Studies in integer programing

Further, we introduce two types of composite inequalities, obtainable by combining elementary ine- qualities according t o specific rules, and some related inequalities obtainable direc

STUDIES IN INTEGER PROGRAMMING

Managing Editor

Peter L HAMMER, University of Waterloo, Ont., Canada

Advisory Editors

C BERGE, UniversitC de Paris, France

M.A HARRISON, University of California, Berkeley, CA, U.S.A

V KLEE, University of Washington, Seattle, WA, U.S.A

J.H VAN LINT, California Institute of Technology, Pasadena, CA, U.S.A

G.-C ROTA, Massachusetts Institute of Technology, Cambridge, MA, U.S.A

Based on material presented at theworkshop on Integer Programming, Bonn, 8-12 September 1975, organised by the Institute of Operations Research (Sonderforschungsbereich 21), University of Bonn

Sponsored by IBM Germany

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK* OXFORD

ANNALS OF DISCRETE MATHEMATICS I

STUDIES IN

INTEGER PROGRAMMING

Edited by

P.L HAMMER, University of Waterloo, Ont., Canada

E.L JOHNSON, 1BM Research, Yorktown Heights, NY, U.S.A

B.H KORTE, University of Bonn, Federal Republic of Germany

G.L NEMHAUSER, Cornell University, Ithaca, NY, U.S.A

1977

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD

@ NORTH-HOLLAND PUBLISHING COMPANY - 1 9 7 7

All rights reserved N o part of this publication niay he reproduced, stored in a retrieval systen?

or transmitted, in any f o r m or by any means, electronic, mechanical, photocop.ving, recording

or otherwise, without the prior permission of the copyright owner

Reprinted f r o m t h e journal .4nnals o f Discrete Mathematics Volume I

North-Holland ISBN for this Volume: 0 7204 0765 6

Published by:

NORTH-HOLLAND PUBLISHING COMPANY

AMSTERDAM NEW YORK OX1:ORD

Sole distributors for t h e U.S.A a n d Canada:

k.LSEVIER NORTH-HOLLAND, INC

5 2 VANDERBILT AVENUE

NEW Y O R K , NY 1 0 0 1 7

Printed in T h e Netherlands

PREFACE

This volume constitutes the proceedings of the Workshop on Integer Program- ming that was held in Bonn, September 8-12, 1975 The Workshop was organized

by the Institute of Operations Research (Sonderforschungsbereich 21), University

of Bonn and was generously sponsored by IBM Germany In all, 71 participants

frnm 13 different countries took part in the Workshop

Integer programming is one of the most fascinating and difficult areas of mathematical optimization There are a great many real-world problems of large dimension that urgently need to be solved, but there is a large gap between the practical requirements and the theoretical development Since combinatorial problems in general are among the most difficult in mathematics, a great deal of theoretical research is necessary before substantial advances in the practical solution of problems can be expected Nevertheless the rapid progress of research

in this field has produced mathematical results significant in their own right and has also borne substantial fruit for practical applications We believe that this will be adequately demonstrated by the papers in this volume

The 37 papers appearing in this volume cover a wide spectrum of topics in integer programming The volume includes works on the theoretical foundations of integer programming, on algorithmic aspects of discrete optimization, on specific types of integer programming problems, as well as on some related questions on polytopes and on graphs and networks

All the papers have been carefully referred We express our sincere thanks to all authors for their cooperation, to the referees for their useful support, t o numerous participants for stimulating discussions, and to the editors of the Annals of Discrete Mathematics for their willingness to include this volume in their new series

P Schweitzer

IBM Germany

P.L Hammer E.L Johnson B.H Korte G.L Nemhauser

CONTENTS

Preface

Con tents

A BACHEM, Reduction and decomposition of integer programs over cones

E BALAS, Some valid inequalities for t h e set partitioning problem

M BALL and R.M V A N SLYKE, Backtracking algorithms for network reliabil-

C BERGE and E.L JOHNSON, Coloring the edges of a hypergraph and linear

0 BILDE and J KRARUP, Sharp lower bounds and efficient algorithms for the V.J BOWMAN, JR and J.H STARR, Partial orderings in implicit enumeration

C.-A BURDET and E.L JOHNSON, A subadditive approach to solve linear

V CHVATAL and P,L HAMMER, Aggregation of inequalities in integer

G CORNUEJOLS, M FISHER and G.L NEMHAUSER, On the uncapacitated

D DE WERRA, Some coloring techniques

J EDMONDS and R GILES, A min-max relation for submodular functions on A.M GEOFFRION, How can specialized discrete and convex optimization

D GRANOT and F GRANOT, O n integer and mixed integer fractional

M GROTSCHEL, Graphs with cycles containing given paths

M GUIGNARD and K SPIELBERG, Algorithms for exploiting the structure of

M GUIGNARD and K SPIELBERG, Reduction methods for state enumeration

P HANSEN, Subdegrees and chromatic numbers of hypergraphs

R.G JEROSLOW, Cutting-plane theory: disjunctive methods

E.L LAWLER, A ‘pseudopolynomial’ algorithm for sequencing jobs to minim- J.K LENSTRA, A.H.G RINNOOY KAN and P BRUCKER, Complexity of machine

L LOVASZ, Certain duality principles in integer programming

R.E MARSTEN and T.L MORIN, Parametric integer programming: the right-

Contents uii

J.F MAURRAS, An example of dual polytopes in the unit hypercube

P MEVERT and U SUHL, Implicit enumeration with generalized upper bounds

I MICHAELI and M.A POLLATSCHEK, O n some nonlinear knapsack problems

J ORLIN, T h e minimal integral separator of a threshold graph

M.W PADBERG, On the complexity of set packing polyhedra

U.N PELED, Properties of facets of binary polytopes

D.S RUBIN, Vertex generation methods for problems with logical constraints J.F SHAPIRO, Sensitivity analysis in integer programming

T.H.C SMITH and G.L THOMPSON, A lifo implicit enumeration search algorithm for the symmetric traveling salesman problem using Held and Karp’s 1-tree relaxation

T.H.C SMITH, V SRINIVASAN and G.L THOMPSON, Computational perfor- mance of three subtour elimination algorithms for solving asymmetric traveling salesman problems

J TIND, On antiblocking sets and polyhedra

L.E TROTTER, O n the generality of multi-terminal flow theory

L.A WOLSEY, Valid inequalities, covering problems and discrete dynamic

U ZIMMERMAN, Some partial orders related to boolean optimization and the

S ZIONTS, Integer linear programming with multiple objectives

This Page Intentionally Left Blank

Annals of Discrete Mathematics 1 (1977) 1-11

@ North-Holland Publishing Company

PROGRAMS OVER CONES

matrix N has special structure

where N is an ( m , r ) and B an ( m , n ) integer matrix As B is an arbitrary ( m , n )

integer matrix, the convex hull of the feasible set of (1.1) is a generalized corner polyhedron, that is an equality restricted integer program, where the nonnegativity restriction of some of the variables are relaxed To give a group representation of the problem, we reformulate (1.1) as a congruence problem,

min c ' x

s.t Nx = b m o d B

x E N'

1

2 A Bachem

where we define Nx = b (mod B ) , iff there is a A E Z", such that Nx - b = BA holds T o set this definition in a more general framework we have to introduce the concepts of Smith and Hermite normal form

Definition If B is an (m, n ) integer matrix, we denote by S ( B ) and H(B) the Smith

and Hermite normal form of B, S * ( B ) and H*(B) denotes the nonsingular part of

form are denoted by U,, KB and the projection matrices, which eliminate the

nonsingular part S * ( B ) of S ( B ) are denoted by WE, VB Thus we have S * ( B ) =

Sometimes it is advantageous to look at congruences from an algebraic point of

view, that is to look at the definition of a : = x ( = m o d a ) l as an image of the function a : = h , ( x ) = x - a [ x / a ] (where "[x]" denotes the integer part of x) For

formula and we get the generalized form as

h E ( x ) : = x - B [Btx]

where B denotes the Hermite form H(B)VB of B (the zero colums of H(B) are omitted) and where B denotes the Moore-Penrose inverse of B In fact we have

Proposition (1.3) Let G be an additive subgroup of Z" The map hB : G -+ he ( G ) is

a homomorphism onto ( h e ( G ) , @ ) with kernel ( h B ) = {x E G I x = BA, A E Z " } , and

Reduction and decomposition of integer programs 3

So we conclude

hence h, is a homomorphism Let x E kernel(h,), that means x = B[Btx] If we

denote b : = [ B t x ] E Z' and a : = (b',Oh-,)' we conclude x = H ( B ) a and x = Bc where c = Ka, here K denotes the unimodular right multiplicator of H ( B ) Let

now x = Ba with a E Z", that means x = Bb, b E Z' With B t x = b we conclude

Clearly problem (1.5) is a group problem over the group G ( B ) , which is not

necessarily of finite order (it depends obviously on the rank of B) If we follow the usual definition of equivalent matrices (cf (5)), that is the ( m , n ) integer matrix A and the ( r , s ) integer matrix B are equivalent iff they have the same invariant factors (apart from units), we get a slight generalization of a well known fact:

Remark (1.6) The groups G ( A ) and G ( B ) are isomorphic, iff the matrices A and

B are equivalent and m- r a n k(A) = r-rank(B)holds

Using this result it is easy to give a formula for the number of different (nonisomorphic) groups G ( B ) , where the product of invariant factors of the (rn, n )

matrices B is fixed This number is well known for regular ( m , n ) integer matrices

B Here we are going to treat the general case

Definition Let B be an ( m , n ) integer matrix We call the product of the invariant factors of B the invariant of B (inv (B)) which coincides with the determinant of B

in case B is a square nonsingular matrix

If d = n;=, % P > is a representation of d = inv(B) as a product of prime factors

and p a function from NZ into N defined recursively as

p ( O , m ) : = l , p ( n , O ) : = O ( n , m E N ) , we define

Proposition (1.7) The number of nonisomorphic groups G ( B ) , where B varies over

equals the integer number K ( d )

Bachem

Notice that K ( d ) is a finite number, though we consider all ( m , n ) integer matrices B with m, n E N If we compute the numbers K ( d ) and L ( d , m ) for d ' s

between 1 and lo5, we note that 0 S K ( d ) 5 10 in 95% of the cases, that is the group

G ( B ) is more o r less determined by d = inv(B)

Proof of Proposition (1.7) Two groups are isomorphic iff the generating matrices are equivalent and the rank condition holds (cf Remark (1.6)) Proving the first part

of t he proposition we have only t o deal with maximal row rank matrices and using Remark (1.4) we can restrict ourselves to square matrices, because h , ( x ) is defined

in terms of H*(B) and this an ( m , n ) integer matrix with d e t H * ( B ) = inv(B) Because of the divisibility property of the invariant factors of an ( m , m ) integer matrix it suffices now t o compute th e number of different representations of the exponents of a prime factor presentation of the determinant d = det B as a sum of

m nonnegative integers In fact this number equals p ( q , m ) (cf ( 2 ) ) and moreover H(d) is finite because

2 Minimal group representation

W e have seen that (1.5) is a group problem, namely of the group G ( B ) In fact this is the group which will usually be considered in the asymptotic integer programming approach (cf (3)), whereas the actual underlying group of (1.5) is the group

G ( N / B ) : = { h , ( x ) / x = N A , A E Z'}

which is a subgroup of G ( B ) generated by the columns of the matrix N From a computational point of view the group G ( N / B ) is more difficult to handle than the group G ( B ) (though it has less elements), because there is n o proper respresenta- tion of G ( N / B ) From this reason here we are going to find a 6 E N" which will be defined in terms of N and B, such that the group G ( N / B ) is isomorphic to

Reduction and decomposition of integer programs 5

G (diag(6)) Clearly this is a minimal group representation of problem (1.5) and as

a corollary we get the order of G ( N / B ) by

First we want to give some results concerning congruences which will be used later, they seem to be of general interest, though

Theorem (2.1) Let B be an ( m , n ) integer matrix with rank ( B ) = m, N an ( m , s)

integer matrix, b E Z" and A : = ( N , B ) The system of congruences

where H:=(K,V,WML, R ) Here we denote b y L : = S * ( A ) - ' U a N , M : =

S*(A)-'U,B and R denotes the last s - k columns of KM, where k : = r a n k ( N )

Proof Without loss of generality we set b = 0 It is easy to see that S*(M, L ) equals

an ( m , m ) identity matrix I"', so we conclude

6 A Bachem

Let y = ( y i , y:)' be a ( k , s - k ) partition of y , then we get

y l , y z integer

Let K i ( i = 1 , ., k ) b e unimodular matrices, which transform the ith row of

completes the proof

Theorem (2.2) With the notations of theorem (2.1) we get

Reduction and decomposition of integer programs 7

Let

where Is-' denotes an ((s - k ) , ( s - k)) identity matrix Because of H =

where Q denotes the first k rows of U,

From the proof of theorem (2.1) we know that

S*(QL) = diag(t,-r+l, ., t m )

so

which completes the proof

Now we are able to give an isomorphic representation of the subgroup G ( N / B )

Theorem (2.3) Let B be an ( m , n ) and N a n ( m , r ) integer matrix with rank(B) =

G ( N / B ) = G ( S * ( E ) ) ,

WM UML and L : = S * ( N , B)-' U(N,B)N, M : = S * ( N , B ) - ' U(N,B)B

8 A Bachem

{I?} / kern el (he )

where kernel(hB) = {x E {I?} 1 x = 0 mod B}

With Theorem (2.1) we conclude

kernel(hB) = {x E Z" 1 x = Ny, y = 0 modKMWMUMLfor a y E Z'}

3 Partitioning of integer programs over cones

T h e computational effort to solve the problem

Reduction and decomposition of integer programs 9

To simplify notation let B = S * ( B ) , i.e B is given as a diagonal matrix (Otherwise

we have to impose some special structure on UB.)

Let us denote the set of feasible solutions of problem (3.1) by

SG(N, b/B):={x E N ‘ 1 Nx - b E kernel(h,)}

Let N be an ( m , r ) integer matrix of form (3.2), let b,(x):=he(b - N,x),,, where I,

corresponds to the row indices of the submatrix N, and let us denote by

if b z ( y ) e G ( N , /B,,),

z(b,(y)): = minc:x,

the optimal value of the subproblems

Proposition (3.4) The programs

[ x E S G ( N , , b , ( y ) / B , ) otherwise,

min c’x

are equivalent

Proof Let r, ( y ) be the minimard corresponding to the optimal value z (b, (y)) Let y

be optimal in (3.6) and assume that there is an f E S G ( N , b/B),

( i # x:=(y, r2(y), ., r,(y)) such that c ‘ f < c’x

Let f : = ( f l , P 2 , , P,), where 9, are the components corresponding to N, Because f, are feasible, we get

c : P, 3 min c,x, = c’X, i = 2 , , r

X, E S G (Nn, b, (9‘ )/B,, )

and the contradiction

c‘P 3 c l j l + 2 c:P, a c ’ x = min ciy + 2 .z(b,(y))l y E N )

proves one part of the proposition, however the reverse direction is trivial

z l ( x z , .,x,):=minc,x,

s.t

Let again N b e an ( m , r ) integer matrix which has form (3.3) and define

z,(xi, .,x,):=mincix, + zi-,(xi, .,x,)

x , E S G ( N i , b i / B , , ) , i = 2 , , r,

as the optimal value of the subproblems

which yields in the same way

€or all i > 1, because

implies

So we get the result

c’X = min c,x, + Z , - ~ ( X , )

x, E S G ( N , b , l B ) ,

which completes the proof

The computational experience with algorithms canonically based on Propositions (3.4) and (3.5) is up to now limited to some of the Bradley-Wahi [l] test examples, which have determinants greater than 1,000,000 The results are very promising in the sense that it is possible to solve “cone problems” of such large order The complete computational results together with comparisons of existing group algorithms will be the subject of a following paper

Reduction and decomposition of integer programs 11

Acknowledgment I wish to acknowledge the interesting discussions I had with E.L Johnson on the subject of this paper T h e paper has been revised substantially while

he was a visiting professor at the University of Bonn

[4] M Marcus and E.E Underwood, A Note on the Multiplicative Property of the Smith Normal Form,

J of Res of the Nat Bureau of Standards-B., 76B (1972) 205-206

[5] M Newman, Integral Matrices (Academic Press, New York, 1972)

[6] M Newman, The Smith Normal Form of a Partitioned Matrix, J of Res of the Nat Bureau of New Haven, December 1969

Standards-B, Vol 78B (1974) 3-6

This Page Intentionally Left Blank

Annals of Discrete Mathematics 1 (1977) 13-47

@ North-Holland Publishing Company

SOME VALID INEQUALITIES FOR THE SET

to enhance orthogonality tests in implicit enumeration or column generating algorithms Further,

we introduce two types of composite inequalities, obtainable by combining elementary ine- qualities according t o specific rules, and some related inequalities obtainable directly from the set partitioning constraints These inequalities provide convenient primal all-integer cutting planes that offer a greater flexibility and are usually stronger than the earlier cuts which d o not use the special structure of the set partitioning problem In the final section we discuss a primal algorithm which uses these cuts in conjunction with implicit enumeration

1 Introduction

Set partitioning is one of those combinatorial optimization problems which have wide-ranging practical applications and for which n o polynomially bounded algorithm is available Though both implicit enumeration and cutting plane algorithms have been reasonably successful o n this problem, the practical impor- tance of solving larger set partitioning models than we can currently handle makes this a very lively research area (see [6] for a recent survey of theoretical results and algorithms, and a bibliography of applications)

In this paper we introduce a family of valid inequalities derived from the logical implications of the set partitioning constraints, and investigate their properties and potential uses We first define some basic concepts, then at the end of this section

we outline the content of t h e paper

The set partitioning problem can be stated as

* This research was supported by the National Science Foundation under Grant # GP 37510x1 and

by the U.S Office of Naval Research under contract N00014-67-A-0314-007NR

13

14 E Balm

min{cx I A X = e, x, = 0 or 1,; E N }

where A = ( a , - ) is an m x n matrix of 0's and l's, e is an rn-vector of l's,

N = (1, ., n } We will denote by a, the j t h column of A , and assume that A has no

zero row and n o zero column Also, we will write M = (1, ., m}

The convex hull and the dimensions of a set S, and the vertex set of a polytope T,

will be denoted by conv S, dim S and vert T respectively

Denoting by "conv" the convex hull, we will call

dim P =G dim LP = n - r(A )

where r(A) is the rank of A

An inequality

satisfied by all x E P is called valid for P A valid inequality (1) such that

for exactly k + 1 affinely independent points x E P, 0 k s dimP, defines a k-dimensional face of P and will itself be called a face (though since dim P < n, a

given face can be defined by more than one inequality) If k < d i m P , the face is proper, otherwise it is improper In the latter case, the hyperplane defined by (1') contains all of P, and is called singular

A valid inequality (1) is a cut, or cutting plane, if it is violated by some x E LP \ P

A face of P, whether proper or not, may or may not be a cutting plane If dim P = dimLP, then the affine hull of P is the same as that of LP; hence any

hyperplane which contains all of P, also contains all of LP, and therefore n o improper face of P is a cutting plane If dim P < dim LP, then improper faces of P

may also be cutting planes

Proper faces of maximal dimension are called facets Evidently, P has faces (hence facets) if and only if di mP 2 1, which implies n > r(A) If dim P = dimLP,

15

then the facets of P are of dimension n - r ( A ) - 1, i.e., each facet contains exactly

n - r ( A ) affinely independent points of P Since 0 P, these affinely independent points are linearly independent vectors

A valid inequality (1) is maximal if for any k E N and any T ; > T k there exists

The following is an outline of the content of this paper

We start (Section 2) with a class of homogeneous canonical inequalities that we call elementary, since all the subsequent inequalities can be built up from these first ones by various composition rules The elementary inequalities, together with the

0-1 condition and the constraints Ax S e, imply the constraints Ax 3 e ; but they also cut off fractional points satisfying Ax = e, x 3 0 We discuss the conditions under which a given elementary inequality is (a) a cutting plane, (b) maximal, (c) a facet or an improper face of P

When a given elementary inequality is not maximal, it can be strengthened In Section 3 we discuss two systematic strengthening procedures for these inequalities

In Section 4 we show that each elementary inequality is equivalent on LP to a set

packing inequality and to each of several set covering inequalities The first one of these equivalences suggests a graph-theoretical interpretation We introduce a

“strong” intersection graph of the matrix A defining P, and show that a set packing inequality is valid for P if and only if it corresponds to a complete subgraph of the strong intersection graph of A ; and it is maximal if and only if this complete subgraph is a clique

The next two sections deal with composite inequalities, obtained by certain rules from the elementary inequalities These composite inequalities have the following property Given an integer basic solution t o the system Ax = e, x a 0 , and a set S of nonbasic variables, none of which can be pivoted into the basis with a value of 1 without making the solution infeasible, there exists a composite inequality which can be used as a primal all-integer cut t o pivot into the basis any of the variables in S

without losing feasibility

Finally, in Section 7 we introduce a class of inequalities which are satisfied by every feasible integer solution better than a given one, and which can be strengthened to a desired degree by performing implicit enumeration on certain subproblems We then discuss a hybrid primal cutting plane/implicit enumeration algorithm based on these results

Throughout the paper, the statements are illustrated on numerical examples

Valid inequalities of the form

where Q C N i k , for some i E G k , will be called elementary They play a central role

as building blocks for all the inequalities discussed in this paper These elementary inequalities are canonical in the sense of [4] (i.e., they have coefficients equal to 0, 1

or - l), hence each of them is parallel to a (n - 1 Q 1 - 1)-dimensional face of the unit cube

Remark 2.1 The slack of an elementary inequality is a 0-1 variable

Proof

cannot exceed 1

Since Q C Nik C Ni for some i E M, the sum of the variables indexed by Q

Proposition 2.1 For every k E N and i E a,, the inequality

Proof From the definition of Nik, for every x E vert P, X I , = 1 implies x, = 1 for at

least one j E Nik But this is precisely the condition expressed by (2); thus (2) is satisfied by all x E v e r t P, hence by all x E P 0

Remark 2.2 The number of distinct inequalities ( 2 ) is at most c k E N ( M kI

Proof

of these inequalities may be identical

There is one inequality (2) for every zero entry of the matrix A, but some

The converse of Proposition 2.1 is not true in general, i.e., a 0-1 point satisfying all inequalities (2) need not be in P, as one can easily see from the counterexample offered by R such that X, = 1, V j E N However, a weaker converse property holds

Some valid inequalities 17

Proposition 2.2

qualities (2), also satisfies Ax 3 e

A n y x E (0, l}", x # 0, which satisfies A x s e and all the ine-

Proof Let X E (0, l}", X # 0, be such that AX s e, AX# e Then there exists i E M

such that X, = 0, V j E N, Further, since X # 0, there exists k E fit such that Xk = 1 Therefore X violates the inequality

since Nik C Ni 0

Corollary 2.2.1

inequality ( 2 ) ; and every inequality (2) cuts off some x E p \ P

Every nonzero vertex of P not contained in P is cut off by some

Proof Every x E p \ P violates A x 2 e ; hence if it is a nonzero vertex of P,

according to Proposition 2.2 it violates some inequality (2) On the other hand, every inequality (2) cuts off the point X € p defined by ?k = 1, Xj =o,

V j E N \ { k } 0

Proposition 2.3 For k E N , i E M k and Q Nik, the inequality

is valid if and only if x E vert P and x k = 1 implies xi = 0, vj E Nik \ Q

Proof

Remark 2.1, x, = 0, v j E Q (since Q C N g k ) , and x violates (3)

because (2) is valid 0

Necessity: if x E vert P and X I , = x, = 1 for some j E Nik \ Q, then from

Sufficiency: if x E vert P and X k = 1 implies x, = 0, v j E Nik \ 0, then (3) is valid

Next we illustrate the elementary inequalities on a numerical example

Example 2.1

polytope with coefficient matrix A (where the blanks are zeroes):

Consider the numerical example of [5], i.e., the set partitioning

3) and therefore each of the sets N31, N 4 and N51 can be replaced by Q = {3}, and

each of the above inequalities can be replaced by

X I - x3 =s 0

In the next section we discuss procedures for strengthening elementary ine-

qualities of the type (3) (which subsumes (2)) by systematically reducing the size of the sets Q subject to the condition of Proposition 2.3

As mentioned in Section 1, a valid inequality may o r may not be a cut, i.e., may

or may not b e violated by some x E LP, P

Proof According to a classical result (see, f o r instance, [20, Theorem 1.4.4]), (3) is

a consequence of the system Ax = e, x 3 0 if and only if there exists 8 E R"

satisfying (4) and (5) If (3) is a consequence of Ax = e, x 3 0, it is clearly not a cut Conversely, if (3) is not a cut, then it is satisfied by all x E LP, hence a consequence

of Ax = e, x 3 0 0

Next we address the question of when a given elementary inequality is

undominated, i.e., maximal First, if for some j E N, x E P implies x j = 0, then clearly the coefficient of x j can be made arbitrarily large without invalidating the

given inequality Therefore, without loss of generality, we can exclude this

degenerate case from our statement

Proposition 2.5 Assume that the inequality (3), where Q C Ni, for some k E N ,

(i) for every j E Q there exists x E vert P such that x j = x k = 1;

(ii) for every j E # { k } there exists x E vert P such that x, = 1 and x k 2 x h ,

Some ualid inequalities 19

Proof This is a specialization of the statement that a valid inequalityrx s ro for a

0-1 polytope T C R" is maximal if and only if for every j E N there exists x E T

such that xJ = 1 and r x = ro 0

If a valid inequality is not maximal, then at least one of its coefficients can be increased without cutting off any x E P In the case of an arbitrary polytope, this is all we know, and it is not true in general that more than one coefficient can be increased without invalidating the inequality In the case of elementary inequalities for P, however, one can say more

Corollary 2.5.1 Assume that for every j E N there exists x E vert P such that

i E M k , and let S1, S2 be the sets of those j E N for which conditions (i) and (ii),

and the inequalities

Proof

To prove the validity of (7), let x E vert P be such that XI, = 1 Then xi = 0,

V j E Sz n Ni \ {k} (hence V j E Sz n T ) , since otherwise from the definition of Sz,

x,, > XI, = 1 for some h E Q, which is impossible Further, from (3), xi = 1 for some

j E Q Hence (7) holds for all x E P such that xk = 1

E 7'; and from

the definition of S2, xj = 1 for some j E Sz f l T implies xk < xh for some h E 0, i.e.,

x h = 1 for at least one h E Q Hence (7) also holds for all x E P such that xk = 0 0

Clearly, if for some S ' C N the nondegeneracy assumption of Proposition 2.5 (and Corollary 2.5.1) is violated for all j E S', then the coefficient of each xi, j E S',

can be made arbitrarily large, in addition to the changes in the coefficients of xi,

From the above Corollary, nonmaximal elementary inequalities can be strengthened, provided we know S In the following sections we give several procedures for identifying subsets of S

Next we turn t o the question of when a maximal elementary inequality is a face of maximal dimension, i.e., a facet or an improper face of P This question is of interest since P is the intersection of the halfspaces defining its facets and improper faces The next proposition gives a sufficient condition for an elementary inequality to be a facet or an improper face of P

The validity of (6) follows from Proposition 2.3 and the definition of S ,

Now let XI, = 0 From the definition of T, xj = 1 for at most one

20 E Balas

Proposition 2.6

P N = P n { x E R " ) x , = O , V j E Q U { k } }

Proof Let d = dimP, d ' = dimPNs Since (3) is maximal, for every j E Q there

exists x' Evert P such that x : = x i = 1 Also, since Q C N,, xl, = 0, V h E Q \ { j }

for each of these q points x ' With each point x', j = 1, ., q, we associate a row

vector y' E R", obtained by permuting the components of x' so that x i comes first,

and the components indexed by Q come next

Further, let z E R"", j = 1, ., d ' + 1, be a maximal set of affinely independent

vertices of PN,, and let yq" E R", j = 1, ., d ' + 1 be row vectors of the form

permutation of components, a vertex of P Then the matrix Y whose rows are the vectors y ' , i = 1, , q + d ' + 1, is of the form

XI I x*

x = [ ;+-]

where X I is the q x ( q + 1) matrix

(tne blanks stand for zeroes), Z is the ( d ' + 1) x ( n - q - 1) matrix whose rows are the vectors ti, j = 1, ., d ' + 1, 0 is the ( d ' + 1) x ( q + 1) zero matrix, and X , is a

Since X and Z are of full row rank, so is Y ; and since Y has q + d ' + 1 rows, it

follows that P contains at least q + d ' + 1 affinely independent points; hence

d 2 d ' + q

affinely independent points of P; and since each of these points satisfies (3) with equality, the same is true of every other point of P Hence in this case (3) is an improper face of P

+ 1, then there exists a point x ' E P which, together with the

d ' + q + 1 points corresponding to the rows of Y , forms an affinely independent set

If x ' also satisfies ( 3 ) with equality, then (3) is an improper face of P; otherwise (3) is

a facet of P

If d = d ' +

are cutting planes, since each of them cuts off the fractional point 2 defined by

X I = Xz = Xs = a, ffs = 1, Xj = 0 otherwise; but they are not maximal, since the conditions of Proposition 2.5 are violated for j = 9,12 in the case of the first

inequality and j = 4 in the case of the second one Therefore, x 1 - x3 S 0 and

x I - x 3 + x s - x I 2 s 0 are both valid (Corollary 2.5.1) T h e inequality x 1 - x3 s 0 is maximal, since the assumption and conditions of Proposition 2.5 are satisfied It is also a facet of P, since the dimensionality condition of Proposition 2.6 is satisfied

and the point X defined by X, = XI4 = XIS = 1, Xj = 0 otherwise, does not lie on

An inequality r r ‘ x s rro is called stronger than rrx G rro, if .rr: 3 rrj for all j , and

r r ; > T for at least one j

In this section we discuss two procedures for replacing a valid elementary inequality which is not maximal, with a stronger valid elementary inequality Th e

first procedure uses information from the other elementary inequalities in which x k

has a positive coefficient; the second one uses information from the elementary inequalities in which x, has a positive coefficient for some j E Q

Proposition 3.1

the inequalities

22 E Balas

Proof From the definition of the sets QU), x E P with x j = 1 implies

for all j E Q,, i E GI, Therefore, if j E T,, then x E P with x, = 1 implies

for some h E f i e ; which implies X I , = 0, since (3‘) holds for z = h

satisfied by all x E P, then so is the system (8)

Hence x E P and xe = 1 implies xi = 0, V j E T, Therefore, if the system (3’) is

0

Proposition 3.1 can be used to strengthen the inequalities (2) by replacing the sets

N,, with Qi = Nik ., T, It can then again be applied t o the strengthened inequalities, and so on, until n o further strengthening is possible on the basis of this proposition alone

Applying the proposition to an inequality of the system (3’) consists of identifying the set T This can be done by bit manipulation and the use of logical “and” and logical “or” The number of operations required is bounded by 1 Qi I X lak I

Example 3.1

of Example 2.1, and let us use Theorem 3.1 to strengthen the inequality

Consider again the set partitioning polytope defined by the matrix

Since {3} is contained in each of N4] and NS1, the inequalities associated with

A second application of Proposition 3.1 brings n o further improvement For k = 2, a2 = {1,5}, N,, = {13,14}, Nsz = {5,13} and none of the two corre-

For k = 5, a5 = {1,2,3,4}; N l s = {1,6,8,9,14}, N 2 S = {1,2,11,15}, N3s = {2,6,8},

these two sets can both be replaced by x 1 - x3 =s 0

sponding inequalities can be strengthened via Proposition 3.1

N4, = {2,8,9,11} Using Proposition 3.1 to strengthen the inequality

Some valid inequalities 23

associated with N , 5 , we set Qh = Nh5, h = 1 , 2 , 3 , 4 ,

Q(1) = {2,6,8,9,11,14,15}, Q(6) = {1,2,8,9,14), Q(8) = {1,2,6,9,11,14}, Q(9) = {1,2,6,8,11,14), Q(14) = {1,6,8,9),

and we find that

Q(1) II Q3, Q(9) 3 0 3 ,

and hence TI = {1,9} and the inequality associated with N I 5 can be replaced b y

xs - x(j- xs - x,4 s 0

When Proposition 3.1 is used to strengthen all rather than just one of the

elementary inequalities in which a certain variable x k has a positive coefficient, it is convenient to work with the set

= { j E Qol Q(j)> Qh for some h € G k )

by looking once at each j E Qo, and then use Qi \ T in place of Q i , T, in (8) The number of operations required for applying Proposition 3.1 once to all elementary inequalities in which xk has a positive coefficient is then bounded by

Next we discuss a second procedure for strengthening elementary inequalities

I a O I x l G k (

Proposition 3.2

inequalities

Proof Let k E N , i E M k , and 1 E u # k Then there exists h E 6% n M l such that

Qhk n Qbl = 0, and therefore adding the two elementary inequalities corresponding

to Qhk and Qhf respectively yields

But then xk = 1 implies x1 = 0 and therefore (3”) can be replaced by (9)

If Proposition 3.2 is applied t o several elementary inequalities, then repeated applications may yield additional improvements like in the case of Proposition 3.1 Applying Proposition 3.2 to an inequality (3”) consists of identifying the set Utk

Again, this can be done by bit manipulation and use of logical “and” and logical

“or” The number of operations required for each j E Q,k is bounded by n

G, 1, hence the total number of operations is bounded by I Qa 1 X I Qk 1, like in the case of Proposition 3.1

to replace both N,, and N4, with smaller sets, after applying Proposition 3.1 to

k = 2 We have A?f, = {1,2,5,6,7}, A?, f l M 2 = {5,6,7} and N5z = (3, lo}, Nh2 = {3,4},

N72 = (4) Applying Proposition 3.1 we find that Q(3) 3 N72; thus T5 = T6 = {3}, and the sets N5*, N6*, can be replaced by Ns2\ Ts = (10) and N6*\ Th = (4) respec- tively

Now writing Q,, = N , , and Q,Z = N,, \ T, for i = 5,6,7, we can apply Proposition 3.2 since

Some valid inequalities 25

Q ~ , n Q~~ = 0

and thus U,, = U,, = (2) Hence N 3 , and N 4 , can be replaced by N31\ U 3 , = {5,7}

and N4, \ U,, = {6,8} respectively

4 Nonhomogeneous equivalents of the elementary inequalities and a graph- theoretical interpretation

In this section we introduce two classes of nonhomogeneous canonical ine-

qualities, which are equivalent on LP to the elementary inequalities (3) One of

these two classes of inequalities lends itself to an interesting graph-theoretical interpretation

Proposition 4.1

Proof Since Q C Nik and ke Ni, (10) can be obtained by adding equation i of

Ax = e to (3) Further, j E Nh implies a,ak# 0, and j E Q implies aiak = 0; therefore Q n Nh = 0 From this and the fact that k E Nh, each inequality (11) can

be obtained by multiplying (3) with - 1 and then adding to it equation h of Ax = e

Since any x E LP satisfies Ax = e, it follows that any x E LP that satisfies (3), also

satisfies (10) and each of the inequalities (11)

Further, (3) can be obtained from (lo), as well as from each of the inequalities

( l l ) , by the reverse of the above operations, therefore any x E LP which satisfies

(lo), or any of the inequalities ( l l ) , also satisfies (3); and, in view of the preceding paragraph, it also satisfies all the other inequalities of (lo), (11) 0

Remark 4.1 Proposition 4.1 remains true if the condition ‘‘x E LP” is replaced

(i) x defined by XI, = 1, x, = 0, j E N , { k } , satisfies (10) but violates (3) and (11);

To illustrate this, we assume Q# 0, Ni \ Q # 0, Nh f l

valid for the set packing polytope

when exactly they are not

Proposition 4.2 The inequality (10) cuts of some x E P \ P i f and only i f Q # Ntk

Proof Necessity If Q = N a k , then aka, # 0, V j E N, \ Q Also, aha, # 0, V h , j E

thogonal, and therefore (10) is satisfied by all x E P

XI, = xh = 1 for some h E N,k \ Q, x, = 0 otherwise, belongs to P, but violates (10)

Example 4.1 The (strengthened) elementary inequality

X I + X I + XlO+ XI3 == 1

From the practical standpoint of an implicit enumeration algorithm, every solution t o the set packing problem defines a partial solution to the associated set partitioning problem In this context the above result has to do with cutting off partial solutions to the set partitioning problem and has the following implication

We say that a set Q C N,, is minimal if n o element of Q can be removed without invalidating the (valid) inequality (3) Also, a partial solution is said to be cut off if its zero completion is cut off

Some valid inequalities 27

Corollary 4.2.1 Let k E N If the sets Qi C Nk, i E M k are minimal, then the

which have no feasible completion

Proof Suppose the sets Q,, i E G k , are minimal, and let .Fk = t = 1, with &ah = 0,

be a partial solution which has n o feasible completion Then x E vert P and X k = 1 implies x h = 0 , and there exists i , E 6 i k such that h E N g e k Let (lo), be the inequality (10) for i = i, From Proposition 2.3, h E Nz.\ Q,.; for otherwise (lo), remains valid when Q g is replaced by Q, { h } , contrary t o the minimality of Q#

But then the zero completion of f k = f h = 1, (i.e., the point obtained by setting

x = 0 , j # k , h ) , violates (lo), 0

Corollary 4.2.1 suggests that the inequalities (10) can be used to enhance the orthogonality tests in implicit enumeration (see [9, 14, 191) or in an all-binary column-generating algorithm [3] The latter possibility is currently being explored The set packing inequalities (10) have a well-known graph-theoretical interpreta- tion in terms of the intersection graph of the matrix A We first discuss this interpretation, then use it to derive a new interpretation on a different graph which incorporates more properties of the set partitioning polytope P For background material, see [15, 16, 17, 211

The intersection graph GA of the 0-1 matrix A has a node j for each column a,,

and an edge ( i , j ) for each pair of columns a,, a, such that a,a, # 0 An inequality of the form

c X j S l

j e V

is valid for the set packing polytope defined on A, i.e., is satisfied by all x E P, if and only if V is the node set of a complete subgraph of GA ; and (12) is a facet of if and only if V is the node set of a clique, i.e., a maximal complete subgraph, of GA [8,

171 Evidently, all those inequalities (10) such that { k } U ( N , \ Q) is the node set of a

complete subgraph of GA, are satisfied by all x E P; and from Proposition 4.2, these inequalities are precisely those for which Q = Ngk The other inequalities (lo), for which Q # N,, have n o interpretation on GA

This suggests the following interpretation on a supergraph of GA We define

G ( A ) , the strong intersection graph of the matrix A, t o have a node for each j E N,

and an edge for each pair i, j E N such that there exists n o x E {0,1}" satisfying

Ax = e, with x, = x, = 1 Clearly, GA is a subgraph of G ( A ) , since GA has an edge for each pair i, j E N such that there exists n o x E {0,1}" satisfying Ax e, with

An equivalent definition of G ( A ) is as follows We shall say that an independent

node set S C N of GA defines a feasible partition of N, if N can be partitioned into

subsets N,, ., N,, such that each N,, i = 1, ., p , induces a complete subgraph on

G, and contains exactly one node of S In these terms, ( i , j ) is an edge of G ( A ) if

x, = x, = 1

28 E Balm

A =

and only if there exists n o independent node set S of GA containing both i and j ,

which defines a feasible partition of N

The inequalities (10) can then be interpreted on the strong intersection graph

Proposition 4.3 (i) T he inequality

is satisfied by all x E P if and only if V is the node set of a complete subgraph G‘ of

(ii) Assume that for each j E V there exists x E P such that x, = 1, and that (12) is satisfied by all x E P Then the inequality (12) is maximal if and only if V is the node set of a clique of G ( A )

G ( A )

Some valid inequalities 29

Fig 1

If A ' = (a',,) is the clique-node incidence matrix of G ( A ) (with a:,= 1 if clique i

contains node j , a:, = 0 otherwise), then the system A 'x e', where e' = (1, ., l),

is satisfied by all x E P Furthermore, each inequality of A'x < e' is maximal If p'

denotes the set packing polytope defined o n A ' , i.e.,

F' = conv {x E R" 1 A'x S e ' , x = 0 or 1, j E N } ,

we have the following obvious consequence of Proposition 4.3:

Corollary 4.3.1

5 Composite inequalities of type 1

In this section we give two composition rules which can be used to combine inequalities in a certain class (which contains as a subclass the elementary inequalities of Sections 2-3) into a new inequality belonging to the same class and stronger than the sum of the inequalities from which it was obtained

The class of inequalities to be considered, which we call composite of type 1, is that of all valid homogeneous inequalities with a single positive coefficient when stated in the form '' c O", and with zero coefficient in all columns j that are not

orthogonal to the column k with the positive coefficient In other words, we are

referring to inequalities of the form

f E S

where aka, = 0, V j E S The subclass of elementary inequalities is distinguished

by the additional property that s C Nk for some i E ak

Th e first composition rule given in the next theorem, generates a new inequality (13) from a pair of inequalities of type (13), such that the positive coefficient of the first inequality corresponds to a zero coefficient of the second one, while the positive coefficient of the second inequality corresponds to a negative coefficient of the first one

30 E Balm

For k E N, we will denote by L ( k ) the index set of those columns orthogonal to

ak, and by L ( k ) its complement; i.e

L ( k ) = { j E N I aiak = 0}, L ( k ) = N\ L ( k )

Proposition 5.1

and the inequalities

For k , h E N, let S, C L ( r ) , r = k , h, be such that h E s k , k E s h ,

are satisfied by all x E P Then all x E P satisfy the inequality

Since xk = 1 implies xi = 0, V j E L ( k ) , S' can be replaced in (16) by S' n L ( k )

Also, since xk s 1, all coefficients 2 in (16) can be replaced by 1 without cutting off any x E P Thus (16) can be replaced by

Example 5.1

N I S , NZ8 and Nz6 respectively, after strengthening via Proposition 3.1, are

Consider again Example 2.1 The inequalities (3) corresponding to

Some v d i d inequalities 31

Using Proposition 5.1 t o combine the first two inequalities, we have k = 5 , h = 8,

Ss = {6,8,14}, Ss = {10,15}, S = ({6,8,14} \ (8)) U (15) [since {lo} @ L ( 5 ) ] , and the resulting composite inequality is

xs - xs- X14- XIS c 0

Since {lo}€ Ss n L(5)# 0, this inequality is stronger than the sum of the inequalities from which it was obtained Combining the new inequality with the last one of the above three inequalities, we have (the new) k = 5, h = 6, Ss = {6,14,15}, Ss= {11,15}, S = {14,15, ll}, and the composite inequality is

xs - XI1 - XI4 - XIS =s 0

Since (15) E Ss n Ss # 0, this inequality is again stronger than the sum of the inequalities from which it was obtained

Next we give a second composition rule, which can be used to obtain all valid inequalities (13) for a certain index k E N from the set of elementary inequalities

(3) corresponding to the same index k

Proposition 5.2

inequalities

Q o = U Q,

1 E .Wk

Then the inequality

Proof (i) Necessity If S G N , { k } does not satisfy (IS), then there exists some

Q C Qo \ S such that X defined by XI = 1, j E { k } U Q, XI = 0 otherwise, belongs to

P But X violates (13)

(ii) Sufficiency We first show that for all x E vert P,