance of the MSFP2. SI: Solve the MSFP2 to obtain ( ( x * h \ h & H ) , (yf \leL),p). (If MSFPg is infeasible, no equilibrium exists.) S2: Solve the problem (11.44) to obtain an equilibrium ((x*h \ h € H), (yf I e L),p*) with largest p*. (If (11.44) is infeasible, no equilibrium exists.) Whereas the above algorithm yields the largest equilibrium price vector, the smallest price vector can be computed by solving (11.45) instead of (11.44) in step S2.

Bibliographical Notes The unified framework for indivisible commodities by means of discrete convex analysis is proposed in Danilov-Koshevoy-Murota [34], [35], to which Theorems 11.14 and 11.15 as well as Notes 11.9 and 11.17 are ascribed. Theorem 11.16 for the structure of equilibrium prices and Note 11.18 are by Murota [147]. The gross substitutes property was introduced by Kelso-Crawford [111] and investigated thoroughly by Gul-Stacchetti [84], in which the equivalence of (GS), (SI), and (NC) is proved. The connection of these conditions to M^-concavity was pointed out by Fujishige-Yang [69] for set functions, with subsequent generalizations by Danilov-Koshevoy-Lang [33] (Theorem 11.6) and Murota-Tamura [160] (Theorems 11.4 and 11.5). See Roth-Sotomayor [180] for more on (GS).

11.5.

Computation of Equilibria

345

The computation of an equilibrium via an M-convex submodular flow problem described in section 11.5 is due to Murota-Tamura [161]. M-convexity is also amenable to the stable marriage problem (stable matching problem) of Gale-Shapley [74], which is one of the most applicable models in economics and game theory. Eguchi-Fujishige [47] formulates a generalization of the stable marriage problem in terms of M^-convex functions and presents an extension of the Gale-Shapley algorithm. Submodularity plays important roles in economics and game theory. We mention here the paper of Shapley [186] as an early contribution and Bilbao [14], Danilov-Koshevoy [31], Milgrom-Shannon [129], and Topkis [203] as recent literature.

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Chapter 12

Application to Systems Analysis by Mixed Matrices This chapter presents an application of discrete convex analysis to systems analysis by mixed matrices. Motivated by a physical observation to distinguish two kinds of numbers appearing in descriptions of physical/engineering systems, the concepts of mixed matrices and mixed polynomial matrices are introduced as mathematical tools for dealing with two kinds of numbers in systems analysis. Discrete convex functions arise naturally in this context and the discrete duality theorems are vital for the analysis of the rank of mixed matrices and the degree of determinants of mixed polynomial matrices.

12.1

Two Kinds of Numbers

A physical/engineering system can be characterized by a set of relations among various kinds of numbers representing physical quantities, parameter values, incidence relations, etc., where it is important to recognize the difference in the nature of the quantities involved in the problem and to establish a mathematical model that reflects the difference. A primitive, yet fruitful, way of classifying numbers is to distinguish nonvanishing elements from zeros. This dichotomy often leads to graph-theoretic methods for systems analysis, where the existence of nonvanishing numbers is represented by a set of arcs in a certain graph. Closer inspection reveals, however, that two different kinds can be distinguished among the nonvanishing numbers; some of the nonvanishing numbers are accurate in value and others are inaccurate in value but independent of one another. We may alternatively refer to the numbers of the first kind as fixed constants and to those of the second kind as system parameters. Accurate numbers (fixed constants): Numbers accounting for various sorts of conservation laws, such as Kirchhoff's laws, which, stemming from the topological incidence relation, are precise in value (often ±1). Inaccurate numbers (system parameters): Numbers representing independent

347

348

Chapter 12. Application to Systems Analysis by Mixed Matrices

Figure 12.1. Electrical network with mutual couplings.

physical parameters, such as resistances in electrical networks and masses in mechanical systems, which, being contaminated with noise and other errors, take values independent of one another. It is emphasized that the distinction between accurate and inaccurate numbers is not a matter in mathematics but in mathematical modeling, i.e., the way in which we recognize the problem. This means in particular that it is impossible in principle to give a mathematical definition to the distinction between the two kinds of numbers. The objective of this section is to explain, by means of typical examples, what is meant by accurate and inaccurate numbers and how numbers of different nature arise in mathematical descriptions of physical/engineering systems. We consider three examples from different disciplines: an electrical network, a chemical process, and a mechanical system. Example 12.1. Consider the electrical network in Fig. 12.1, which consists of five elements: two resistors of resistances Ti (branch i) (i = 1,2), a voltage source (branch 3) controlled by the voltage across branch 1, a current source (branch 4) controlled by the current in branch 2, and an independent voltage source of voltage e (branch 5). The element characteristics are represented as

where & and rn are the current in and the voltage across branch i (i = 1 , . . . , 5) in the directions indicated in Fig. 12.1. We then obtain the following system of equations:

12.1.

Two Kinds of Numbers

349

The upper five equations are the structural equations (Kirchhoff's laws), while the remaining five are the constitutive equations. The nonzero coefficients, ±1, appearing in the structural equations represent the incidence relation in the underlying graph and are certainly accurate in value. The entries of —1 contained in the constitutive equations are also accurate by definition. In contrast, the values of the physical parameters ri, r2, a, and (3 are likely to be inaccurate, being only approximately equal to their nominal values on account of various kinds of noises and errors. The unique solvability of this network amounts to the nonsingularity of the coefficient matrix of (12.1). A direct calculation shows that the determinant of this matrix is equal to r-2 + (1 — a)(l + /3)ri, which is highly probably distinct from zero by the independence of the physical parameters {ri,r2,a,/?}. Thus, the electrical network of this example is solvable in general or, more precisely, solvable generically with respect to the parameter set {ri,r 2 ,a,/?}. The solvability of this system will be treated in Example 12.11 by a systematic combinatorial method (without direct computation of the determinant). • The second example concerns a chemical process simulation. Example 12.2. Consider a hypothetical system (Fig. 12.2) for the production of ethylene dichloride (C2H4C12), which is slightly modified from an example used in the Users' Manual of Generalized Interrelated Flow Simulation of The Service Bureau Company. Feeds to the system are 100 mol/h of pure chlorine (Ck) (stream 1) and 100 mol/h of pure ethylene (C2H 4 ) (stream 2). In the reactor, 90% of the input ethylene is converted into ethylene dichloride according to the reaction formula

At the purification stage, the product ethylene dichloride is recovered and the unreacted chlorine and ethylene are separated for recycling. The degree of purification is described in terms of the component recovery ratios 01, 02, and 03 of chlorine, ethylene, and ethylene dichloride, respectively, which indicate the ratios of the amounts recovered in stream 6 of the respective components over those in stream 5. We consider the following problem:

350

Chapter 12. Application to Systems Analysis by Mixed Matrices

Figure 12.2. Hypothetical ethylene dichloride production system. Given the component recovery ratios ai and a-2 of chlorine and ethylene, determine the recovery ratio x = 03 of ethylene dichloride with which a specified production rate y mol/h of ethylene dichloride is realized. Let un, Ui2, and Ui3 mol/h be the component flow rates of chlorine, ethylene, and ethylene dichloride in stream i, respectively. The system of equations to be solved may be put in the following form, where u is an auxiliary variable in the reactor and r (= 0.90) is the conversion ratio of ethylene:

This is a system of linear/nonlinear equations in the unknown variables x, u, and Uij, where the equation UQS = x u53 in the purification is the only nonlinear equation. We may regard a, (j = 1,2) and r (— 0.90) as inaccurate and independent numbers. The stoichiometric coefficients in the reaction formula (12.2) are accurate numbers. The Jacobian matrix of (12.3), shown in Fig. 12.3, contains five inaccurate numbers, ai, oi2, r, x, and 1^53. The solvability of this problem will be treated in Example 12.12. Example 12.3. Consider the mechanical system in Fig. 12.4 consisting of two masses m\ andTO2,two springs fci and fc2, and a damper /; u is the force exerted

12.1.

Two Kinds of Numbers

351

Figure 12.3. Jacobian matrix in the chemical process simulation. from outside. This system may be described by vectors x = (xi,X2,X3,x 4 ,X5,o; 6 ) and u = (u), where x\ and x2 are the vertical displacements (downward) of masses m\ and m^, #3 and x± are their velocities, £5 is the force by the damper /, and XQ is the relative velocity of the two masses. The governing equation is

We may regard {rni,m2, &i, &2, /} as independent system parameters and other nonvanishing entries (i.e., ±1) of F, A, and B as fixed constants. The Laplace transform67 of the equation (12.4) gives a frequency domain description

where o;(0) = 0 and it(0) = 0 are assumed. The coefficient matrix [A — sF B] is a polynomial matrix in s with coefficients depending on the system parameters. 67

See Chen [23] or Zadeh-Desoer [220] for the Laplace transform.

352

Chapter 12. Application to Systems Analysis by Mixed Matrices

Figure 12.4. Mechanical system.

We have employed a six-dimensional vector x in our description of the system. It is possible, however, to describe this system using a four-dimensional state vector. The minimum dimension of the state vector is known to be equal to the degree in s of the determinant of

Thus the number degdet[A — sF] is an important characteristic, sometimes called the dynamical degree, of the system (12.4). As illustrated by the examples above, accurate numbers often appear in equations for conservation laws such as Kirchhoff's laws; the law of conservation of mass, energy, or momentum; and the principle of action and reaction, where the nonvanishing coefficients are either 1 or — 1, representing the underlying topological incidence relations. Another typical example is integer coefficients (stoichiometric coefficients) in chemical reactions such as

where nonunit integers such as 2 appear. In dealing with dynamical systems, we encounter another example of accurate numbers that represent the defining relations, such as those between velocity v and position x and between current £ and charge Q:

12.2.

Mixed Matrices and Mixed Polynomial Matrices

353

Figure 12.5. Accurate numbers. Typical accurate numbers are illustrated in Fig. 12.5. The rather intuitive concept of two kinds of numbers will be given a mathematical formalism in the next section.

12.2

Mixed Matrices and Mixed Polynomial Matrices

The distinction of two kinds of numbers can be embodied in the concepts of mixed matrices and mixed polynomial matrices. Assume that we are given a pair of fields F and K, where If is a subfield of F. Typically, K is the field Q of rational numbers and F is a field large enough to contain all the numbers appearing in the problem in question. In so doing we intend to model accurate numbers as numbers belonging to K and inaccurate numbers

Chapter 12. Application to Systems Analysis by Mixed Matrices

354

as numbers in F that are algebraically independent over K, where a family of numbers ti,..., tm of F is called algebraically independent over K if there exists no nonzero polynomial p(X\,..., Xm) over K such that p(ti,..., tm) = 0. Informally, algebraically independent numbers are tantamount to free parameters. A matrix A = (A^) over F, i.e., Aij e F, is called a mixed matrix with respect to (K, F) if where (M-Q) Q = (Qij) is a matrix over K and (M-T) T = (Tij) is a matrix over F such that the set of its nonzero entries is algebraically independent over K. We usually assume to make the decomposition (12.7) unique. Example 12.4. In the electrical network of Example 12.1 it is reasonable to regard {TI, r2, a, (3} as independent free parameters. Then the coefficient matrix in (12.1) is a mixed matrix with respect to (K, F) = (Q, Q(ri, r2, a, /?)), where Q(ri, r%, a, j3) means the field of rational functions in r\,r2,a,(3 with coefficients from Q. The decomposition A = Q + T is given by

Consider now a polynomial matrix in s with coefficients from F:

12.2.

Mixed Matrices and Mixed Polynomial Matrices

355

where Ak (k = 0 , 1 , . . . , N) are matrices over F. We say that A(s) is a mixed polynomial matrix with respect to (K, F) if it can be represented as

with

where (MP-Q) Qk (k = 0,1,..., N) are matrices over K and (MP-T) Tk (k = 0 , 1 , . . . , N) are matrices over F such that the set of their nonzero entries is algebraically independent over K. Obviously, the coefficient matrices

are mixed matrices with respect to (K, F). Also note that A(s) is a mixed matrix with respect to (K(s), F ( s ) ) in spite of the occurrence of the variable s in both Q(s) and T(s), where K(s) and F(s) denote the fields of rational functions in variable s with coefficients from K and JF, respectively. Mixed polynomial matrices are useful in dealing with linear time-invariant dynamical systems. The variable s here is primarily intended to denote the variable for the Laplace transform for continuous-time systems, though it could be interpreted as the variable for the z-transformes for discrete-time systems. Example 12.5. In the mechanical system of Example 12.3 it is reasonable to regard {mi,TO2, ki,kz, /} as independent free parameters. Then the matrix A(s) in (12.6) is a mixed polynomial matrix with respect to (K,F) = (Q, Q(rni,m2,fci,/c 2 ,/)). The decomposition A(s) = Q(s) + T(s) is given by

Our intention in the splitting (12.7) or (12.8) is to extract a meaningful combinatorial structure from the matrix A or A(s) by treating the Q-part numerically and the T-part symbolically. This is based on the following observations. 68

See Chen [23] or Zadeh-Desoer [220] for the z-transform.

356

Chapter 12. Application to Systems Analysis by Mixed Matrices

Q-part: As is typical with electrical networks, the Q-part primarily represents the interconnection of the elements. The Q-matrix, however, is not uniquely determined, but is subject to our choice in the mathematical description. In the electrical network of Example 12.1, for instance, the coefficient matrix

for Kirchhoff 's voltage law may well be replaced with

Accordingly, the structure of the Q-part should be treated numerically or linear algebraically. In fact, this is feasible in practice, since the entries of the Q-matrix are usually small integers, causing no serious numerical difficulty in arithmetic operations. T-part: The T-part primarily represents the element characteristics. The nonzero pattern of the T-matrix is relatively stable against our choice in the mathematical description of constitutive equations, and therefore it can be regarded as representing some intrinsic combinatorial structure of the system. It can be treated properly by graph-theoretic concepts and algorithms. Combination: The structural information from the Q-part and the T-part can be combined properly and efficiently by virtue of the fact that each part defines a combinatorial structure with discrete convexity (matroid or valuated matroid, to be more specific). Mathematical and algorithmic results in discrete convex analysis (matroid theory and valuated matroid theory) afford effective methods of systems analysis. We may summarize the above as follows: Q-part by linear algebra T-part by graph theory Combination by matroid theory

12.3

Rank of Mixed Matrices

The rank of a mixed matrix A = Q + T can be expressed in terms of the L^-convex functions (submodular set functions) associated with Q and T. This enables us, for example, to test efficiently for the solvability of the electrical network in Example 12.1 and of the chemical process simulation problem in Example 12.2. Let A = Q + T be a mixed matrix with respect to (K,F). The rank of A is denned with reference to the field F. That is, the rank of A is equal to (i) the maximum number of linearly independent column vectors of A with coefficients taken from F, (ii) the maximum number of linearly independent row vectors of A with coefficients taken from F, and (iii) the maximum size of a submatrix of A for

12.3.

Rank of Mixed Matrices

357

which the determinant does not vanish in jF. The row set and the column set of A are denoted by R and C, respectively. For / C R and J C C, the submatrix of A with row indices in / and column indices in J is designated by A[I, J]. We start with the nonsingularity of a mixed matrix. Proposition 12.6. A square mixed matrix A = Q + T is nonsingular if and only if there exist some I C R and J C C such that both Q[I, J] and T[R \I,C\J] are nonsingular. Proof. It follows from the denning expansion of the determinant that

with e(I, J) e {1, —1}. If A is nonsingular, we have det A^Q and hence det Q[I, J] • det T[R \ /, C \ J} ^ 0 for some / and J. The converse is also true, since no cancellation occurs among nonzero terms on the right-hand side by virtue of the algebraic independence of the nonzero entries of T. The following is a basic rank identity for a mixed matrix. Theorem 12.7. For a mixed matrix A = Q + T,

Proof. Proposition 12.6 applied to submatrices of A establishes (12.9).

D

The right-hand side of the identity (12.9) is a maximization over all pairs (/, J), the number of which is as large as 2' fl l + l c 'l, too large for an exhaustive search for maximization. Fortunately, however, it is possible to design an efficient algorithm to compute this maximum on the basis of the following facts: • The function p ( I , J) = rankQ[7, J] can be evaluated easily by Gaussian elimination. • The function r(I, J) = rankTf/, J] can be evaluated easily by finding a maximum matching in a bipartite graph representing the nonzero pattern of T. • The maximization can be converted, with the aid of Edmonds's intersection theorem for matroids (a special case of Theorem 4.18), to the minimum of an L''-convex function. To state the main theorem of this section we need another function 7 : 2R x 2° -> Z denned by

Note that 7(7, J) represents the number of nonzero rows of the submatrix T[I, J}.

358

Chapter 12. Application to Systems Analysis by Mixed Matrices

Theorem 12.8. For a mixed matrix A = Q + T,

Proof. (12.10) can be proved from (12.9) with the aid of Edmonds's intersection theorem for matroids (a special case of Theorem 4.18) and (12.11) can be derived from (12.10) using the formula which is a version of the fundamental min-max relation between maximum matchings and minimum covers. (12.12) follows easily from (12.11). For details see the proofs of Theorem 4.2.11 and Corollary 4.2.12 of Murota [146]. We mention the following theorem as an immediate corollary of the third identity (12.12). Note the duality nature of this theorem. Theorem 12.9 (Konig-Egervary theorem for mixed matrices). For a mixed matrix A = Q + T, there exist I C R and J C C such that (i) |/| + |J| -rankQ[7, J] = \R\ + \C\ -rankA, and (ii) rankT[J, J] =0. Proof. Take (/, J) that attains the minimum in (12.12).

D

To see the connection of the above rank formulas to L''-convexity, we define three functions gp,gr^9-y '• ZR(JC —> Z U {+00}, with domgp = domgr = domg7 {0,l}*uc,by

Proposition 12.10. gp, gT, and g^ are Lfi-convex functions. Proof. It is easy to see that p(I(JJ) = p(R\I, J) +1/| is a submodular set function on R U C (see (2.70)). This is equivalent to the L''-convexity of gp by Theorem 7.1, and similarly for gT and
12.4.

Degree of Determinant of Mixed Polynomial Matrices

359

where 1 e ZRuC. Note that both gp + gT and gp + #7 are lAconvex and therefore (gp + gT)' and (gp + g^)9 are M^-convex. As for (12.12) we observe that

is an L'-convex set and

This shows that the right-hand side of (12.12) is the minimum of an lAconvex function over an L^-convex set. The discrete convexity implicit in Theorem 12.8 is thus exposed. A concrete algorithmic procedure for computing the rank of A = Q + T is described in section 4.2 of Murota [146]. Example 12.11. The unique solvability of the electrical network in Example 12.1 can be shown by Theorem 12.7 applied to A = Q + T in Example 12.4. In (12.9) the maximum value of 10 is attained by / = {1,2,3,4,5,7,10} and J = {3,4,5,7,8,9,10}. In Theorem 12.8 the right-hand sides of (12.10), (12.11), and (12.12) are equal to 10 with the minima attained by / = R and J = 0. Example 12.12. The generic solvability of the chemical process simulation problem in Example 12.2 is denied by Theorem 12.8 applied to the Jacobian matrix in Fig. 12.3. In (12.12) the minimum is attained by

Note that 7(7, J) = 0, |J| = |J| = 10, and p ( I , J ) = 3, for which p ( I , J ) - \I\ \J\ + \R\ + \C\ = 3 - 10 - 10 + 16 + 16 = 15 < \R\ = \C\ = 16. This shows that t Jacobian matrix in Fig. 12.3 is singular, and hence the simulation problem is not solvable in general.

12.4

Degree of Determinant of Mixed Polynomial Matrices

The degree of determinant of a mixed polynomial matrix A(s) = Q(s)+T(s) can be expressed in terms of the infimal convolution of two M-convex functions associated with T(s) and Q(s). This enables us, for example, to compute the dynamical degree of the mechanical system in Example 12.3 in an efficient way by solving an M-convex submodular flow problem. Let A(s) = (Aij(s)) be a polynomial matrix with each entry being a polynomial in s with coefficients from a certain field F. We denote by R and C the row set and the column set of A(s). The degree of minors (subdeterminants) is an important characteristic of A(s). For example, the sequence of 6k (k = 1, 2 , . . . ) of the highest degree in s of a minor of order fc,

360

Chapter 12. Application to Systems Analysis by Mixed Matrices

determines the Smith-McMillan form at infinity as well as the structural indices of the Kronecker form (see section 5.1 of Murota [146]). Here the function to be maximized in (12.13) is essentially M-concave, since w : 2,RUC —> Z U {—00} defined by o>(/ U J) = 5(R \ I, J) for / C R and J C C is a valuated matroid (see (2.74) and (2.77) as well as Example 5.2.15 of Murota [146]). The following is the basic identity for the degree of the determinant of a mixed polynomial matrix. Theorem 12.13. For a square mixed polynomial matrix A(s) = Q(s) + T(s), degdet^ = max{degdetQ[7, J]+degdetT[R\I,C\

J] \ \I\ = \ J\,I C R, J C C}, (12.14)

where both sides are equal to —oo if A is singular. Proof. It follows from the denning expansion of the determinant that

with e ( I , J ) 6 {!,—!}. Since the degree of a sum is bounded by the maximum degree of a summand, we obtain

The inequality turns into an equality if the highest degree terms do not cancel one another. The algebraic independence of the nonzero coefficients in T(s) ensures this. D The right-hand side of the identity (12.14) is a maximization over all pairs (7, J), the number of which is as large as '2\R\+\C\, too large for an exhaustive search for maximization. Fortunately, however, it is possible to compute this maximum efficiently by reducing this maximization problem to the M-convex submodular flow problem. To see the connection to M-convexity, we define functions /Q,/T : ZRuC —> Z U {+00} with dom/ Q ,dom/ T C {0,1} RUC by

Both fq and fa are M-convex functions. The right-hand side of (12.14) can now be identified as the negative of an integer infimal convolution of these M-convex functions. Namely,

12.4.

Degree of Determinant of Mixed Polynomial Matrices

361

where 1 € ZRUC. This reveals the discrete convexity implicit in Theorem 12.13 and also shows an efficient way to compute the degree of determinant of a mixed polynomial matrix A(s) = Q(s) + T(s), since the infimal convolution of M-convex functions can be computed efficiently by solving an M-convex submodular flow problem, as we have seen in Note 9.30. Such an algorithm is described in detail in section 6.2 of Murota [146]. Example 12.14. The dynamical degree of the mechanical system in Example 12.3 can be computed by Theorem 12.13 applied to A(s) = Q(s) + T(s) in Example 12.5. In (12.14) the maximum value of 4 is attained by I = {1,2,5,6} and J = {1,2,5,6}. Hence the dynamical degree is equal to four. •

Bibliographical Notes This chapter is largely based on Murota [146]. The observation on two kinds of numbers and the concept of mixed matrices are due to Murota-Iri [149], [150], in which Theorem 12.7 is given. Theorem 12.8 is taken from [146]. The Konig-Egervary theorem for mixed matrices (Theorem 12.9) is due to Bapat [7] and Hartfiel-Loewy [86]. The connection between mixed polynomial matrices and M-convexity explained in section 12.4 is due to Murota [143] and a related topic can be found in Iwata-Murota [104]. Applications of matroid theory to electrical networks are fully expounded in Iri [95] and Recski [175]. When gyrators are involved in electrical networks, a generalization of mixed matrices to mixed skew-symmetric matrices is useful, as is explained in section 7.3 of Murota [146]. See Geelen-Iwata [75] and Geelen-IwataMurota [76] for recent results on mixed skew-symmetric matrices. Matroid theory also finds applications in statics and scene analysis (GraverServatius-Servatius [80], Recski [175], Sugihara [195], Whiteley [215], [216], [217]). For planar truss structures, in particular, a necessary and sufficient condition for generic (infinitesimal) rigidity can be expressed in terms of unions of graphic matroids, where a matroid union is a special case of the Minkowski sum of two Mconvex sets. It is noted that the rigidity of a truss structure can be represented by a rank condition on a matrix associated with the truss, but that this matrix does not fall into the category of mixed matrices. Recent results on rigidity in nongeneric cases are surveyed in Radics-Recski [174].

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Index entering, 53 leaving, 52 augmenting path, 60, 273, 274 6-, 297 auxiliary network, 252, 263

accurate number, 347 active triple, 297 acyclic, 107 admissible potential, 122 affine hull, 78 agent, 324 aggregate cost function, 335 aggregation of function to subset, 143, 162 by network transformation, 272 algebraically independent, 354 algorithm competitive equilibrium, 344 conjugate scaling, 320 cycle-canceling, 313 domain reduction, 284 domain reduction scaling, 287 fully combinatorial, 290 greedy, 3, 108 IFF fixing, 300 IFF scaling, 299 L-convex function minimization, 305, 306, 308 M-convex function minimization, 281, 283, 284, 287 primal-dual, 315 pseudopolynomial, 288 Schrijver's, 293 steepest descent, 281, 305, 306 steepest descent scaling, 283, 308 strongly polynomial, 288 submodular function minimization, 293, 299, 300 successive shortest path, 312 two-stage, 310 weakly polynomial, 288 arc, 52

base, 105 extreme, 105 matrix, 69 matroid, 70 base family matrix, 69 matroid, 70 valuated matroid, 72 base polyhedron, 18, 105 integral, 18 biconjugate function, 82 integer, 212 bipartite graph, 89 bipartite matching, 89 Birkhoff's representation theorem, 292 Boolean lattice, 104 boundary, 53 branch, 52 budget set, 324 certificate of optimality, 12 chain, 88 characteristic curve, 54, 251 discrete, 57 characteristic vector, 16 chemical process, 349 Choquet integral, 16, 104 closed convex function, 79 closed convex hull, 78 closed interval, 77 closure 379

380

concave function, 216 convex function, 93 convex set, 78 coboundary, 53, 248 another convention, 253 cocontent, 55 combinatorial optimization, 3 commodity divisible, 327 indivisible, 323 compartmental matrix, 43 competitive economy, 324 competitive equilibrium, 325 complementarity, 88 complements, 62 concave closure, 216 concave conjugate, 11, 81 discrete, 212 concave extensible, 93 concave extension, 93 concave function, 9, 78 quasi-separable, 334 separable, 333 conductance, 41 cone, 78 convex, 78 L-convex, 131 M-convex, 119 polar, 82 conformal decomposition, 64 conjugacy theorem closed proper M-/L-convex, 210 in convex analysis, 11, 82 discrete M-/L-convex, 30, 212 polyhedral M-/L-convex, 209 conjugate function concave, 81, 212 convex, 81, 212 conjugate scaling, 319 conjugate scaling algorithm, 320 conservation law, 54 constitutive equation, 54, 349 constraint, 1 consumer, 323 consumption, 323 content, 55

Index

contraction normal, 45 unit, 45 convex closure function, 93 set, 78 convex combination, 78 convex cone, 78 convex conjugate, 10, 81 discrete, 212 convex extensible, 93 convex extension, 93 local, 93 convex function, 2, 9, 77 closed, 79 dual-integral polyhedral, 161 integral polyhedral, 161 laminar, 141 polyhedral, 80 positively homogeneous, 82 proper, 77 quadratic, 40 quasi, 168 quasi-separable, 140 separable, 10, 95, 140, 182 strictly, 77 univariate, 10 convex hull, 78 closed, 78 convex polyhedron, 78 convex program, 2 M-, 235 convex set, 2, 78 convexity discrete midpoint, 23, 129, 180 function, 77 in intersection, 92 midpoint, 9 in Minkowski sum, 92 quasi, 168 set, 78 convolution infimal, 80 integer infimal, 143 by network transformation, 272 cost function

Index

aggregate, 335 flow, 53, 246, 255, 256 flow boundary, 256 producer's, 324 reduced, 249 tension, 53 current, 41, 53 current potential, 55 cut capacity function, 247 cycle negative, 122, 252, 263 simple, 62 cycle-canceling algorithm, 313 decreasing marginal return, 330 demand correspondence, 325 set, 325 descent direction, 147 diagonal dominance, 41 directed graph, 52, 88 directional derivative, 80 Dirichlet form, 45 discrete Legendre-Fenchel transformation, 13, 212 discrete midpoint convexity function, 23, 180 set, 129 discrete separation theorem generic form, 13, 216 L-convex function, 218 L-convex set, 36, 126 M-convex function, 217 M-convex set, 36, 114 submodular function, 17, 111 submodular function (as special case of L-separation), 33, 224 discreteness in direction, 10 in value, 13 distance function, 122 distributive lattice, 292 distributive law, 292 divisible commodity, 327 domain reduction algorithm, 284

381

domain reduction scaling algorithm, 287 dual-integral polyhedral convex function, 161 L-convex function, 191 M-convex function, 161 dual integrality intersection theorem, 20, 114 linear programming, 89 minimum cost flow problem, 252 polyhedral convex function, 161 polyhedral L-convex function, 191 polyhedral M-convex function, 161 submodular flow problem, 261 dual linear program, 87 dual problem, 87 dual variable, 53 duality, 2, 11 Edmonds's intersection theorem, 20 Fenchel, 85 Fenchel-type, 222, 225 L-separation, 218 linear programming, 87 M-separation, 217 matroid intersection, 225 separation for convex functions, 84 separation for convex sets, 35, 83 separation for L-convex functions, 218 separation for M-convex functions, 217 separation for submodular functions, 17, 111 strong, 87 valuated matroid intersection, 225 weak, 87 weight splitting, 34, 225 dynamical degree, 352 economy of Arrow-Debreu type, 323 Edmonds's intersection theorem, 3, 20, 112

382

(as special case of Fenchel-type duality), 34, 224 effective domain function over Rn, 9, 21, 77 function over Zn, 21 set function, 103 electrical network, 41, 43, 348 multiterminal, 53 elementary vector, 64 entering arc, 53 epigraph, 79 equilibrium competitive, 325 economy, 325 electrical network, 55 exchange axiom (B-EXC[R]), 118 (B-EXC+pRJ), 118 (B-EXC[Zj), 18, 101 (B-EXC+[Z]), 102 (B-EXC_[Z]), 103 (B-EXC W [ZJ), 103 (B>>-EXC[R]), 118 (B"-EXC[Z]), 117 local, 135 M-convex function, 26, 58, 133 M-convex polyhedron, 118 M-convex set, 18, 101 M^-convex function, 27, 134 M^-convex polyhedron, 118 M^-convex set, 117 (M-EXC[R]), 29, 56, 160 (M-EXC'[R]), 160 (M-EXC[Z]), 26, 58, 133 (M-EXC'[Z]), 26, 133 (M-EXCioc[Z]), 135 (M-EXC w [Zj), 137 (M>i-EXC[R]), 29, 47, 162 (M^-EXC'[R]), 162 (M^-EXC+[R]), 48 (M^-EXC[Z]), 27, 134 (Mb-EXC'[Z]), 134 (-M"-EXC[Z]), 330 matroid, 69 multiple, 333

Index

polyhedral M-convex function, 29, 56, 160 polyhedral M^-convex function, 29, 47, 162 simultaneous, 69 weak, 137 exchange capacity, 284, 312 exchange economy, 327 extension concave, 93 convex, 93 distance function, 165 local convex, 93 Lovasz, 16, 104, 111 partial order, 108 set function, 16, 104 extreme base, 105 Farkas lemma, 50, 87 feasible 6-, 296

dual problem, 236 flow, 247, 258 minimum cost flow problem, 247 potential, 122 primal problem, 235 set, 1 submodular flow problem, 258 Fenchel duality, 12, 85 Fenchel transformation, 81 Fenchel-type duality generic form, 13 L-convex function, 32, 222 M-convex function, 32, 222 submodular function, 225 fixed constant, 347 flow, 53 ^-feasible, 296 feasible, 247, 258 Frank's discrete separation theorem, 17, 111 (as special case of L-separation), 33, 224 Frank's weight-splitting theorem, 34, 225 fully combinatorial algorithm, 290

Index

fundamental circuit, 149 g-polymatroid, 117 generalized polymatroid, 117 generator, 45 global minimizer, 79 global optimality, 2 global optimum, 9 goods divisible, 327 indivisible, 323 gradient, 80 graph acyclic, 107 bipartite, 89 directed, 52, 88 Grassmann-Pliicker relation, 69 greedy algorithm, 3, 108 gross substitutes property, 153, 331 stepwise, 155, 331 ground set, 70 gyrator, 361 Hamiltonian path problem, 257 hole free, 90 ideal, 107 IFF fixing algorithm, 300 IFF scaling algorithm, 298, 299 in kilter, 314 inaccurate number, 347 incidence chain, 88 graph, 88 topological, 347 income, 324 independent set matrix, 68 matroid, 70 indicator function, 79, 90 indivisible commodity, 323 indivisible goods, 323 infimal convolution, 80 integer, 143 by network transformation, 272 initial endowment, 324

383

total, 326 initial vertex, 53 inner product, 79 integer biconjugate, 212 integer infimal convolution, 143 integer interval, 92 integer subdifferential, 166 integral base polyhedron, 18 integral L-convex polyhedron, 131 integral M-convex polyhedron, 118 integral neighborhood, 93 integral polyhedral convex function, 161 L-convex function, 191 L''-convex function, 192 M-convex function, 161 M^-convex function, 162 integral polyhedron, 90 integrality dual, 161, 252, 261 linear programming, 89 minimum cost flow problem, 252 polyhedral convex function, 161 polyhedral L-convex function, 191 polyhedral M-convex function, 161 polyhedron, 90 primal, 252, 261 submodular flow problem, 261 integrally concave function, 94 integrally convex function, 7, 94 submodular, 189 integrally convex set, 96 intersection convexity in, 92 M-convex, 219 matroid, 3 submodular polyhedron, 20 valuated matroid, 225 intersection theorem Edmonds's, 3, 20, 112 Edmonds's (as special case of Fencheltype duality), 34, 224 M-convex, 219 valuated matroid, 225 weighted matroid, 225 interval

Index

384

closed, 77 integer, 92 open, 77 jump system, 120 kilter diagram, 53, 251 in, 314 out of, 314 Kirchhoff's law, 349 current, 353 voltage, 353 Konig-Egervary theorem for mixed matrix, 358 L-concave function, 22 polyhedral, 190 L-convex cone, 131 L-convex function, 8, 22, 177 dual-integral polyhedral, 191 integral polyhedral, 191 polyhedral, 190 positively homogeneous, 193 quadratic, 52, 182 quasi, 199 semistrictly quasi, 199 L-convex polyhedron, 123, 131 integral, 131 L-convex set, 22, 121 L-optimality criterion, 185, 193 quasi, 201 L-proximity theorem, 186 quasi, 201 L-separation theorem, 33, 218 L2-optimality criterion, 232 L2-proximity theorem, 232 L2-convex function, 229 L2-convex set, 128 L^-convex function, 229 L2-convex set, 129 L''-convex function, 8, 23, 178 integral polyhedral, 192 polyhedral, 192 quadratic, 48, 52, 182 L^-convex polyhedron, 129, 131

L^-convex set, 121, 128 Lagrange duality, 234 Lagrangian function, 236 dual, 242 laminar convex function, 141 by network transformation, 273 laminar family, 141 Laplace transform, 351 lattice, 292 distributive, 292 sub-, 104 leading principal minor, 40 leading principal submatrix, 40 leaving arc, 52 Legendre-Fenchel transform concave, 11, 81 convex, 10, 81 discrete, 13, 212 Legendre-Fenchel transformation concave, 11, 81 convex, 10, 81 discrete, 13, 212 Legendre transformation, 81 level set, 172 linear extension partial order, 108 set function, 16, 104 linear order, 108 linear program, 87 dual problem, 87 primal problem, 87 linear programming, 86 duality, 87 linearity in direction 1, 177, 190 local convex extension, 93 local optimality, 2 local optimum, 10 Lovasz extension, 16, 104, 111 LP, 87 duality, 87 M-concave function, 8, 26 polyhedral, 160 M-convex cone, 119 M-convex function, 8, 26, 133 dual-integral polyhedral, 161

Index integral polyhedral, 161 polyhedral, 160 positively homogeneous, 164 quadratic, 52, 139 quasi, 169 semistrictly quasi, 169 M-convex intersection problem, 219, 264 theorem, 219 M-convex polyhedron, 108, 118 integral, 118 M-convex program, 235 M-convex set, 27, 101 M-convex submodular flow problem, 256 economic equilibrium, 341 M-matrix, 42 M-minimizer cut, 149 with scaling, 158 M-optimality criterion, 148, 163 quasi, 173 M-proximity theorem, 156 quasi, 174 M-separation theorem, 33, 217 M2-optimality criterion, 227, 228 M2-proximity theorem, 228 M2-convex function, 226 Mg-convex set, 116 M2-convex function, 226 M^-convex set, 117 M^-convex function, 8, 27, 134 integral polyhedral, 162 polyhedral, 161 quadratic, 48, 52, 139 M^-convex polyhedron, 117, 118 M^-convex set, 102, 117 Markovian, 45 matching, 89 bipartite, 89 perfect, 89 weighted, 89, 266 mathematical programming, 1 matrix compartmental, 43 incidence (chain), 88

385

incidence (graph), 88 M-, 42 mixed, 354 mixed polynomial, 355 mixed skew-symmetric, 361 node admittance, 42 polynomial, 71, 354 positive-definite, 39 positive-semidefinite, 39 principal sub-, 40 totally unimodular, 88 matroid, 70 induction through a graph, 270 intersection problem, 34, 225 valuated, 72 max-flow min-cut theorem for submodular flow, 259 maximum submodular flow problem, 259 maximum weight circulation problem, 61 mechanical system, 350 midpoint convexity, 9 discrete function, 23, 180 discrete set, 129 Miller's discrete convex function, 98 min-max relation, 2 minimizer, 79 global, 9, 79 integrally convex function, 94 L-convex function, 185, 305 L2-convex function, 232 local, 10 M-convex function, 148, 281 M2-convex function, 227, 228 maximal, 290, 291, 304, 307 minimal, 290, 291, 305, 307 submodular set function, 288 minimizer cut M-convex function, 149 M-convex function with scaling, 158 quasi M-convex function, 174 quasi M-convex function with scaling, 175 minimum cost flow problem, 53, 245

386

integer flow, 246 minimum cut, 316 minimum spanning tree problem, 149 Minkowski sum, 80, 90 convexity in, 92 discrete, 90 integral, 90 minor, 40, 359 leading principal, 40 principal, 40 mixed matrix, 354 mixed polynomial matrix, 355 mixed skew-symmetric matrix, 361 money, 323 monotonicity, 54 multimodular function, 183 multiple exchange axiom, 333 multiterminal electrical network, 53 negative cycle, 122, 252, 263 criterion, 252, 263, 264 negative support, 18 neighborhood, integral, 93 network, 53 auxiliary, 252, 263 electrical, 53 transformation by, 270 network flow duality, 268 electrical network, 41, 43 L-convexity, 24, 31, 56, 58, 270 M-convexity, 28, 31, 56, 58, 270 maximum weight circulation, 61 minimum cost flow, 245 multiterminal, 53 submodular flow, 255 no complementarities property, 332 strong, 332 node, 52 node admittance matrix, 42 normal contraction, 45 objective function, 1 off-diagonal nonpositivity, 41 open interval, 77 optimal potential, 89, 251

Index

optimal value function, 236 optimality global, 2, 9 local, 2, 10 optimality criterion integrally convex function, 94, 95 L-convex function, 185, 193 L2-convex function, 232 M-convex function, 148, 163 M-convex submodular flow, 262264 M2-convex function, 219, 227, 228 minimum cost flow, 249, 252 by negative cycle, 252, 263, 264 by potential, 249, 260, 262 quasi L-convex function, 201 quasi M-convex function, 173 submodular flow, 260 submodular set function, 185 sum of M-convex functions, 219 valuated matroid intersection, 225 weighted matroid intersection, 225 optimization combinatorial, 3 continuous, 1 discrete, 3 optimum global, 9 local, 10 out of kilter, 314 pairing, 79 parallel, 62 partial order acyclic graph, 107 extreme base, 108 perfect matching, 89 minimum weight, 89, 266 Poisson equation, 41, 43, 47 polar cone, 82 polyhedral convex function, 25, 80 L-concave function, 190 L-convex function, 190 L^-convex function, 192 M-concave function, 160

Index

M-convex function, 160 M^-convex function, 161 method, 8 polyhedron base, 105 convex, 78 integral, 90 integral L-convex, 131 integral M-convex, 118 L-convex, 123, 131 L^-convex, 129, 131 M-convex, 108, 118 M^-convex, 117, 118 rational, 90 submodular, 112 polynomial matrix, 71, 354 mixed, 355 polytope, 90 positive definite, 39 positive semidefinite, 39 positive support, 18 positively homogeneous function, 7, 82 L-convex function, 193 M-convex function, 164 potential, 41, 53, 89, 248 criterion, 249, 260, 262 optimal, 251 primal-dual algorithm, 315 primal integrality, 252, 261 intersection theorem, 20, 114 linear programming, 89 primal problem, 87 principal minor, 40 principal submatrix, 40 leading, 40 producer, 323 production, 323 profit, 324 function, 324 projection base polyhedron, 117 function to subset, 143, 162 M-convex function, 134 M-convex polyhedron, 118 M-convex set, 102

387

M2-convex function, 226 Mg-convex set, 117 polyhedral M-convex function, 161 proper convex function, 77 proximity theorem, 156 L-convex function, 186 L2-convex function, 232 M-convex function, 156 M2-convex function, 228 quasi L-convex function, 201 quasi M-convex function, 174 pseudopolynomial algorithm, 288 quadratic form, 39 function, 39 L-convex function, 182 L^-convex function, 48, 52, 182 M-convex function, 139 M-convex function, 48, 52, 139 quasi convex, 168 semistrictly, 168 quasi L-convex function, 199 quasi L-optimality criterion, 201 quasi L-proximity theorem, 201 quasi linear, 324 quasi M-convex function, 169 quasi M-minimizer cut, 174 with scaling, 175 quasi M-optimality criterion, 173 quasi M-proximity theorem, 174 quasi-separable concave function, 334 convex function, 140 quasi submodular, 198 semistrictly, 198 rank function matrix, 69 matroid, 70 rational polyhedron, 90 reduced cost, 249, 251 relative interior, 78 reservation value function, 325 resistance, 41 resolvent, 45

388

resource allocation problem, 4, 176 restriction function to interval, 92 function to subset, 143, 162 L-convex function, 178 L-convex polyhedron, 131 L-convex set, 121 L2-convex function, 229 L2-convex set, 129 polyhedral L-convex function, 192 rigidity, 361 ring family, 104, 107, 292 saddle-point theorem, 238 scaling, 145 conjugate, 319 cost, 318 domain, 145 nonlinear, 170, 199 scaling algorithm L-convex function, 308 M-convex function, 283, 287 M-convex submodular flow, 320 semigroup, 45 semistrictly quasi convex, 168 semistrictly quasi L-convex, 199 semistrictly quasi M-convex, 169 semistrictly quasi submodular, 198 separable concave function, 333 quasi, 334 separable convex function, 10, 95, 140, 182 with chain condition, 182 quasi, 140 separation theorem convex function, 2, 11, 84 convex set, 35, 83 generic discrete, 13, 216 L-convex function, 218 L-convex set, 36, 126 M-convex function, 217 M-convex set, 36, 114 submodular function, 17, 111 series, 62 simple cycle, 62 single improvement property, 332

Index

spanning tree, 149 stable marriage problem, 345 stable matching problem, 345 steepest descent algorithm L-convex function, 305, 306 M-convex function, 281 steepest descent scaling algorithm L-convex function, 308 M-convex function, 283 stepwise gross substitutes property, 155, 331 stoichiometric coefficient, 350 strictly convex function, 77 strong duality, 87 strong no complementarities property, 332

strongly polynomial algorithm, 288 structural equation, 54, 349 subdeterminant, 40, 359 subdifferential, 80 concave function, 217 discrete function, 166 integer, 166 subgradient, 80 discrete function, 166 sublattice, 104 submatrix leading principal, 40 submodular, 62, 206 function, 16, 44, 70, 104 function on distributive lattice, 292 integrally convex-function, 7, 189 polyhedron, 20, 112 utility function, 330 submodular flow problem, 255 economic equilibrium, 341 feasibility theorem, 258 M-convex, 256 maximum, 259 submodular function minimization IFF fixing algorithm, 300 IFF scaling algorithm, 298, 299 Schrijver's algorithm, 293 submodularity, 44, 177, 190 inequality, 16, 44, 177 local, 180

Index

substitutes, 62 successive shortest path algorithm, 312 sum

of functions, 80 of M-convex functions, 226 supermodular, 16, 62, 105, 145, 206 function 105 supply correspondence, 324 set, 324 support function, 82 negative, 18 positive, 18 system parameter, 347 tension, 53 another convention, 253 terminal vertex, 52, 53 tight set, 108 transformation by network, 269 of flow type, 269 of potential type, 269 transitive, 107, 119 translation submodularity, 23, 44, 178 triangle inequality, 24, 122 two-stage algorithm, 310 unimodular totally, 88 unique-min condition, 266 unit contraction, 45 unit demand preference, 334 univariate function, 10 discrete convex, 95 polyhedral convex, 80 utility function, 324 valuated matroid, 7, 72, 225 intersection problem, 225 valuation, 72 variational formulation, 43, 55 vertex, 52 initial, 53 terminal, 53 voltage, 41, 53

389

voltage potential, 55 weak duality, 87 exchange axiom, 137 weakly polynomial algorithm, 288 weight splitting matroid intersection, 34, 225 valuated matroid intersection, 225

weak

z-transform, 355