Representations and characterizations of vertices of bounded-shape partition polytopes

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LINEAR ALGEBRA AND ITS APPLICATIONS ELSEWER Linear Algebra and its Applications 278 (1998) 263-284

Representations

and characterizations

of

vertices of bounded-shape

partition polytopes

Frank K. Hwang a, Shmuel Onn b>l, Uriel G. Rothblum

b3*,2

“ Depurtment of’ Applied Mathematics, Chiaotung Uniwrsity, Hsinchu. Taiwan 30050, ROC ’ Faculty qf’htdustrial Engineering and Manqement, Technion-Israel Institute of Technology,

Ha$a 32000, Israel

Received 1 May 1997; accepted 11 November 1997 Submitted by R.A. Brualdi

Abstract

Consider a finite set whose elements are associated with vectors of common dimension. A partition of such a set is associated with a matrix whose columns are the sums of the vectors corresponding to each part. The partition polytope associated with a class of partitions (that share the number of parts) is then the convex hull of the corresponding matrices. We derive representations and characterizations of these polytopes and their vertices. 0 1998 Elsevier Science Inc. All rights reserved.

Keywords: Partitions; Polytopes; Vertices

1. Introduction

Following Barnes et al. (1992), hereafter referred to as BHR, we study partitions where each element of the partitioned set is associated with a (fixed) number of numerical attributes. So, vectors A], . ,A” are given, say of

* Corresponding author.

’ Research of this author was supported by the Mathematical Sciences Research Institute at Berkeley, California, by the Fund for the Promotion of Research at the Technion and by a VPR Grant at the Technion.

’ Research of this author was supported by the E. and .I. Bishop Research Fund at the Technion and by ONR Grant NOOO14-92-51142.

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264 FK. Hwang et al. I Linear Algebra and its Applications 278 (19%‘) 263-284

dimension k, and we consider (ordered) partitions TX = (~1,. ,np) of N E { 1,. . , n}. Given a partition 7~ = (71,). . . , n,), we refer to the integer p as the size of rc and to the integer vector (17~~ 1, . . . , jn,/) as the shape of 71. Also, 7~ is associated with the k x p matrix A” E (C_, A’, . , )& A’). The partition polytope associated with a set I7 of partitions sharing a common size p is de- fined as the convex hull of all k x p matrices A” corresponding to partitions ‘/t in Ii’. Of particular interest are constrained-shupe partition polytopes where n is determined by constraints over shapes; if such constraints are in terms of lower and upper bounds, we refer to bounded-shape partition polytopes.

In this paper we explore vertices of partition polytopes. A motivation for one’s interest in the vertices of convex hulls of finite sets is the following standard result.

Proposition 1.1. Let Y be a finite set of vectors of common size and P E conv Y. Then Y contains all vertices of P. Further, if/z(.) is a convex function on P, u maximum of h(.) over P is attained at a vertex of P and such a vertex maximizes h(.) over Y. 3

Proposition 1.1 is relevant to the study of maximization problems over sets of partitions where the objective F(n) associated with a partition 71 has the representation F(Z) = h(A”) with h(.) as a real-valued convex function on the convex hulls of the A”‘s; see Hwang and Rothblum (in progress), Gao et al. (1998) and references therein for specific applications of such partitioning problems in diverse fields that include clustering, statistics, scheduling, reliability, inventory and system assembly. Specifically, Proposition 1.1 suggests that these partitioning problems be embedded in the problem of maximizing h(.) over the corre- sponding partition polytope, or restricted to optimization over partitions corresponding to vertices of that polytope. Our study of the vertices of partition polytopes is motivated by the second approach.

We mention that the above embedding and restriction of partitioning problems correspond to the two fundamental approaches used in the study of polytopes and optimization problems thereupon ~ one focusing on facets, that is, on defining systems for linear inequalities, while the other focusing on vertices. Nei- ther approach necessarily dominates the other, and moving from one to the other is generally nontrivial computationally. The solution of optimization problems over partitions is explicitly addressed in Hwang et al. (unpublished manuscript).

3 It is well-known that the conclusions of Proposition 1.1 extend to functions which arc yuasi- convex, that is, functions h(.) satisfying h[cta + (1 - c()b] < max{h(a), h(b)} for vectors a and b in their domain and 0 < z < 1. Further, these results were recently extended to an even larger class of functions called edge-quasi-convex; see Hwang and Rothblum (1996).

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E K. Hltung et d. I Linccw Algehrtr trnd its App1icwtion.c 27X i 1998~ X3 2X4 26.5 BHR explored partitioning problems and partitioning polytopes under a nondegeneracy assumption asserting that the columns of the underlying matrix A are nonzero and distinct. Some of their results were extended in Hwang et al. (submitted) our goal herein is to extend the results of BHR with full general- ization to degenerate cases.

One issue of interest concerns uniqueness of the representation of vertices. For non-degenerate bounded-shape partition polytopes the issue was settled in Theorem 5 of BHR as follows:

Proposition 1.2. Let P he a hounded-shupe partition pol>~tope 11here the colun~m of’ the Lmderl~~ing rnatris A are nonzero ad distinct md let V he II rertrs of’ P.

Then V 1~1s II unique representation NS A’ Ipith TI LI prrr-tition in the underlining .wt of’purtitions.

Obviously. the unique representation of vertices does not hold in generate cases where A’s columns include repeated vectors and/or zero vectors. For example, if all the columns of A coincide and are all nonzero. there is a one-to- one correspondence between the potential shapes of partitions and the associ- ated vectors, but, with p > 2 and II > p there will be multiple partitions with any given shape. More generally, we have that switches of indices with identical corresponding vectors between the parts as well as shifts of zero vectors between the parts will not change the associated vector. In Section 3 (Theorem 3.5) we demonstrate that the above are the only degrees of freedom in multiple representations of vertices of bounded-shape partition polytopes. We also demonstrate that Proposition 1.2 cannot be extended to constrained-shape partition polytopes.

A (geometric) necessary condition and an (algebraic) necessary and sufficient condition for vertices of bounded-shape partition polytope were obtained in BHR. The necessary condition was generalized to degenerate cases and to constrained-shape polytopes in Hwang et al. (submitted). The extended result is given in Proposition 1.3.

Proposition 1.3. Let P he a constrained-shupe partition polj*tope uYth wrtc.y .1” inhere T = (~1. . rep) is a corresponding partition. For t = 1. .p. let CT, 5 conv {A;: ,j E z,}. Then ji)r each pair of’ distinct intk.v I’..! E {

I. %p}

~ CT,.

n CJ$

is either empty, or contains u singk point \~%icii is ci conmon wrtes of CT,. and 0,; sucl~ N conmon rertc.v i.v rw~e.s.sari!)~ one

qf’ A ‘.Y

columns.

The sufficient condition for vertices given in Proposition 1.3 is appealing because of its geometric expression. Further, the condition is used in Hwang et al. (submitted) to enumerate, in polynomial time, the vertices of constrained-shape polytopes. Still, an example in BHR demonstrates that, even

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266 F. K. Hwang et al. I Linear Algebra and its Applications 278 (I 998) 263-284

in nondegenerate cases, the condition is not necessary. A condition which is both necessary and sufficient for vertices of nondegenerate bounded-shape partition polytopes in terms of solvability of linear systems was developed in BHR (it is included in Theorem 3.1). In Section 4 (Theorem 4.2) we extend the characterization to degenerate cases and demonstrate that its verification can be ex- ecuted in effort which is polynomial in the parameters of the problem; the result facilitates a polynomial test for vertices.

Partition polytopes and preliminaries are formally introduced in Section 2. Representations of partition polytopes and their vertices are given in Section 3, with particular emphasis on the degrees of freedom in such representations. Fi- nally, an algebraic characterization of the vertices is provided in Section 4.

2. Preliminaries: Partition polytopes

Throughout, we let k and n be positive integers. These parameters will be fixed throughout this section.

Superscripts are used to denote columns of matrices, subscripts for rows and double indices for elements, e.g., U’, Ui and U;. The vector of l’s of appropriate dimension is denoted e. For matrices U and V of common dimension, say m x p, the inner product of U and V is defined by (U, V) s CL, CT=, iJ/Ff, We recall that for matrices U, V and W of dimension m x p, m x q and q x p, respectively, we have that (U, VW) = (V’U, W) = (UWT, V).

A partition is an ordered collection of sets z = (71, ! . . . , q,), where 7c1, . . . , q, are disjoint, nonempty subsets of N whose union is N. Given such a partition 7t, we refer top as its size and to the sets nl , . . . , zp as its parts. Also, if the number of elements in the parts of a partition 7~ =

(rci, .

. . , T-C,) are nl, . . , nP, respective- ly, we refer to (121, . . . , nP) as the shape of n; of course, in this case CT=, nj = INI = n. Partitions of size p are called p-partitions and partitions of shape (nl , . . ! nP) are called (ni , . . . , n,)-partitions.

Sets of partitions of particular interest are those whose shape is constrained to be in a prescribed set. Specifically. if r is a set of positive integer p-vectors with coordinate-sum n (that is, I’ is a set of potential shapes of p-partitions) we refer to the set of all p-partitions whose shape is in r as the set of r-shape pur- titions; at convenience, we suppress the explicit dependence on r and refer generically to constrained-shape partition-sets. If L and U are positive integer p-vectors satisfying L < U and CT=, Lj < n < CT=, Uj, the (nonempty) set of positive integer p-vectors (n, , . . , n,) with coordinate-sum n that satisfy Lj<nj<U,foreachj= l,...,pisdenotedr (L.u); the corresponding set of partitions is denoted LICLx”) and, with the dependence of L and U suppressed, referred to as a hounded-shape partition-set.

Let A be a k x n real matrix. For a p-partition rr = (rci , . . , np) we define the n-summation-matrix of A, denoted A”, by

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E K. Hwang et al. I Linear Algebra and its Applications 278 (I 998) 263-284 ‘67

A”- [p,...,p] E

WL~.

(2.1)

With e’. . . ek as the unit vectors in Rp, we note that

=

f--+(e”)r.

(2.2)

/‘I IEn,

We recall that a polytope is the convex hull of a finite set. For a matrix A E lRkxn and a set of p-partitions lI, the purtition polytopr with dutu-nmtri.\- A corresponding to II, denoted P,“, is the convex hull of {A”: rc E f7} 2 [WA”“. While the notational dependence of 9,” on A and II is always preserved. we sometimes refer to partition polytopes or to the partition poll’topes correspond- ing to I7. If I7 = ZI@“) for corresponding positive integer p-vectors L and U we use the notation P(L.U’ for P$’ and refer to this polytope _A as a hounded-shape pur- tition polytope.

We recall that a vertex of a polytope P is a point 11 in P having the property that the only representation of u as i (u + 6) with a. b E P has a = h. It is well known that V E P is a vertex of P if and only if there is a linear function that attains a unique maximum over P at V, and we use this property interchange- ably with the above definition. The important role vertices play in convex maximization problems is discussed in the introduction.

We next consider the case where the data-matrix is the identity I E R”‘“. In this case, for each p-partition n, I” E RnX” is given by

(I”); E 1 iftErc,, 0 otherwise.

where e’ for t = 1,. . n and eJ for j =: 1. %p denote the unit vectors in R” and [w”, respectively; consequently (using (Eq. (2.2)) with I as the underlying matrix)

= AxA’(&)T = A”. /=I ltn,

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268 FK. Hwung et al. I Linear Algebra and its Applications 278 (1998) 263-284

An explicit representing systems of linear inequalities is next derived for bounded-shape partition polytopes with the identity as the data-matrix.

Lemma 2.1. Let L and U be positive integer p-vectors satisfying L < U and J& Li <n < CT=, U,, and let Il be the set of r cL’u)-partitions. Then P/” is the solution set of the linear system:

X: 20 fort= l,..., nandj= l,... :p: (2.5a)

A*= 1 fort= l,...,n,

j=i

(2.5b)

for j = 1, . . . ?p. (2.5~)

Proof. Let K be the solution set of (2.5). Trivially, I” E K for each 7~ E II, implying that the convex hull of these matrices, namely Pin, is contained in K. Next, standard results (that rely on the fact that the constraint matrix of the inequality system (2.5) is totally unimodular) assure that the vertices of K are integer solutions of (2.5) (cf., Schrijver, 1986); as integer solutions of (2.5) correspond to p-partitions in n, that is, have representation as I” for some rr E II, each vertex of K is in P/“. By another standard result, K is the convex hull of its vertices, and consequently K is contained in Pf’. 0

3. Vertex representation

In the current section we derive representations of partition polytopes and their vertices. We recall Proposition 1.2 which asserts unique representations as A” of the vertices of bounded-shape partition polytopes when the vectors A’, . . , A” are nonzero and distinct. The next example demonstrates that with the columns of A nonzero and distinct, multiple representations of interior vectors of bounded-shape partition polytopes and of vertices of constrained-shape partition polytopes (which are not bounded-shape) are possible.

Example. Let k = 1, n = 4, A = (-2, - l( 1,2) and p = 2. For positive indices i and j with i + j = 4, we let Zj’(‘.j) be the partitions with shape (i> j) and we let Pci,i) be the corresponding partition polytopes. Now, rr* = ({ 1,4}, {2,3}) and x2 = ({2,3),{1,4)) are two distinct partitions in fl(2;2) that satisfy A”’ = AK’ = (0,O). Of course, (0,O) is not a vertex of Pc2.*) = {(x, -z): -3 < CI < 3) (in fact, in view of Proposition 1.2, (0,O) is not a vertex of any bounded-shape partition polytope). Also, {A”: 7~ E Ii’(‘.3)} = {A”: 7~ E Ii’(3.1)} =

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F K. Mwcmg rr (11. I Linear Algehru uncl its Applic~ations 278 (19%‘) 263-284 xc)

{(-2.2),(-1.1). (l.-1),(2-2)) = {AK: no 17i’.“)‘!‘7”‘}. So, the vertices of the partition polytope corresponding to 117(‘,3) u 177(3.‘1 are (-2,2) and (2. -2) and each is realizable by two partitions. Of course, IIi’.3i U d3.‘) is a set of partitions which is constrained-shaped, but not bounded-shape.

The next result provides three necessary and sufficient conditions for vectors corresponding to given partitions to be vertices of bounded-shape partition polytopes. One of these (condition (d) in Theorem 3.1) was introduced in BHR; another (condition (b)) tightens the necessary condition for being a vertex stated in Proposition 1.2.

Theorem 3.1. Let A E RXxn have nonzero und distinct colurnn.v, let L,). . L,,. c/l. . r/, he positive integers suti.fj,ing L, < C: ,for ,i = 1 1 , p uncl Cy_, L, < II 6 CT=, Uj, and let 71 E 17 WJ, i Then the fbllo,ring mre qku- . lent.

(a) A” is II vertex of’??““,

(b) A” = AI” is u unique representation qf A” us A” = AX Irith X E P,!“” ‘. (c) {Y E P;’ [‘I: AY = A” und (I”; Y) < n - l} = 8, und

(d) there esists cc matrix C E RkXP und vector r E W such thut:

(1) (C” ~ C”)TA’ > c(, - ‘q. jbr r. s E { 1. p} ,rith r # .r und t E q.. (2) x,- 6 0 if /7c,I > L,., and

(3) clF 3 0 if l-r,-1 < u,..

Proof. Let I! = 17(L.u). We recall from Eq. (2.4) that A” = Al” for each (T E Ii’. (a) + (b): Suppose A” = AI” is a vertex of PF’L ‘. Then A” is the unique max- imizer over !‘(r,c’ of some linear functional, say one that is represented by the matrix C E EJxp. Now, suppose that A” = AX with x’ E Pj’.“’ and we will show that X = I”. As X E @“, there exist partitions 7~‘. . IT“ in n and positive coefficients ~1.. ~ xc, which sum to 1 such that X = XI_, cc,I”‘; in particular,

(C,il”) = (C,AX) = Cz=, r,y(C,AI”‘) = Cy=, x,(C,A”‘). As A” is the unique maximizer over & (‘~J’) of the linear function represented by C, as the 2,‘s are positive and sum to 1 and as the A”“s are in qiL”), it follows that for each .)‘ = I % 3 q. (C, A”‘) = (C. A”) and AT’ = A”. Thus Proposition 1.2 implies that all 71”s coincide with 71 and therefore X = Cz=, xJK’ = I”.

(b) + (c): Suppose A” = AI” is a unique representation of A” as A” = AX with X E P:“L’i. Let Y be a matrix in P/(““) satisfying AY = A’. It then follows that Y = I”. implying that (Y.Z”) = (I”, I”) = II. Thus Pj’.“’ f’ {Y t W”‘“:

AY=A”and(I”.Y)<n-1}=8.

(c) + (a): Assume that A” is not a vertex of 9:“’ ‘. From Proposition 1.1, each vertex of ~~~“” IS m the set {A”: 71 E IT} and a standard result implies that A” has a representation as a convex combination of vertices of Z’:““‘. Hence. there exist partitions 71’ ~ . : 7~“ in i7, all distinct from 71, and positive coefficients xl.. . a(, which sum to 1 such that A” = cy , a,A”‘. As A”‘ = AI”‘ for

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270 FK. Hwang et al. I Linear Algebra and its Applications 278 (1998) 263-284

s = 1,. . , q, hence, A” = Cz=‘=, a,AIti = A(C:=, cc,Z”‘). Also, the convexity of P(L*U) assures that Cz=, c(,I* E P,(L.U’. Now, as (I”,Z”) 6 n - 1 for each p-parti- _I tion cr that is distinct from rc, we have that (I”, Cz==, a,Z”‘) = Cz=, CI,(I”,Y’) < Cz==, a,(n - 1) = 12 - 1. So, X = Cz=, c@’ E @‘) satisfies .4X = A” and (Z”,X) < n - 1, demonstrating that {X E P,@“): AX = A” and

(I”,X) < IZ - 1) is not empty.

(a) w (d): This equivalence is established in Theorem 5 of BHR. 0 The three necessary and sufficient conditions for being a vertex of a bounded-shape partition polytope given in Theorem 3.1 yield computational methods with polynomial complexity in n,p and k; further discussion of such methods is deferred till the end of the current section, at which point the restrictive assumption that A’s columns are nonzero and distinct is relaxed.

Condition (a) of Theorem 3.1 does not imply condition (b) when A’s columns include repeated vectors and/or zero vectors; for example, if all columns of A coincide, each single-shape partition polytope contains a single point which is a vertex of the polytope and this vertex has multiple representations as A” when p > 1 (in fact, this example neither satisfies the conclusion of Propo- sition 1.2). In Theorem 3.5 we identify variants of condition (b) which charac- terize vertices of bounded-shape partition polytopes without the assumption that A’s columns are nonzero and distinct. A modification of condition (d) which is necessary for (a) and applies for the general case is developed in Section 4. We are not aware of a corresponding modification of condition (c). A few additional definitions are needed before we are ready to explore general bounded-shape partition polytopes. Let fi be the number of nonzero distinct columns of A. We will consider matrices with 2 + 1 rows or ii + 1 columns where these rows/columns are indexed by 0, 1, . , ii. Further, when a matrix has rl + 1 rows indexed by 0, 1, . ii, we use underlining to denote the submatrix obtained by truncating the O-row, so, if B E R(n+‘)xn, then & E IWlixn.

Given a k x n matrix A, wet let k’ be the k x h submatrix of A obtained by deleting zero and multiple columns that appear in A, for uniqueness we assume that the first of any group of repeated columns of A is preserved while the others are deleted, and the order of k;s columns is induced from A. Also, let 2 be the k x (1 + 2) matrix obtained from 2 by adding the zero vector as the O-col- umn. Of course, never does k’ have a zero column but 2 always does. Finally, let J E @if+” _{be the (0, 1 }-matrix} _{with J’ = e’ (the s-unit} _vector _{in rW”> if} A’ = k”(# 0) and J’ = e” (the O-unit vector in [WC”+‘)) if A’ = 0. For example, if

1 0 3 3 1

1 0 0 0 1

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F. K. Hwang et al. I Linear Algebra and its Applications 278 i IY98) 263 -2X4 271 0 1 0 0 0 1 0 0 0 1 J= 1 0 0 0 1 and J_ = 00110’ 00110

The use of underlining to denote truncation of the zero row of a matrix having 1 + ri rows (introduced in the above paragraph) and the construction of 2 from k’ by augmenting it with a zero vector imply that for each matrix Y with 1 + iz rows we have that

AY =& in particular,

(3.1)

~J=A=&. (3.3)

When A has no zero columns, the forthcoming development can be carried out without the use of 2, but solely with the use of k’. In particular, when A’s columns are nonzero and distinct, k’ = A and J_ = 1 E R”““.

We will use J,J_ and A E Rkxn as data-matrices. In particular, for a y- partition rt Eq. (2.4) implies that JZ” = .J”,J_Z” = z”. AI” = A” and, by Eq. (3.2)

A” = AI” = (iJ)I” = ;1(JI”) zz AJ’ (3.3)

and

A” = Al” =

(i&Z” = ,i(JZ)n = &.

(3.4)

The next lemma shows that for a set of partitions L’, Pfl> PJ”% Pl and P,” form a sequence of polytopes where each is a projection of its predecessors; further. the composite projection of P,!” onto Pf is given by X --) AX. The decomposi- tions we are about to establish are demonstrated in Fig. 1. We shall refer to P,! as the generalized transportation polytope corresponding to 17.

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272 I? K. Hwang et al. I Linear Algebra and its Applicutions 278 (I 998) 263-284 Lemma 3.2. Let A E Rkxn and let Il be a set ofpartitions. Then:

(a) Py = {JX: X E Py C Wxp}, (b) PJ” = (1: Y E P; c R(‘+n)xp}, (c) P; = {ki: z E PJ” c W”}.

Further, the composiie projections oj’P7 onto PF, of P,?y ontf PA” and of Py onto PA” are given, respectively, by X + JX;X + AX and Y + AY.

Proof. A standard argument about convex hulls shows that

{JX: X E PF} = {JX: X E conv{Z”: rr E Z7}} = conv{JI”: 7-r E II}

= conv{J”: 7c E II} = 9:: _(3.5)

proving (a). The same argument combines with Eq. (3.4) to show that {ki: Z E Py} = {ki: Z E conv{J_“: 71 t II}}

= conv{&n: rt E II} = conv{A”: n E II} = P,“, _(3.6) proving (c). Next, to establish (b), observe that the projection mapping Y E [W(‘+“)“P into 1 E [Wnxp by eliminating the 0 row is a linear operator; this operator is representable by a matrix, say E E R”‘(‘+“) with EY = 1 for each Y E R(lf”)xp. As in Eq. (3.6) we then get that

(1: Y E Py} = {EY: Y E Py} = {EY: Y E conv{J”: 71 E II}}

= conv{EJ”: rr E II} = conv{C: z E Z7} = P/. _(3.7)

Finally, the composite projections of Pf’ onto PJ”, of P: 0nto.P: and of Py onto PA” are given respectively, by X + Jx i JX, X + A(JX) = k’@X) = (kJ)X = AX (here we use Eq. (3.2)) and Y + k’x = AY (here we use (3.1)). 0

Corollary 3.3. Let A E Rkxn and let Ll be a set of p-partitions. Then the partition polytope PA” is the image of the generalized transportation-polytope PF under the

linear function mapping X E Py C Rnxp into AX E Rkxp. In particular, Jbr every p-partition rt,A’ = AIX.

We next obtain an explicit representation, through a system of linear inequalities, for bounded-shape partition polytopes when the data matrix is J. The result resembles Lemma 2.1 which concerns the case where the data matrix is I. Lemma 3.4. Let L and U be positive integer p-vectors satisfying L < U and xy=1Lj<n< CT=, Uj. Then PJ ‘L~u’ ir the solution set of the linear system .

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F K. H~+ang et al. I Linear Algebru and its Applicntions 278 (1998) 263-284 273

kk;‘=(Je)y fors=0,1,...,5, j-1

(3.8b)

(3.k)

btlith e as the vector (1, , l)T E R”.

Proof. Trivially, each of the matrices J” for 7c E H(L,‘) satisfies (3.8), hence, the convex hull of these matrices, namely PJ (‘.‘I, is contained in the solution set of (3.8) which we denote by K.

Adding variables ‘;” for j = 1,. . >p, replacing (3.8~) by the constraints Y/!’ = c;=, Y! and L, < q! < Uj for j = 1,. . . :p and adding the constraint CT=, q? = Ci=O(Je)s = n, th e mear system (3.8) is expanded 1’ to a network flow problem with integer lower and upper bounds on arc-flows (see Fig. 2); in particular, standard results (that rely on the fact that the constraint matrix of the defining linear system is totally unimodular) assure that the polytope associated with the network flow problem has integral vertices (see Schrijver, 1986). The constructed expansion defines a one-to-one lineur map from K onto this polytope, under which vertices are mapped onto vertices, and we conclude that all vertices of K are integer matrices.

Let Y* be a vertex of K; it then follows from the above paragraph that Y’ is an integer matrix. For each s=O;l, . . . . ii, let H,={t=l,..., a: A’=k”}: these sets partition N and for each s, Cyz, (Y’): = (Je),s = /HJ. The latter to- gether with the integrality of Y* implies that for s = 0, 1. ? ii, there exists a (not necessarily unique) partition of H,_ into sets cry,, . . , CT+ such that

10.~~ 1 = (Y*)I. For j = 1,. . . ,p, let rc, s UfzO g,,. It follows that ?I], . . n,,

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274 I? K. Hwang et al. I Linear Algebra and its Applications 278 (1998) 263-284

partition N; also, for j = 1,.

. . ,P, bjl = C”, IQ = C:&‘*): so (Eq.

(3.8~))

implies that

Lj < lnjl < Uj.

Thus, 7~ E n @auf. Next, for s=O,l,...,

ii and j=

1,.

. ,P,C~~~,P); = bs,l = (Y*):, h ence, (.I”): = (JP)Js = CtCH, (I”): = (Y)i,

implying that Y’ =

J” E Py,“). So,

each vertex of

K

is in P’,‘). By another

standard result,

K

is the convex hull of its vertices, implying that

K

is contained

in PylU’.

0 The explicit representation of P’%‘) in Lemma 3.4 does not generally extend

to Py,‘); specifically, when zero vectors exist, they have to be accounted for the

lower and upper bounds on the cardinality of the parts. But, part (b) of Lemma

3.2 shows that P’,U’ is a projection of P’,‘). When A has no zero column,

(Je)O = 0

and all solutions of (3.8) have Y/ = 0 for j = 1,.

. . ,p;

by eliminating

these variables, we get from (3.8) a characterization

of Py,U).

The next theorem provides three necessary and sufficient conditions for vec-

tors corresponding

to partitions to be vertices of bounded-shape

partition

polytopes, without the assumption that A’s columns are nonzero and distinct.

The three conditions look cumbersome and repetitive, but, they have distinct

uses. Condition (b) extends the uniqueness of the representation

of vertices

when A’s columns are nonzero and distinct (condition (b) of Theorem 3.1),

while condition (d) explains the potential degrees of freedom in multiple repre-

sentations of extreme points of the partition polytopes (see the discussions fol-

lowing the theorem and Corollary 3.6). Conditions (c) and (d) concern the

polytopes Py,U’) and

P,(L,‘)

which have explicit linear inequalities representa-

tions (Lemma 2.1 and Corollary 3.3

)

and are therefore useful for computable

tests; in fact, the representation of

PJL.‘/)

is used to establish the (most difficult)

implication (a) * (c).

Theorem 3.5. Let A E Rkxn, let L1, . . . , Lp,, U1, . . . , Up be positive integers

satisfying L < U and CT=, Lj 6 n 6 c$ Uj, and let 71 E Il(L,U). Then the following are equivalent:

(a)

A” is a vertex of Pf,“),

(b) {Z E

Py? ki = A”} = {J”} and J_” is a vertex of Py”/), and

(c) {Y E P:“,“‘: AY =A”} = {Y E Py.“): Y = J_“} and s”is a vertex of Py’“), _{_} and

(d) bTL :,pr(L,u): AX =

A”} = {X E P,@“): LX = J_“} and J_” is a vertex of

I .

Proof.

(a) +

(c): Suppose

A”

is a vertex

of

Pf’“‘.

To see that

{Y E

Pj% 1 = J_“} G {Y E Py’“): AY = A”},

let Y E

Py’(/)

satisfy 1 = J’.

It then follows from Eq. (3.1) and Eq. (3.4) that

AY = iI= k’s” = A”.

To see

the reverse inclusion, let Y E

Py’“”

satisfy 2 Y =

A”.

Consider the network flow

(with integer lower and upper bounds on arc-flows) expansion of the linear

system (3.8) as described in the proof of Lemma 3.4; see Fig. 2. Lemma 3.4 and

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F.K. Hwang et al. I Linear Algebra and its Applications 278 (1998) 263-284 215

its proof show that

Py’“’ IS

m one-to-one correspondence with the solution-set

.

of this network flow problem with vertices mapped onto vertices. Thus, we may

and will identify vectors in

Py,‘),

that is, solutions of (3.8) with their

expansion to solutions of the network flow problem; in particular, this is the

case for Y and

J”.

Also, a circuit z is a nonzero, normalized, minimal support

solution of the (homogenous) system

ez;

=

0, /=I

z; -

24.-O

forj=

l,..., p,

3=0

(3.9a)

(3.9b)

P

c

z{=O

fors=O,l,...,

ii,

(3.9c)

j=l

where

normalized

means that (Izll, = 1 (with 11 I/m denoting the

1,

norm) and

minimal support

means that the set of nonzero variables of no solution of (3.9)

is strictly contained in that of z. Standard results show that the coordinates of a

circuit z are all - 1,O and 1 and that each node of the network presented in

Fig. 2 appears in either 0 or 2 indices corresponding to nonzero coordinates

of z. Also, each column of a circuit z hasoat most tvvo nonzero elements which

can take only the values -1 and 1; as

A = 0

and

A , . ,A”

are nonzero and

distinct, it follows that

circuit z satisfies ;z = 0 if and only if z = 0.

(3.10)

As Eq. (3.9b) determines the zy’s of a circuit z from the remaining coordi-

nates, we identify such a circuit with its projection z E Rnxp.

As

A”

is a vertex of

Pf,“) , A”

is the unique maximizer over

Pfsu)

of some lin-

ear function; let such a linear function be represented by the matrix C E Rkxp.

Also, as Y and

J”

are solutions of the network flow problem, a standard result

about network flows (e.g., Denardo, 1982 p. 99) implies that Y -J” can be de-

composed into a sum Cy=, p,z’ where for each t = 1,

. . , q, z’

is a circuit of the

network flow problem, p, is a positive number and

J” + j3,z’

is a feasible solu-

tjon of the network flow problem, that is,

J” + /?,ti E Py,‘),

and by Lemma 3.2,

AJ” + &A2 = A(J” + &z’) E PA (‘L’) Now .

Eq. (3.3) and the unique optimality

of

A”

over

Py3u’

under the linear function represented by C implies that for

t= I,... ,q, (C,A”) B [C,;J” t &h) = (C,A”) + /$(C,k)

with

equality

holding if and on!y if

AJ” + /I42 = A”,

that is, (C.

Af) < 0

with equality hold-

ing if and only if

AZ’ = 0,

and by Eq. (3.10) this is the case if and only if Z’ = 0.

As we are assuming that

AY = Aa

and Y =

J” + CR1 ptz’, we

have that (C,

A”) = (C:AY) = (C,AJ” +c;=, &Az’) = (C,A”) + c;=, &(C, AZ’ ) < (C. A”);

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276 F.K. Hwang et al. I Linear Algebra and its Applications 278 (1998) 263-284

it follows that the inequalities in the above string hold as equalities. Thus for t= l,... , q, (C, ,&) = 0 and consequently z’ = 0. Thus r = J_” + CR1 &$ = J_“.

It remains to show that J_” is a vertex of @“I. Indeed, assume that J” has a representation J” = j3Zi + (1 - p)Z2 for some 0 < B < 1 and Z' , Z2 E Py”/), and we will show that Z’ = Z2 = J”. By Eq. (3.4) A” = kJR, and by Lemma 3.2 ki’ and ki2 are in Pf,‘). As A” is assumed to be a vertex of Pf,‘) and A” = k’s” = /?ki’ + (1 - p)ki2, we conclude that A” = ki’ = ki2. By part (b) of Lemma 3.2 there exist matrices Y’ and Y” in Pp.‘) with 1’ = Z’ and x2 = Z2, and Eq. (3.1) implies that AY’ =ix’ =ki’ =A” and 2Y’ =2x2 = AZ2 = A”. Hence, the established conclusion {Y E PF.“‘): >Y = A”} = {Y E Py: I= p} implies that Z’ = _si’ = p and Z2 = x2 = I”.

(c) + (b): It suffices to show that if {Y E Py”):iY = A”} = {Y E Py? jf = r}, then {Z E Py”‘): ki = A”} = {J_“}. So, assume that the first equality holds. By Eq. (3.4), k’s” = A”; as J” E Py’“), it follows that J” E .(z E Py? ki = A*}. To establish the reverse inclusion let Z E Pf.‘) sat- isfy AZ = A”. It then follows from part (b) of Lemma 3.2 that Z = Y_ for some Y E P@.“). for such Y we have from Eq. (3.1) that 2Y = iI= ki = A” and, by assumption, this implies that 1 = J”, that is, Z = I= J_“.

(b) + (d): From Eqs. (3.2) and (3.4), if X E PF,‘) and JX = p then AX = (k’J)X = k’(JX) = ,@ = A=. So, {X E P,(L+ JX = s”} C {X E P,(L’“): AX = A”}. Thus, it suffices to show that if {X E P,(L,‘): JX = p} c

{X E Ppc AX = A”} then there exists a matrix Z E Py,“) with k> = A” and Z # J”. So, suppose that X E P,(L’“) satisfies AX = A”and JX # p, and let Z s JX. Then Z = JX #J-y. and by Lemma 3.2 and Eq. (3.2), respectively, z = JX E Pys”) and AZ = A(JX) = (AJ)X = AX = A”.

(d) + (a): Suppose condition (d) holds. We will assume that A” is not a ver- tex of Pf’“’ and establish a contradiction. From Proposition 1.1, each vertex of P@“) is in the set {A”: n E II} and a standard result assures that A” has a rep- kentation as a convex combination of vertices of Pf.“). Hence, there exist partitions

rc’

,

. . . ,714 in II and positive coefficients czl, , clq which sum to 1 such that A” = Cyzl cqA” and each A”’ is a vertex of Pf,‘). As A”’ = AZ”’ for t = 1, . , q, implying that A” = cr=, arAn’ = c:=, a,AZXt = A(CT=, QZ”‘). As P(L’u) is convex, X = CT=, QZ~’ E P,(L)‘). So, X E P,(L*“) and AX = A”; hence, cbndition (d) implies that JX = J”, that is, J_” = JX = J(c;=, QZK’) = Cbi txlJZn’ = CTz1 QJ_R’. As J_” is assumed to be a vertex of Pyx”), as all the tll’s are positive and as all the J”‘s are in PFxu), we conclude that for t= l,... , q, J_“’ = J_” and therefore A7 = kj” = k’p = A”. As each A”’ is as- sumed to be a vertex of PL (w) whereas A” G not, we reached a contradiction which proves the asserted implication. IJ

The equivalence of conditions (a) and (d) in Theorem 3.5 shows that for multiple representations of vertices of Pfxu) in the form AX where

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F.K. Hwung et al. I Linear Algeh and ifs Applicuiions 27X (1998) 263-284 217 X E P,(L’u),JX is unique. Restricting this condition to vectors associated with

partitions we get the following corollary of Theorem 3.5 which extends Propo- sition 1.2 to situations where A’s columns may include repeated vectors and/or zero vectors. The corollary shows that the freedom of selecting a partition corresponding to a particular vertex of F’f,U) reduces to the exchange indices associated with common vectors and to the shift indices associated with the zero vector.

Corollary 3.6. Let A E Rkxn, let LI.. ,Lp! UI,. . . Up be positive integers satisfjiing L < U and C,“=, Lj < n < C,“=, Uj, and let V he a vertex

Of

qjL."j.

Then J_” coincide-for partitions II E lIiTIL.c’) w’ith A” = V.

Proof. The implication (a) + (b) in Theorem 3.5 implies that if V = A” = A”’ for 71,~’ E n, then J_” = J_*‘. 0

Condition (c) of Theorem 3.5 is next used to describe a test for vectors associated with partitions to be vertices of given bounded-shape partition polytopes. An alternative method is described in Hwang et al. (unpublished manuscript), and a test for vertices of partition polytopes determined by arbi- trary shape-constraints (not necessarily through lower and upper bounds) are provided in Hwang et al. (submitted).

Testing ij’a vector A” is a vertex of' the bounded-shape partition polytope: Let A: L? U and rc be as in Theorem 3.5. Our test for determining whether or not A” is a vertex of F$“‘) has two parts.

The first part of the test determines whether or not J_” is a vertex of F’jL.“. We observe that J” is a vertex of PJ (‘.‘) if and only if for every representation of _L” as i (Y’ + Y’) with Y’ and Y” in Pp.G’) we have that Y’ = Y”. This condi- tion holds if and only if for each s = 0: l! , ii and .i = 1: .p the maximum of (Y’ - Y”): over Y’ and Y” satisfying Y’ E P:““, Y” E PjL”‘) and

Y = 4 (Y’ + Y”) is zero, which is verifiable by solving Zp linear programs where each-has fip variables and 2(ii i-p) + fip constraints. When A has no zero col- umns, J = J_ and a test for being a vertex of F’y’“’ = F’jL,‘) can be developed from the explicit representation of PjL’L” available from the Lemma 3.4.

The second part of the test determines whether or not 1 = J” for each matrix Y satisfying Y E PjL”” and AY = A”. In view of Lemma 3.4 (&h ” the expansion discussed in the proof of the lemma), the assertion that Y E Py”’ is character- ized by a linear system having (ii + l)(p + 1) variables, (ii + 1) +p constraints. nonnegativity constraints and lower and upper bounds on p variable; also, the requirement 2 Y = A” reduces to another kp constraints. Testing whether or not each solution Y to the joint system satisfies I= J” can be accomplished by maximizing and minimizing I’,’ for j = 1, . . . , p and s = 1. , ii over the joint system, that is, by solving 2Fip corresponding linear programs.

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278 FK. Hwang et al. I Linear Algebra and its Applications 278 (1998) 263-284

The above method for testing whether or not a vector A” is a vertex of a

bounded-shape

partition polytope depends on condition (c) of Theorem 3.5

which reduces to condition (b) of Theorem 3.1 when A’s columns are nonzero

and distinct. We observe that conditions (c) and (d) of Theorem 3.1 yield alter-

native computational

methods under the restricted assumption of that theo-

rem. Indeed, condition (c) concerns solvability of a (sparse) linear system

with np

(0, 1}-variables

and (n + l)(k + 1) equality and weak inequality con-

straints, and condition (d) concerns solvability of a linear system having

(k + 1)~ variables and (p - l)n strict inequality constraints. Each of these tests

is obviously polynomial in k, R and

p.

4. Vertex characterization

In this section we tighten the necessary condition of Proposition 1.3 for be-

ing a vertex of a constrained-shape

partition polytope to obtain a condition

which is both necessary and sufficient. The result extends Theorem 5 of

BHR by relaxing the assumption that the columns of A are nonzero and dis-

tinct; see Theorem 3.1. Our analysis is carried out in two steps. First, we tighten

the necessary condition of Proposition 1.3 to obtain a stronger necessary con-

dition, then, we modify this tighter condition to obtain a condition which is

both necessary and sufficient.

We recall that argmaxXeA f(x) refers to the set of maximizers of the function

f(.) over A.

Theorem 4.1.

Let Ll,. . . ,Lp, VI,. . . , Up be positive integers satisfying L < U and

ziieLj 5; < iFTz, Uj, let Il be the set of I’(LJ)-shape partitions, let z E IZ vertex

NS E {A”: u E qa” {AU:

of Pfl”’ and for T,SE {l,...,p} let

u E 71,). Then for some matrix C E Rkxp and vector a E Rp,

(a)

a, < 0 for r =

1,

. . . , p satisfying L, < Inr(,

(b)

a,. 2 0 for r =

1,

. . . ,p satisfying U, > 1~~1,

(c) (Cd- 1?)~,4” < a, - a, for distinct indices r, s E {

1,.

. . ,p} and u E 7c,,

(d)

if (C” - C’)TA” = a, - cl, for distinct indices r, sE {l,...,P) and

UE~., then:

(i)

Au E Ars if A” # 0, and

(ii)

A” = 0, (n,l > L, and lzS,l < US if Ars = 0.

Proof. Our proof modifies the arguments proving the necessity of the condition

characterizing

a vertex of bounded-shape

partition polytopes as given in

Theorem 5 of BHR under the assumption that the columns of

A

are nonzero

and distinct.

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I? K. Hwang et al. I Linear Algebra and its Applications 278 (1998) 263-284 219

As A” is a vertex of Pf it is the unique maximizer over P,” of some linear function, say one that is determined by the matrix C E [Wkxp; so

(C,X) < (C,A”) for each X E F’,” \ {A”}. _(4.1)

As in BHR, consider the linear assignment problem with indices 0, 1, . . . p and cost-coefficients

I f

m~~x{(C’ - C’)TA” if I’,S 3 1, d,, =

-cx

i

if I’ = 0,s 3 1 and lrrV1 = L,. (4.2) --x if s = 0,r 3 1 and /7c,1 = U,.. 0 otherwise.

Circuits (with respect to this problem) are then nonzero, normalized, minimal support solution of $& =0 fors=O,l,...,p (4.3) r=O and 2ZY., = 0 forr=O,l,..., p, (4.4) V=O

where normalized means that max{ Iz,I: r,s = 0, 1, . . . ,p} = 1 and minimal support means that if z’ # 0 satisfies Eqs. (4.3) and (4.4) then the set of nonzero coordinates of z’ is not strictly included in that of z.

With I as the (p + 1) x (p + 1) identity, it is shown in BHR that under the assumption that the columns of A are nonzero and distinct, if z is a circuit with Z+z>O, then

(4.5) and when (4.5) holds as equality, z., = 0 for all indices r,s = 1,. . :p. While the arguments of BHR establishing the inequalities of (4.5) are applicable when the assumption that the columns of A are nonzero and distinct is relaxed, the conclusions from equality in (4.5) need not hold and such situations are exam- ined in the next paragraph. Still, as in BHR, the (weak) inequalities of (4.5) suf- fice to show that the identity is optimal for our linear assignment problem, and further, Linear Programming Duality and the Weak and Strong Complement- arity Theorems imply the existence of a vector SI = (‘x0. CI~ : , a,) satisfying (a), (b) and

& G u, - a., for all distinct indices r, s E { 1, ,p}, (4.6) with strict inequality holding in (4.6) if and only if z, = 0 for all circuits z satisfying (4.5) with equality.

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We next examine circuits z for which I + z > 0 and (4.5) holds as equality. Let z be such a circuit. As in BHR, it follows that there exist distinct indices rl,r2,. . ,rq where q > 2 such that, with r,+i = r-1,

(

1 for (r,s) = (r,,r,+l), t= l,..., q

zrs = -1 for (r,s) = (rt,ry), t = 1,. . . ,q; _(4.7) 0 otherwise.

Without loss of generality assume that q = min{r,: t = 1, , q} and for t= l,... , q with r, 3 1 let ut be a maximizer or (Cc+’ - Crl)TA” over u E x,,. Now, if (4.5) holds as equality and r1 3 1, the arguments of BHR yield a partition rr’ obtained by cyclic shifts of the vectors A”1 ,A”>, . . . ,A+ with (C,A”) = (C,A”). It then follows from (4.1) that A” = A”, implying that, with uq+l - ul, Au1 = A”+‘, for t = 1, . , q; in particular, in this case the common vector A”’ = A”2 = . . = A!+ is in l-J;=, (Ir’,rr+l. Also, if (4.5) holds as equality and t-1 = 0, the arguments of BHR show that L,, < lx,> 1, ITC,.~/ < Uvq and the ex- istence of a partition rr’ obtained by shifts of the vectors A”*, A”3, . . , A”yml, with

(C,A”) = (C,A”). F rom (4.1) we then have that A” = A”, implying that Au2 = 0 and Au! = A”!+( for t = 2,3,. . ,q - 2. So, we have that Lrz < ITC,.~, l-rvqI < Urq and O = A”’ = A”3 = . . . = A+‘. in particular 0 E n’1’ nr’n+i. _t-2 Of course, some of these conclusion are vacuous if q = 2 or if q = 3.

Let r,s E {I!. . ,p} with r # s and let u E 71,. The definition of the &‘s in Eq. (4.2) and (4.6) then imply that (C’ - C’)A” <d, < c(, - a,, establishing (c); further, if (C’ - C’)A” = cx, - ccs, then u E argmax,,,+((e - C’)A” and d, = cc, - cc,. We establish (d) by further analysis of the case where (CS - C’)A” = CI, - c(, and considering two cases:

Case I: Ars # 0. In this case there exist w E n, and v E 71, with A” = A”. As we already demonstrated that u E argmax,,,r(@ - C’)A”, we have that (CS - C’)A” 3 (C” - C’)A”’ = (C” - C’)A”. We will prove by contradiction that A” = A“. Suppose A” # A”. Let G be the partition obtained from 7c by switching v from rcY to rc, and u from rc, to rc,. Then 0 E ZZ and A” = A” + (A” - A”)(& - er)T #A”. So, A” E Pa \ {A”}, and (4.1) implies that (C,A”) > (C,A”) = (C,A” + (A” -A”)(8 - er)T); hence,

0 > (C, (A” - A”)(e” - er)T) = (C(G? - e’), (A” -A”))

= ((Cs - C’), (A” -A”)) = (Cs - C’JTA” - (Cs - C”)TA”,

in contradiction to the assertion (CS - C’)A” 2 (C” - C’)A”. The contradiction proves that A” = A”, in particular, A” E AK’.

Case II: A” # 0. We have already concluded that d, = a, - cc, and therefore there exists a circuit z that satisfies z + Z 2 0, (4.5) with equality and z, # 0; it then follows that z has the representation (4.7) further, without loss of generality we assume that rl = min,=i....,~ r,. The assertion ii” # 0 combines with the analysis of equality in (4.5) to imply that necessarily q = 3, rl = 0,

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E K. Hwang et al. I Linear Algehru and its Applicutions 278 jlU%) 263-284 281

(7,s) = (YZ.O)> I% > L, and 17~~1 < U/, and A”’ = 0; further. as u E argmax.,E,r(C.’ -

C’)A”;

A”’ can be selected as A” in the construction of the partition 7~’ from 71 and z, implying that A” = A”’ = 0. 0

It is easy to verify that the necessary condition for being a vertex of a bounded-shape partition polytope asserted in Theorem 4.1 implies the necessary condition of Proposition 1.3 (which applies to the more general constrained-shape partition polytopes). The necessary condition of Theorem 4.1 is next modified to obtain a condition which is both necessary and sufficient (cf., Theorem 5 of BHR). The task is accomplished by tightening the conclusions from equalities (C” - C”)A’ = 2, - CI, in (d). But, the new condition lacks a simple geometric/ algebraic motivation of the (necessary) condition of Theorem 4.1; further, verification of the polynomial test for vertices described in Section 3 in simpler. Theorem 4.2. Let LI ~ , Lp. (/I, . . Up he positire integers satisfying L < U und J$, L, < n 6 Cyz, Uj, let Il be the set qf r (L.C’)-shape partitions and let 71 C Ll. Then A” is u 1:erte.x of’

9,” if

and on/y if-for some matri.v C E Rxxl’ and mctol x t w.

(a) IX,- < 0 ,for r = 1~. . ,p satisfying L, < /xJ. and (b) cr,.>O~forr=l,...,psatisjjvingU,.>~n,.l;

(c) (C - C’)‘A” < ry - M, for distinct indices t*‘. s E { I ~ , p} and II E 71, : (d, ) if’ q 3 2. rl , . rq are distinct indices in { 1. . p} , ~1. ~ uy are indices

in {l,... _n} with u, E 71, and (with r,+l = 1)

(@;-I - Crr)TA1l~ = M,-~ - u,,_, f& t = 1.. . .q: then A”1 = A”’ = = A’%; (dz) if‘ q z 2.~1,. . . rq are distinct indices in { 1. ,p} withl,, > In,., / and

U,.<, < /z’.~~ I ( and ~1, . . . % r/,-l are indices in { 1. . , n} with u, E n, and (@;+I - Crr)’ A”) = I, - c(?!+, for t = 1.. q - 1. then A”1 = A”’ z :Z

A% ’ z 0.

Proof. Necessity: Assume that A” is a vertex of PA” and C and CI are constructed as in the proof of Theorem 4.1. In particular, (a)-(c) are satisfied. the coefficients & for r,s = 1, ,p given by Eq. (4.2) satisfy (4.6), and (from the arguments of the proof of Theorem 4.1) if z is a circuit (for the corresponding assignment problem) with I + z 3 0. with Eq. (4.7) in force and with equality holding in (4.5), then:

(i) if t-1 > 0 then argmax,,,?, (fi+l - C’I)~A” is invariant of t = 1~. . y and the common set consists of a single element, and

(ii) if rl = 0 then argmax,,,, (CQ+l - C”I)~A” = (0) for t = 1, , q - 1. Supposeq32,r,!..., ry are distinct indices in { 1. .p} and ul, . . uq are indices in { 1~ , n} with uI E ic, and (with r,,,, = l)(p+l - C’i)T,4tfl = % - CG,., for t = l! . , q. For each t, (c) and (4.6) show that (C-+l - C’.I)‘A”i 6 maxXE,,, (Crf-’ - C’oTA” = c&.,,.,+, < rr, - T,_, ,(@-I-’ ~ C’Q)TA”’ and therefore A”’ E argmax,,,, (CrlL’ - C’i)TA’ and d,.,.,., = x,, - x,.,+, It

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follows that the vector z defined from ~1,.

. . , rq

by the right-hand side of

(Eq. (4.7)) is a circuit satisfying

P P 9

cc drszrs = c dr,,rl+, = ear, - a,,, = 0,

r=O so I=1 I=1

As z is a circuit satisfying (4.5) with equality, assertion (i) of the above pa-

ragraph shows that argmax,,,rl (P+l - C’I)~A” is invariant of

t =

1,

. , q

and

the common

set consists

of a single element,

hence,

the fact that

AR E argmax xEn,,

(C’t+l - CroTA” for

t =

1,

. . , q

implies that A”’ = A”2 =

. . . = A’q, and the proof of (d), is complete.

A similar line of argument applies for the case considered under d-2 except

that in order to obtain a circuit of the assignment problem it is necessary to

augment rl,

. . . , rq

with r,+i = 0.

Sufikiency: Suppose C and a satisfy conditions (a)-(d). We show that A” is a

vertex of Pf by showing that (C,X) < (C, A”) for each X E Pf and equality

holds only if X = A”.

By Lemma 2.1, PI” is the solution set of (2.5). Adding variables q? for

j=

l,...,p,

replacing

(2.5~)

by

the

constraints

q! = Cf=, qj

and

Lj 5 $? < Uj for

j = 1,.

. .

,p

and

adding

the

constraint

CT=, 59 =

CzZo(Je)s = ~1, th e mear system (2.5) is expanded to a network flow problem

1’

with integer lower and upper bounds on arc-flows (see the more detailed con-

struction in Lemma 3.4 and the corresponding Fig. 1 which applies to Py rath-

er than P,“); as the augmented variables are uniquely determined by the original

ones, we identify feasible flows of the network and elements in PI” (see the

proof of Theorem 3.5 for details that apply to P$‘). We consider circuits of

the network which are nonzero, normalized, minimal support solution of

(2.5) (see the proof of Lemma 3.4 and Theorem 4.1 for more detailed defini-

tions of circuits for other network flows). A circuit z for which I” + z E P,!’

may have one of two representation.

It is either identified with sequences

r-1,.

. . ,

r4 of distinct indices in { 1,

. .

,p} and

uI, . . . , uq

of distinct indices in

{l,... ,n}suchthatq>2,

~(~n,fort=l,...,

qand,withr,+,=l,

Z UT =

(

1 for (u,r) = (ut,rr+l), t = 1,.

. ,q,

-1

for(z4,r)=(Ut,r1),

t=l,...,

q,

_(4.8)

0 otherwise,

or with sequences r-1,.

. . ,

r4 of distinct indices in { 1,

. . .

,p} and uI,

. . . , ql

of

distinct

indices

in

{l,...,n}

such

that

922, &En,

for

t= l,...,q-

1, lx,,1 >L,,,

1~~~1

< ur, and

1 _{for (u, r) = (u,, Y,+I}

_{), t = 2,}

. . , 4,

Z ur =

-1

for(U,r)=(Ur,r,),

t=l,...,

q-l,

(4.9)

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F.K. Hwang et al. I Linear Algebra and its ApplicutionJ 278 11998) 263-284 283 In particular, each such a circuit corresponds to a unique partition G(Z) with I”(‘) zz 1” + z.

A”! (&i I _ ,b)T

A circuit with representation (4.8) satisfies AZ = CT_, and

and similarly ‘4% (e”-’ _ &i)T

a circuit with representation (4.9) satisfies AZ = CT:, and

(C.Az) =

9-l Y-1

C(C” _{- C”)TA”l e E(&!} _{- a,,.,) = x,, - xry < 0,} (4. I I )

1=I I ~1

where the last inequality follows as (a) with /z,, 1 > L,, imply that r,., 6 0 and (b) with lrcrql < U,.(, imply that M,.~ > 0. We conclude that if (C,Az) = 0, then ((C% - C).‘)TAU, = z, - a,,_, for all relevant indices t, and therefore (d) implies that AZ = 0. So, AZ =‘O for each circuit z satisfying (C,

AZ)

= 0.

Let Y E p,“. By Corollary 3.3 Y = Ax for some X E P/“. Next, a standard result about network flows (used in the proofs of Lemma 3.4 and Theorem 4.1) assures that X - I” can be decomposed into a sum CR, Bk2 where for each k = l?

.

, q, 2 is a circuit of our network flow problem, pk is a positive number and Z” + ok2 E P,“, the latter implying (again, by Corollary 3.3) that A” + &Azk = A(I” + /@) E l’4 “. From Eqs. (4.10) and (4.11) we have that (C,

Ap)

< 0 for each k, implying that (C, A” + fikAti) 6 (C, A”); hence, (C, Y) - (C, A”) = (C, A(X - I”)) = Cb, Pk(C, AZk) < 0 with equality holding only if (C. A2) = 0 for each k. As the above paragraph shows that A,” = 0 whenever (C, AZk) = 0, we conclude that if (C. Y) = (C, A”) then Ati = 0 for each k, implying that Y = AX = A(I” + Cz_, /i&) = A” + cz:, /?,A2 = A”. So, indeed, we established that (C, X) < (C.A”) for each X E Pi’ with equality holding only for X = A”. 0

References

Barnes, E.R., Hoffman, A.J., Rothblum, U.G., 1992. On optimal partitions having disjoint convex and conic hulls. Math. Programming 54, 69-86.

Denardo, E.V., 1982. Dynamic Programming: Models and Applications. Prentice-Hall, Englewood Cliffs, New Jersey.

Gao, B., Hwang, F.K., Li, W.C.W., Rothblum, U.G., 1998. Partition polytopes over I- dimensional points, Mathematical Programming, to appear.

Hwang, F.K., Onn, S., Rothblum, U.G., Optimization over partitions: Linear programming and explicit solution, unpublished manuscript.

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284 I;: K. Hwang et al. I Linear Algebra and its Applications 278 (1998) 263-284 Hwang, F.K., Onn, S., Rothblum, U.G., Polynomial enumeration of the vertices of constrained-

shape partition polytopes, submitted.

Hwang, F.K., Rothblum, U.G., 1996. Directional-quasi-convexity, asymmetric Schur-convexity and optimality of consecutive partitions. Mathematics of Operations Research 21, 54&554. Hwang, F.K., Rothblum, U.G., Partitions: Clustering and optimality, in progress.