Polyhedra with submodular support functions and their unbalanced simultaneous exchangeability

Discrete Applied Mathematics 131 (2003) 433 – 448

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Polyhedra with submodular support functions and their unbalanced simultaneous exchangeability Kenji Kashiwabaraa , Takashi Takabatakeb;∗ a Department

of Systems Science, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan b Division of Systems Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-machi, Toyonaka, Osaka 560-8531, Japan Received 1 September 2000; received in revised form 2 July 2001; accepted 2 July 2001

Abstract We discuss matroid-likeness of polyhedra whose facets have non-01-vectors as their normal vectors. We propose, as a generalized class of submodular polyhedra, the class of down-monotone polyhedra whose support functions satisfy submodularity on non-negative vectors. The sets of feasible out2ows of certain bipartite generalized networks are examples of such polyhedra. We prove that such polyhedra have certain unbalanced simultaneous exchangeability between two axes. This property gives a simple criterion of optimality for a linear objective function on these polyhedra. We also prove that this simultaneous exchangeability characterizes this generalized class of polyhedra, while a non-simultaneous version of this exchangeability does not. ? 2003 Elsevier B.V. All rights reserved. Keywords: Submodular polyhedra; Matroids; Simultaneous exchange; Generalized 2ow

1. Introduction The notion of matroids, which is equivalent to the success of certain greedy algorithm, explains basic discrete structures in combinatorial optimization. A matroid is a pair (S; I) of a 8nite non-empty set S and I ⊆ 2s which satis8es: (I1) I is not empty; ∗

Corresponding author. E-mail addresses: [email protected] (K. Kashiwabara), [email protected] (T. Takabatake). 0166-218X/03/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. PII: S 0 1 6 6 - 2 1 8 X ( 0 2 ) 0 0 4 6 6 - 3

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(I2) for all X ∈ I, every subset of X belongs to I; (I3) for all X; Y ∈ I with |X | ¡ |Y |, there exists s ∈ Y \ X such that X ∪ {s} ∈ I. From this de8nition, a matroid (S; I) satis8es the following simultaneous exchangeability: (SIMEX) For all X; Y ∈ I and for every s ∈ Y \ X , either X ∪ {s}; Y \ {s} ∈ I or there exists t ∈ X \ Y such that (X ∪ {s}) \ {t}; (Y \ {s}) ∪ {t} ∈ I, which guarantees some greedy algorithm to succeed for a kind of optimization problems on matroids. It should be noted that (SIMEX) is equivalent, under (I1) and (I2), to the following non-simultaneous exchangeability (EX), since the condition (I3) may be replaced with (EX): (EX) For all X; Y ∈ I and for every s ∈ Y \ X , either X ∪ {s} ∈ I or there exists t ∈ X \ Y such that (X ∪ {s}) \ {t} ∈ I. See [13,14,16] for more details of matroids. Several classes of polyhedra have been introduced via submodular functions as generalizations of matroids. The class of submodular polyhedra is one of such classes. A submodular polyhedron P ⊆ RS is a polyhedron which can be represented as 
 S x(i) 6 f(X ) for all X ⊆ S P = x∈R | i∈X

with some function f : 2S → R ∪ {+∞} satisfying f(∅) = 0; f(S) ¡ + ∞, and the following submodularity for set functions: f(X ) + f(Y ) ¿ f(X ∪ Y ) + f(X ∩ Y )

for all X; Y ⊆ S;

(1)

where we assume +∞ ¿ + ∞ as a convention. See [6] for more details about submodular polyhedra. Submodular polyhedra, and other generalized polyhedra, such as polymatroids [4], generalized polymatroids [5], bisubmodular polyhedra [3,2,1], have only facets which may be determined by their normal vectors composed of 0; 1; −1. In this paper, we propose, as a generalizations of the matroid, a class of polyhedra named extended submodular polyhedra, which may have facets with arbitrary non-negative normal vectors. This class includes the class of submodular polyhedra. Extended submodular polyhedra are capable of describing the sets of feasible out2ows of certain generalized bipartite networks. Here, facets determined by non-01-vectors are derived from gain factors. We de8ne extended submodular polyhedra as down-monotone polyhedra whose support functions, 1 denoted by f : RS → R ∪ {+∞} here, satisfy f(p) + f(q) ¿ f(p ∨ q) + f(p ∧ q)

for all p; q ∈ RS :

(2)

In (2), R+ denotes the set of non-negative reals, and for p; q ∈ RS+ , the symbols p ∨ q and p ∧ q denote the vectors in RS+ de8ned by (p ∨ q)(i) := max{p(i); q(i)}; 1

(p ∧ q)(i) := min{p(i); q(i)} (i ∈ S):

See Section 2 for the de8nition of support functions.

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It is known that the support functions of submodular polyhedra satisfy the same submodularity as (2) and this submodularity is used to de8ne the L-convex functions [11] and the L] -convex functions [7], which are regarded as discrete versions of convex functions. In Section 4, we show certain unbalanced simultaneous exchangeability of extended submodular polyhedra. We also prove in Section 5 that this unbalanced simultaneous exchangeability characterizes the class of extended submodular polyhedra. These results generalize the fact that the class of submodular polyhedra is characterized by a balanced version of simultaneous exchangeability, for the complete proof of which we refer to [12]. These results justify our generalization. In Section 6, we point out that simultaneous exchangeability is not equivalent to non-simultaneous one for extended submodular polyhedra, while their equivalence is established for matroids and several other generalized structures including submodular polyhedra [10–12].

2. Preliminaries For a vector p ∈ [−∞; +∞]S , we write supp(p) and supp+ (p) for {i ∈ S | p(i) = 0} and {i ∈ S | p(i) ¿ 0}, respectively. For i ∈ S, the symbol i denotes i’s characteristic vector in RS , i.e., i (i) = 1 and i (j) = 0 for each j ∈ S \ {i}. A set P ⊆ RS is called down-monotone if for each y ∈ P, every vector x ∈ RS with x 6 y belongs to P. Note that every submodular polyhedron is down-monotone. The support function ∗C : RS → R ∪ {+∞} of a convex set C ⊆ RS is de8ned by ∗C (p) := sup{p; x | x ∈ C}

(p ∈ RS ):

Every support function ∗C is convex and positively homogeneous, i.e., ∗C (kp)=k∗C (p) for all k ¿ 0 and all p ∈ RS . For a down-monotone convex set, its support function may have 8nite values only for non-negative vectors. Therefore, we sometimes treat the support functions of a down-monotone polyhedron as a function de8ned on non-negative vectors. Therefore, we also call restrictions of support functions for such down-monotone polyhedra to the non-negative vectors just support functions, in this paper. For a polyhedron P ⊆ RS and a point x ∈ P, we say that a vector p ∈ RS is x-tight if p; x = ∗P (p). For a full-dimensional polyhedron P, we use the notation P(P) for the set of the normal vectors of facets of P which are normalized so that the  l1 -norm of each vector in P(P) is equal to 1. (The l1 -norm of p ∈ RS is p1 := i∈S |p(i)|:) Then P = {x ∈ RS | p; x 6 ∗P (p) for all p ∈ P(P)} holds. We also use the notation Px (P) de8ned by Px (P) := {p ∈ P(P) | p; x = ∗P (p)}

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for each x ∈ P. Note that P(P) is a 8nite set for a full-dimensional polyhedron P. If the polyhedron P is down-monotone, all vectors in P(P) are non-negative. 3. Extended submodular polyhedra We propose the following class of polyhedra as a generalized class of submodular polyhedra. Denition 1. A down-monotone polyhedron P ⊆ RS is an extended submodular polyhedron if its support function ∗P : RS → R ∪ {+∞} satis8es: ∗P (p) + ∗P (q) ¿ ∗P (p ∨ q) + ∗P (p ∧ q)

for all p; q ∈ RS+ :

(3)

Remark 2. By down-monotonicity, the support function of an extended submodular polyhedron satis8es not only “submodularity for non-negative directions” (3) but also the following “submodularity for all directions”: ∗P (p) + ∗P (q) ¿ ∗P (p ∨ q) + ∗P (p ∧ q)

for all p; q ∈ RS :

Since the support function of a submodular polyhedron satis8es (3), every submodular polyhedron is an extended submodular polyhedron. We list several operations that preserve the class of extended submodular polyhedra. Proposition 3. For any extended submodular polyhedra P; Q ⊆ RS , any vectors a, u ∈ RS , any diagonal |S| × |S|-matrix A whose diagonals are positive, any subset S  of S, any j ∈ S, and any b ∈ R, the following polyhedra are also extended submodular polyhedra: [Translation] P + a = {x + a ∈ RS | x ∈ P}; [Axiswise scaling] AP = {Ax ∈ RS | x ∈ P}; [Vector sum] P + Q = {x + y ∈ RS | x ∈ P; y ∈ Q};
[Projection] P|S  = {x ∈ RS | y ∈ P; x(i) = y(i) for all i ∈ S  }; [Fixing a component] Pj:b ={x ∈ RS\{j} | y ∈ P; y(j)=b; x(i)=y(i) for all i ∈ S \{j}}; [Reduction by a vector] P u = {x ∈ RS | x ∈ P; x 6 u}. Proof. “Translation” and “Axiswise scaling” are obvious. “Vector sum” follows from the fact that the support function of the vector sum of two convex sets is the sum of two respective support functions (see[15, p. 133] for example). “Projection”, “Fixing a component”, and “Reduction by a vector” follow from Theorem 14 proved in Section 5. Fig. 1 shows an example of extended submodular polyhedra. A class of extended submodular polyhedra is obtained from certain generalized bipartite network 2ows.

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Fig. 1. An extended submodular polyhedron.

Let N = ((S + ; S − ; A); c; M b; +) be a bipartite network with gain, where its underlying graph D =(S + ; S − ; A) is a simple bipartite digraph with bipartition of the source set S + and the sink set S − and with the arc set A ⊆ S + × S − ; cM : A → R is its upper capacity function, b : S + → R is a supply function, and + : A → (0; +∞) is a gain function. The boundary of a 2ow ’ ∈ RA on S + ∪ S − is the function @’ : S + ∪ S − → R de8ned by   ’(i; t) if i ∈ S + ;    t∈S − @’(i) :=    − +(s; i)’(s; i) if i ∈ S − :   s∈S +

We de8ne that a feasible 2ow of N is a function ’ : A → R which satis8es the following conditions: ’(a) 6 c(a) M @’(v) 6 b(v)

for every a ∈ A; for every v ∈ S + :

Then the set of the negatives of the boundaries on S − of the feasible 2ows makes an − extended submodular polyhedron in RS .

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This fact is shown as follows. For each element of s ∈ S + , consider the network Ns of ordinary type (i.e. without gain) with its source set {s} and its sink set S − . If we de8ne feasibility of 2ow for this single-source network Ns in the same way for N, the set of the negative of feasible boundaries makes a submodular polyhedron [9]. When we consider gain for Ns in addition, the resulting polyhedron is obtained from that for Ns by axiswise scaling. The polyhedron for N is the vector sum of such polyhedra corresponding to each s ∈ S + . 4. Exchangeability of extended submodular polyhedra In this section we show that extended submodular polyhedra have a certain generalized kind of simultaneous exchangeability. For any points x; y in a polyhedron P and any element s in supp+ (y − x), we say that the pair (x; y) is a 2-axis exchangeable pair of P along s, if either (1ex) or (2ex) holds: (1ex) x := x + /s ; y := y − /s ∈ P for some / ¿ 0, (2ex) x := x + /(s − rt ); y := y − /(s − rt ) ∈ P for some t ∈ supp+ (x − y); r ¿ 0, and / ¿ 0. We also say that (y; x) is 2-axis exchangeable along s for such x; y; s. We say that (x; y) is 2-axis exchangeable if (x; y) is 2-axis exchangeable along every i ∈ supp(x−y). A polyhedron P is called 2-axis exchangeble if every pair (x; y) of points x; y in P is 2-axis exchangeable. Remark 4. Note that the condition (1ex) in the de8nition above is equivalent to the following condition: (1ex ) There exists  ¿ 0 such that x + /s ; y − /s ∈ P for all / ∈ [0; ]. One may also consider a similar condition equivalent to (2ex). Thus x − y 1 ¡ x − y1 holds for suPciently small / ¿ 0 in either (lex) or (2ex). The next lemma shows a basic property of this exchangeability. Lemma 5. Let (x; y) be a 2-axis exchangeable pair in a polyhedron P. If both x and y are on a face F of P, then points x ; y in either (1ex) or (2ex) are also on F. Equivalently if a; x = a; y = ∗P (a) for some a ∈ RS+ , then a; x  = a; y  = ∗P (a). Proof. We 8rst prove Lemma 5 assuming the exchange is of type (1ex). Then inequality 2∗P (a) = a; x + a; y = a; x + y = a; x + /s  + a; y − /s  = a; x  + a; y  6 2∗P (a) holds. The lemma follows from a; x  6 ∗P (a) and a; y  6 ∗P (a). One can similarly prove this lemma for the exchange of type (2ex). We claim that extended submodular polyhedra are 2-axis exchangeable.

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Lemma 6. An extended submodular polyhedron P ⊆ RS and every face F of P are 2-axis exchangeable. Before proving Lemma 6 we introduce several notations and lemmas. Lemma 7. Let P ⊆ RS be an extended submodular polyhedron and x be a point in P. If p; q ∈ RS are x-tight, then p ∨ q; p ∧ q are also x-tight. Proof. If p; q are x-tight, the value of ∗P (p) + ∗P (q) is 8nite. Then ∗P (p) + ∗P (q) = p; x + q; x = p + q; x = p ∨ q; x + p ∧ q; x 6 ∗P (p ∨ q) + ∗P (p ∧ q) holds. By submodularity inequality (3) we have ∗P (p) + ∗P (q) = p ∨ q; x + p ∧ q; x = ∗P (p ∨ q) + ∗P (p ∧ q): Since both p ∨ q; x 6 ∗P (p ∨ q) and p ∧ q; x 6 ∗P (p ∧ q) hold, we see that p ∨ q and p ∧ q are x-tight. For an extended submodular polyhedron P ⊆ RS , we de8ne the function d : P × S × S → [0; +∞] by
p(i) min{ p(s) | p ∈ Px (P); p(s) ¿ 0} if {p ∈ Px (P) | p(s) ¿ 0} = ∅; d(x; s; i) := +∞ otherwise: For 8xed x and s, the function d(x; s; ·) : S → [0; +∞] may be regarded as a vector in [0; +∞]S . We write d(x; s) for this vector. Lemma 8. Let x be a point in an extended submodular polyhedron P ⊆ RS and let s be an element in S with d(x; s) ∈ RS . Then for each x-tight vector p ∈ RS+ with p(s) ¿ 0, we have d(x; s) 6 (1=p(s))p. Particularly, for every i ∈ S, such that both d(x; i) ∈ RS+ and d(x; i; s) ¿ 0 hold, we have d(x; s) 6 (1=d(x; i; s))d(x; i).  Proof. Let p be an x-tight vector. Then x is some non-negative combination i=1; :::; k 2i pi of vectors p1 ; : : : ; pk in Px (P), where k is some positive integer and 2i ¿ 0 for i = 1; : : : ; k. The inequality d(x; s) 6 pi =pi (s) holds for each pi with pi (s) ¿ 0. Let . p = {2i pi | i = 1; : : : ; k; pi (s) ¿ 0}. Then d(x; s) 6 p =p (s) 6 p=p(s) holds. Lemma 9. Let x be a point in a down-monotone polyhedron P ⊆ RS , let s; t be elements in S, and let r be a positive real number. Then (a) – (c) are equivalent: (a) x + /(s − rt ) ∈ P for some / ¿ 0; (b) every p ∈ Px (P) satis:es p(s) 6 rp(t); (c) rd(x; s; t) ¿ 1. Proof. If rd(x; s; t)¡1, there exists an x-tight vector p ∈ Px (P) satisfying rp(t)¡p(s). Then p; x + /(s − rt ) ¿ p; x = ∗P (p) for any / ¿ 0. Thus, x + /(s − rt ) ∈ P for

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any / ¿ 0. We have shown “(a)⇒(c)”, “(c)⇒(b)” is trivial from the de8nition of d. “(b)⇒(a)” is proved by setting

∗ P (p) − p; x | p ∈ P(P); p(s) ¿ rp(t) ¿ 0; / := min p(s) − rp(t) where we assume the minimum over the empty set is in8nite as a convention. Lemma 10. Let P ⊆ RS be an extended submodular polyhedron. Let x; y be points in P and s be an element in supp+ (y − x). The pair (x; y) is not 2-axis exchangeable along s if and only if both (i) d(x; s) is an x-tight vector in Rs+ , and (ii) d(x; s; j) ¡ 1=d(y; j; s) holds for all j ∈ supp+ (x − y) ∩ supp+ (d(x; s)). Moreover (x; y) is not 2-axis exchangeable along s only if (iii) supp+ (x − y) ∩ supp+ (d(x; s)) = ∅ holds. Proof. We 8rst show the “only if” part. Assume that (x; y) is not 2-axis exchangeable along s. Since (x; y) is not 1-axis exchangeable along s, there exists p ∈ Px (P) with p(s) ¿ 0. Then d(x; s; i) 6 p(i)=p(s)  ¡ + ∞ holds for all i ∈ S and therefore d(x; s) ∈ RS+ . It is easy to see d(x; s) = {(1=p(s))p ∈ RS+ | p ∈ Px (P); p(s) ¿ 0}. By Lemma 7, d(x; s) is x-tight. Thus d(x; s) is a vector in RS+ . We have (iii) from d(x; s); x − d(x; s); y ¿ 0. For each j ∈ supp+ (x − y) with d(x; s; j) ¿ 0, there exists / ¿ 0 such that x + /(s − (1=d(x; s; j))j ) ∈ P by Lemma 9. Since (x; y) is not 2-axis exchangeable along s with respect to j at 1=d(x; s; j), there exists q ∈ Py (P) with q(j)=d(x; s; j) ¿ q(s). Then we have d(y; j; s) 6 q(s)=q(j) ¡ 1=d(x; s; j). Now we show the “if” part. Assume that (x; y) is 2-axis exchangeable along s. If (x; y) is 1-axis exchangeable along s, (i) does not hold. We assume that (x; y) is 2-axis exchangeable along s with respect to some t  ∈ supp+ (x − y) at some r ¿ 0. Then we have rd(x; s; t  ) ¿ 1 and (1=r)d(y; t  ; s) ¿ 1 by Lemma 9. Thus (ii) does not hold. Lemma 11. Let P ⊆ RS be an extended submodular polyhedron and x; y be points in P. If supp+ (y − x) consists of one element s, then (x; y) is 2-axis exchangeable along s. Proof. Suppose that (x; y) is not 2-axis exchangeable along s. By Lemma 10, d(x; s) is an x-tight vector in RS+ . From d(x; s); y 6 d(x; s); x, we have  d(x; s; j)(x(j) − y(j)) (4) y(s) − x(s) 6 j∈supp+ (x−y)

and (4) implies that supp+ (x −y)∩supp+ (d(x; s)) is not empty. For each j ∈ supp+ (x − y) ∩ supp+ (d(x; s)), there exists a y-tight vector qj such that qj (s) ¡ qj (j)=d(x; s; j). Each qj satis8es qj (s) ¿ 0 since otherwise qj ; x ¿ qj ; y would hold. Let q denote {qj =qj (s) | j ∈ supp+ (x − y) ∩ supp+ (d(x; s))}. By Lemma 7, q is a y-tight vector and it satis8es q(s) = 1 and q(j) ¿ d(x; s; j). From q; y ¿ q; x, we have  q(j)(x(j) − y(j)): (5) y(s) − x(s) ¿ j∈supp+ (x−y)

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From (4) and (5), the following contradiction is led:  y(s) − x(s) 6 d(x; s; j)(x(j) − y(j)) j∈supp+ (x−y)

¡



q(j)(x(j) − y(j))

j∈supp+ (x−y)

6 y(s) − x(s): Thus, (x; y) is 2-axis exchangeable along s. Now we are ready to prove Lemma 6. Proof of Lemma 6. We only have to prove the lemma for the face F = P. Then this lemma for an arbitrary face F follows from Lemma 5. Suppose that the lemma does not hold and take a non-2-axis-exchangeable pair (x0 ; y0 ) ∈ P × P that minimizes |supp(x − y)| among all non-2-axis-exchangeable pairs (x; y) ∈ P × P. We may assume without loss of generality that supp+ (y0 − x0 ) contains an element s along which (x0 ; y0 ) is not 2-axis exchangeable. We de8ne X0 := {x ∈ P | supp+ (y0 − x) = supp+ (y0 − x0 ); supp+ (x − y0 ) = supp+ (x0 − y0 ); (x; y0 ): not 2-axis exchangeable pair along s}:  Let x1 ∈ X0 minimize h1 (z) := j∈supp+ (x0 −y0 ) d(z; s; j) among all elements of X0 . Such minimum does exist since {d(z; s) ∈ RS+ | z ∈ P} is a  8nite set from the de8nition of d. Since x0 belongs to X0 , this minimum is at most j∈supp+ (x0 −y0 ) d(x0 ; s; j), which is 8nite by Lemma 10. Note that d(x1 ; s; j) ¡ 1=d(y0 ; j; s)

for all j ∈ supp+ (x1 − y0 ) ∩ supp+ (d(x1 ; s))

(6)

holds also by Lemma 10. By Lemma 11, supp+ (y0 − x1 ) contains an element other than s. Let s1 denote such an element. We de8ne X1 := {x ∈ X0 | h1 (x) = h1 (x1 ); d(x; s1 ; i) ¡ + ∞ for all i ∈ S}: We see that X1 contains at least one element as follows. Let x1 denote x1 + 4s1 , where 4 = min{y0 (s1 ) − x1 (s1 ); max{ ¿ 0 | x1 + s1 ∈ P}}. The vector d(x1 ; s) is also x1 -tight since either d(x1 ; s; s1 ) = 0 or 4 = 0 holds. We have d(x1 ; s; i) 6 d(x1 ; s; i) for all i ∈ S and then also d(x1 ; s; j) ¡ 1=d(y0 ; j; s)

for all j ∈ supp+ (x1 − y0 ) ∩ supp+ (d(x1 ; s))

(7)

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from (6). The pair (x1 ; y0 ) is not 2-axis exchangeable along s by Lemma 10. From the choice of x0 , we have 4 = max{ ¿ 0 | x1 + s1 ∈ P} ¡ y0 (s1 ) − x1 (s1 ). It implies that d(x1 ; s1 ) is a 8nite vector and that x1 belongs to X0 . We have h1 (x1 ) 6 h1 (x1 ) from  (7) and then h1 (x1 ) = h1 (x1 ) from  the choice of x1 . Hence x1 belongs to X1 . Let x2 ∈ X1 minimize h2 (z) = j∈supp+ (x0 −y0 ) d(z; s1 ; j) among all elements of X1 . Since d(x2 ; s1 ); x2  ¿ d(x2 ; s1 ); y0  holds, we have  d(x2 ; s1 ; j)(x2 (j) − y0 (j)) j∈supp+ (x2 −y0 )



¿

d(x2 ; s1 ; i)(y0 (i) − x2 (i)) ¿ 0;

i∈supp+ (y0 −x2 )

where the strict inequality follows from d(x2 ; s1 ; s1 ) = 1. It implies that at least one element j ∈ supp+ (x2 − y0 ) satis8es d(x2 ; s1 ; j) ¿ 0. Let t be an element in supp+ (d(x2 ; s1 )) ∩ supp+ (x2 −y0 ) which minimizes d(x2 ; s; j)= d(x2 ; s1 ; j) for j ∈ supp+ (d(x2 ; s1 )) ∩ supp+ (x2 − y0 ). Then we consider the following vector q de8ned by d(x2 ; s; t) d(x2 ; s1 ): q := d(x2 ; s) ∨ d(x2 ; s1 ; t) Note that q is x2 -tight. Set 5 := min y0 (s1 ) − x2 (s1 ); (x2 (t) − y0 (t))d(x2 ; s1 ; t);



1 t ∈ P sup / ¿ 0 | x2 + / s1 − d(x2 ; s1 ; t) and

x2 := x2 + 5 s1 −

(8)

 1 t : d(x2 ; s1 ; t)

Since P is a polyhedron, the sup in (8) may be replaced with max. The point x2 therefore belongs to P. We claim (i) q is x2 -tight, (ii) q(s) = 1 and (iii) q(j) = d(x2 ; s; j) for all j ∈ supp+ (x2 − y0 ). Our proofs of these claims are as follows. We 8rst show q; x2  = q; x2  for (i). The de8nitions of t and q directly imply q(t) = d(x2 ; s; t). Then (i) is proved by showing q(s1 ) = d(x2 ; s; t)=d(x2 ; s1 ; t). If d(x2 ; s; s1 ) = 0 holds, then q(s1 ) = d(x2 ; s; t)=d(x2 ; s1 ; t) is trivial. If d(x2 ; s; s1 ) ¿ 0 holds, we have d(x2 ; s1 ; t) 6 d(x2 ; s; t)=d(x2 ; s; s1 ) by Lemma 8. Thus q(s1 ) = d(x2 ; s; t)=d(x2 ; s1 ; t) holds. For (ii), we have nothing to show if d(x2 ; s1 ; s)=0. If d(x2 ; s1 ; s) ¿ 0 holds, then (ii) follows from the inequality d(x2 ; s; t) 6 d(x2 ; s1 ; t)=d(x2 ; s1 ; s) by Lemma 8. (iii) is a direct consequence of the de8nitions of t and q. From these claims, we have 0 ¡ d(x2 ; s; j) 6 q(j)=q(s) = d(x2 ; s; j) ¡ 1=d(y0 ; j; s) for all j ∈ supp

+

(x2

− y0 ) ∩ supp

+

(d(x2 ; s)).

(9)

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By Lemma 10, (x2 ; y0 ) is not 2-axis exchangeable along s. Then the choice of (x0 ; y0 ) implies supp(x2 − y0 ) = supp(x0 − y0 ) and also implies 5 = max{/ ¿ 0 | x2 + /(s1 − (1=d(x2 ; s1 ; t))t ) ∈ P}. From h1 (x1 )=h1 (x2 ) 6 h1 (x2 ) and d(x2 ; s; j) 6 d(x2 ; s; j) in (9), we have d(x2 ; s; j) = d(x2 ; s; j) for all j ∈ supp+ (x2 − y0 ) = supp+ (x1 − y0 ). Thus x2 belongs to X1 . The vector d(x2 ; s1 ) is x2 -tight and we have d(x2 ; s1 ; i) 6 d(x2 ; s1 ; i) for all i ∈ S. By Lemma 9, d(x2 ; s1 ; t) ¡ d(x2 ; s1 ; t) holds. Now we have h2 (x2 ) ¡ h2 (x2 ), which is a contradiction. This simultaneous exchangeability gives a criterion of optimality for linear functions over extended submodular polyhedra as follows. Proposition 12. Let P ⊆ RS be an extended submodular polyhedron. For each c ∈ RS+ with ∗P (c) ¡ + ∞ and every x ∈ P, (A) and (B) are equivalent: (A) c; x = ∗P (c); (B) x + /s ∈ P holds for all s ∈ S with c(s) ¿ 0 and all / ¿ 0, and x + /(s − rt ) ∈ P holds for all s; t ∈ S with s = t, all r ¿ 0 with c(s) ¿ rc(t) and all / ¿ 0. Proof. We only show “(B) ⇒ (A)” since the other direction is trivial. Suppose that (A) does not hold while (B) does. Let xopt be a point in the set F(c) := {y ∈ P | c; y = ∗P (c)}; which minimizes l(z) := z − x1 over F(c). (Such minimum is attained since l is a continuous function on closed set F(c).) There exists some element s in S which satis8es xopt (s) − x(s) ¿ 0, since c; xopt  ¿ c; x holds. By Lemma 6 we have a pair of points (x ; y ) ∈ P × P from (x; xopt ) either by (1ex) or (2ex). By (B), we have c; x  6 c; x. By c; x  + c; y  = c; x + c; xopt  and optimality of xopt , we have c; x =c; x and c; y =c; xopt . Then y , an element of F(c), attains l(y ) ¡ l(xopt ), which is a contradiction.

5. Equivalence between submodularity and simultaneous exchangeability In this section we establish the equivalence between submodularity and simultaneous exchangeability. Lemma 13. The support function ∗P : RS+ → R ∪ {+∞} of any 2-axis exchangeable down-monotone polyhedron P ⊆ RS satis:es (3). Proof. Let S be {1; 2; : : : ; n}. If n = 1, the lemma is trivial. If n = 2, we only have to show ∗P (p) + ∗P (q) ¿ ∗P (p ∨ q) + ∗P (p ∧ q)

(10)

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for p; q with p(1) ¿ q(1) and p(2) ¡ q(2). By the convexity and the positive homogeneity of support functions, we have ∗P (p) + ∗P (q) = (6∗P (p) + 7∗P (q)) + (8∗P (p) + 9∗P p(q)) ¿ ∗P (6p + 7q) + ∗P (8p + 9q) = ∗P (p ∨ q) + ∗P (p ∧ q); where 6; 7; 8; 9, respectively, represent q(2)(p(1) − q(1))=(p(1)q(2) − p(2)q(1)); p(1)(q(2) − p(2))=(p(1)q(2) − p(2)q(1)); q(1)(q(2) − p(2))=(p(1)q(2) − p(2)q(1)), and p(2)(p(1) − q(1))=(p(1)q(2) − p(2)q(1)). We assume n ¿ 3 in the rest of this proof. We only have to show (10) for p; q such that the cardinalities of supp+ (p − q) and supp+ (q − p) equal one, since otherwise we can use induction on these cardinalities. (The submodularity inequality with the pair (p; q) of vectors in its left-hand side can be decomposed into two submodularity inequalities with (p − (p(i) − q(i))i ; q) and (p ∨ q − (p(i) − q(i))i ; p) in their left-hand sides, where i ∈ supp+ (p − q).) Without loss of generality, we assume p(1) ¿ q(1); p(2) ¡ q(2), and p(i) = q(i) for i = 3; : : : ; n. We may also assume (p ∧ q)(i) ¿ 0 for some i ∈ S since otherwise p ∨ q coincides with p + q and (10) follows from the convexity of support functions. We 8rst show (10) for p and q with the following condition: (A) No vector p ∈ P(P) and no j ∈ {3; : : : ; n} ∩ supp+ (p ∧ q) satisfy p (j) ¿ 0 and q(1)=q(j) ¡ p (1)=p (j) ¡ p(1)=p(j). Suppose ∗P (p ∨ q) + ∗P (p ∧ q) ¿ ∗P (p) + ∗P (q). Let M1 ; M2 be real numbers which satisfy M1 +M2 ¿ ∗P (p)+∗P (q); ∅ = Fp∧q := {x ∈ P | p∧q; x=M1 } and ∅ = Fp∨q := {x ∈ P | p ∨ q; x = M2 }. Let (x0 ; y0 ) ∈ Fp∧q × Fp∨q minimize the l1 -distance function d1 (x; y) := x − y1 among all pairs in Fp∧q × Fp∨q . For arbitrary x ∈ Fp∧q and y ∈ Fp∨q , we have y(1) ¿ x(1) and y(2) ¿ x(2) from the following inequalities: 0 ¡ M1 + M2 − (∗P (p) + ∗P (q)) 6 p ∧ q; x + p ∨ q; y − (p; x + q; y) = (p(1) − q(1))(y(1) − x(1)); 0 ¡ M1 + M2 − (∗P (p) + ∗P (q)) 6 p ∧ q; x + p ∨ q; y − (p; y + q; x) = (q(2) − p(2))(y(2) − x(2)): We claim here x0 +/1 ∈ P for any / ¿ 0. If the set supp+ (p∧q)∩supp+ (x0 −y0 ) contains an element, denoted by j0 here, this claim does hold, since otherwise Fp∧q would contain x1 := x0 + /(1 − (p ∧ q)(1)=(p ∧ q)(j0 )j0 ) attaining d1 (x1 ; y0 ) ¡ d1 (x0 ; y0 ). Suppose that supp+ (p ∧ q) ∩ supp+ (x0 − y0 ) is empty. Then p ∧ q; y0  ¿ M1 would

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hold and, for an arbitrary element j in supp+ (p ∧ q); Fp∧q would contain p ∧ q; y0  − M1 y0 := y0 − j (p ∧ q)(j) satisfying either y0 (1) = y0 (1) or y0 (2) = y0 (2), which is a contradiction. Therefore, supp+ (p ∧ q) ∩ supp+ (x0 − y0 ) is not empty. Since x0 + /1 ∈ P for any / ¿ 0; P contains x0 + /(1 − rt ) and y0 − /(1 − rt ) for some t ∈ supp+ (x0 − y0 ); r ¿ 0, and / ¿ 0. Note that t ∈ supp+ (x0 − y0 ) implies t = 1; 2 and that (p ∧ q)(t) = (p ∨ q)(t) ¿ 0 holds, for x0 (y0 ) belongs to Fp∧q (Fp∨q ), respectively. We have (p ∨ q)(1) ¿ r(p ∨ q)(t) since otherwise Fp∨q would contain a point y0 − /(1 − rt ) − t for some  ¿ 0, which is closer to x0 in l1 -metric. Then every x0 -tight vector w in P(P) satis8es either w(1)=w(t) 6 r ¡ (p ∨ q)(1)=(p ∨ q)(t) or w(1) = w(t) = 0. From (A), w(1)=w(t) 6 (p ∧ q)(1)=(p ∧ q)(t) holds unless w(1) = 0. It implies that Fp∧q contains x0 + / (1 − (p ∧ q)(1)=(p ∧ q)(t)t ) for some / ¿ 0. This point is closer to y0 than x0 , which is a contradiction. Thus (10) holds for any p; q satisfying (A). We now show (10) without assuming (A). Given p; q ∈ RS+ with ∗P (p); ∗P (q) ¡+∞, we consider a sequence of vectors p=p0 ; p1 ; : : : ; pk =q such that pi =p+2i (q−p) for some 0 = 20 ¡ 21 ¡ · · · ¡ 2k = 1 and that no vector p ∈ P(P) and no j ∈ {3; : : : ; n} ∩ supp+ (p ∧ q) satisfy pi+1 (1)=pi+1 (j) ¡ p (1)=p (j) ¡ pi (1)=pi (j) for i = 0; : : : ; k − 1. From the convexity of the support functions, ∗P (pi ) is 8nite for every i ∈ {0; : : : ; k}. Then ∗P (pi ) + ∗P (pi+1 ) ¿ ∗P (pi ∨ pi+1 ) + ∗P (pi ∧ pi+1 )

(11)

holds for i = 0; : : : ; k − 1 with both sides 8nite. By repeating similar arguments, one can show the following submodularity inequalities one by one: ∗P (pi ∨ pj ) + ∗P (pi+1 ∨ pj+1 ) ¿ ∗P ((pi ∨ pj ) ∨ (pi+1 ∨ pj+1 )) + ∗P ((pi ∨ pj ) ∧ (pi+1 ∨ pj+1 )) =∗P (pi ∨ pj+1 ) + ∗P (pi+1 ∨ pj );

(12)

∗P (pi ∧ pj ) + ∗P (pi+1 ∧ pj+1 ) ¿ ∗P ((pi ∧ pj ) ∨ (pi+1 ∧ pj+1 )) + ∗P ((pi ∧ pj ) ∧ (pi+1 ∧ pj+1 )) =∗P (pi+1 ∧ pj ) + ∗P (pi ∧ pj+1 )

(13)

for i=0; : : : ; k −2 and j=i+1; : : : ; k −1. It implies that both sides of all these inequalities are 8nite. Adding up (11) over i = 0; : : : ; k − 1, (12) over 0 6 i ¡ j 6 k − 1, and (13) over 0 6 i ¡ j 6 k − 1 results in (10). Combining Lemmas 6 and 13, we have the following theorem, which is the main result of this paper. Theorem 14. A down-monotone polyhedron is 2-axis exchangeable if and only if P is an extended submodular polyhedron.

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6. Simultaneous and non-simultaneous exchangeability The aim of this section is to point out that non-simultaneous exchangeability is not enough to characterize the class of extended submodular polyhedra. There exists a non-2-axis exchangeable down-monotone polyhedron P ⊆ RS which satis8es the following unbalanced non-simultaneous exchangeability: (UEX) For all x; y ∈ P and for every s ∈ supp+ (y − x), either (a) or (b) holds: (a) There exists / ¿ 0 which satis8es x + /s ∈ P. (b) There exist t ∈ supp+ (x − y); r ¿ 0, and / ¿ 0 which satisfy x + /(s − rt ) ∈ P. Such a polyhedron, introduced in [8], is obtained with a positively homogeneous function f : {0; 1; 2}3 → R de8ned as follows: f(1; 0; 0) = f(0; 1; 0) = f(0; 0; 1) = 16; f(1; 1; 0) = f(0; 1; 1) = f(1; 0; 1) = 27; f(1; 1; 1) = 31; f(2; 1; 0) = f(0; 2; 1) = f(1; 0; 2) = 42; f(1; 2; 0) = f(0; 1; 2) = f(2; 0; 1) = 42; f(2; 1; 1) = f(1; 2; 1) = f(1; 1; 2) = 46; f(2; 2; 1) = f(1; 2; 2) = f(2; 1; 2) = 55: The values of the other 012-vectors are uniquely determined by positive homogeneity. It is easy to check that the polyhedron P(f) associated with f by P(f) := {x ∈ R3 | p; x 6 f(p) for all p ∈ {0; 1; 2}3 } satis8es (UEX). Fig. 2 shows this P(f). This P(f), however, is not 2-axis exchangeable. Choose, for example, (16,10,3) as x; (15; 12; 1) as y, and the second axis as s to see this fact. It should be noted that the support function ∗P(f) satis8es the submodularity of P(f) among 012-vectors, i.e., ∗P(f) (p) + ∗P(f) (q) ¿ ∗P(f) (p ∨ q) + ∗P(f) (p ∧ q)

for all p; q ∈ {0; 1; 2}3 ;

while it does not satisfy (3), as implied by Lemma 6.

Acknowledgements We are indebted to K. Ando and a referee for pointing out some incompleteness of our previous proofs of Lemmas 6 and 13. We express our gratitude to S. Fujishige, K. Murota, M. Nakamura, A. Shioura, M. Hachimori and Y. Okamoto for their helpful

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Fig. 2. Polyhedron with only non-simultaneous exchangeability.

comments. We are also thankful to anonymous referees for their useful comments, which substantially improved the readability of this paper. References [1] K. Ando, S. Fujishige, On structures of bisubmodular polyhedra, Math. Programming 74 (1996) 293–317. [2] A. Bouchet, W.H. Cunningham, Delta-matroids, jump systems, and bisubmodular polyhedra, SIAM J. Discrete Math. 8 (1995) 17–32. [3] R. Chandrasekaran, S.N. Kabadi, Pseudomatroids, Discrete Math. 71 (1988) 205–217. [4] J. Edmonds, Submodular functions, matroids, and certain polyhedra, in: N. Sauer, R. Guy, H. Hanani, J. SchSonheim (Eds.), Combinatorial Structures and Their Application, Gordon and Breach, New York, 1970, pp. 69–87. T Tardos, Generalized polymatroids and submodular 2ows, Math. Programming 42 (1988) [5] A. Frank, E. 489–563. [6] S. Fujishige, Submodular Functions and Optimization, in: Annal of Discrete Mathematics, Vol. 47, North-Holland, Amsterdam, 1991. [7] S. Fujishige, K. Murota, Notes on L-/M-convex functions and the separation theorems, Math. Programming 88 (2000) 129–146.

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[8] K. Kashiwabara, M. Nakamura, T. Takabatake, Integral polyhedra associated with certain submodular functions de8ned on 012-vectors, in: G. Woeginger, G. CornuTejols, R. Burkard (Eds.), Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, Vol. 1610, Springer, Berlin, 1999, pp. 289–303. [9] N. Megiddo, Optimal 2ows in networks with multiple sources and sinks, Math. Programming 7 (1974) 97–107. [10] K. Murota, On exchange axioms for valuated matroids and valuated delta-matroids, Combinatorica 16 (1996) 591–596. [11] K. Murota, Discrete convex analysis, Math. Programming 83 (1998) 313–371. [12] K. Murota, A. Shioura, Extension of M-convexity and L-convexity to polyhedral convex functions, Adv. App1. Math. 25 (2000) 352–427. [13] J.G. Oxley, Matroid Theory, Oxford University Press, New York, 1992. [14] A. Recski, Matroid Theory and its Applications in Electric Network Theory and in Statics, Springer, Berlin, 1989. [15] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. [16] D.J.A. Welsh, Matroid Theory, Academic Press, New York, 1976.