Studies in integer programing

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 (Sonderforschungsbereich21), University of Bonn.
Sponsored by IBM Germany.

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK* OXFORD


ANNALS OF DISCRETE MATHEMATICS

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

I


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1977

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Reprinted from the journal .4nnals of Discrete Mathematics. Volume I

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PREFACE

This volume constitutes the proceedings of the Workshop on Integer Programming 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, to 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.

The Program Committee

Bonn, 1976
P. Schweitzer
IBM Germany

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

V


CONTENTS
Preface
Con tents

V

vi

A . BACHEM,
Reduction and decomposition of integer programs over cones
E. BALAS,Some valid inequalities for t h e set partitioning problem
M. BALLand R.M. V A N SLYKE,
Backtracking algorithms for network reliability analysis
Coloring the edges of a hypergraph and linear
C. BERGEand E.L. JOHNSON,
programming techniques

0. BILDEand J. KRARUP,
Sharp lower bounds and efficient algorithms for the
simple plant location problem
V.J. BOWMAN,
JR. and J.H. STARR,
Partial orderings in implicit enumeration
A subadditive approach to solve linear
C.-A. BURDETand E.L. JOHNSON,
integer programs
V. CHVATALand P,L. HAMMER,Aggregation of inequalities in integer
programming
On the uncapacitated
G . CORNUEJOLS,
M. FISHERand G.L. NEMHAUSER,
location problem
D . DE WERRA,Some coloring techniques
and R. GILES,A min-max relation for submodular functions on
J. EDMONDS
graphs
A.M. GEOFFRION,
How can specialized discrete and convex optimization
methods be married
D. GRANOTand F. GRANOT,On integer and mixed integer fractional
programming problems
M. GROTSCHEL,
Graphs with cycles containing given paths
Algorithms for exploiting the structure of
M. GUIGNARD
and K. SPIELBERG,
the simple plant location problem

M. GUIGNARD
and K. SPIELBERG,
Reduction methods for state enumeration
integer programming
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 minimize total tardiness
J.K. LENSTRA,
A.H.G. RINNOOY
KANand P. BRUCKER,
Complexity of machine
scheduling problems
L. LOVASZ,Certain duality principles in integer programming
R.E. MARSTEN
and T.L. MORIN,Parametric integer programming: the righthand-side case
vi

1
13

49
65
79
99
117
145
163

179
185
205
22 1
233
247
273
287
293
331
343
363
375


Contents

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,
On some nonlinear knapsack problems
J. ORLIN,The 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. SMITHand 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 performance of three subtour elimination algorithms for solving asymmetric
traveling salesman problems
J. TIND,On antiblocking sets and polyhedra
L.E. TROTTER, On the generality of multi-terminal flow theory
L.A. WOLSEY,Valid inequalities, covering problems and discrete dynamic
programs
U. ZIMMERMAN,
Some partial orders related to boolean optimization and the
Greedy-algorithm
S. ZIONTS,Integer linear programming with multiple objectives

uii

391
393
403
415
42 1
435
457
467

479

495
507
517
527
539
55 1


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Annals of Discrete Mathematics 1 (1977) 1-11
@ North-Holland Publishing Company

REDUCTION AND DECOMPOSITION OF INTEGER
PROGRAMS OVER CONES
Achim BACHEM
Institut fur Okonometrie und Operations Research, Universitat Bonn, Nassestrape 2, 0-53 Bonn,
F.R.G.
Received: 1 August 1975
Revised: 1 November 1975
We consider the problem
min c'x

(t)

s.t. Nx + B y

=

b,

X E N , yEZ"

where N is an ( m , r ) , B an ( m , n ) integer matrix, and b E 2". In Section 2 we characterize all
solutions x E 2' of (t) by an explicit formula and give as a corollary a minimal group
representation of equality restricted integer programs, where some of the nonnegativity restrictions are relaxed. In Section 3 we discuss decomposing integer programs over cones in case the
matrix N has special structure.

1. Introduction
We consider the problem
min c'x
s.t. Nx

+ By = b

x E N', y E Z"

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

modB

x E N'

1


A. Bachem

2

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
S(B), H(B) resp. The unimodulaz matrices which transform B into Smith normal
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 ) =
WBUBB KBVB.

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
( m , n ) matrices B with rank (B) E { m , n} the scalar a is replaced in the above
formula and we get the generalized form as
hE(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 x = BA, A E Z " } , and.
X @ Y : = he(X + y ) .

I

Remark (1.4). Obviously

a

=x(

= modB)

-u-x=BA

forsome A E Z "

a - x E kernel(hB) holds
and so problem (1.1) is equivalent to
min c ' x

6 he(N).& = he(b),
n=1

(1.5)

x, E N ,

where N, denotes the ith column of the matrix N and " = " is the group equation in
the group G ( B ) : = h e ( Z " ) .

Proof of Proposition (1.3). Since B has maximal column range, B ' B is regular, and
we have
1 '.-,
.- means that the left side of the equation will be defined.


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
h B ( x )= x - B [ B t x ]= Bb - Bb = 0 which completes the proof.
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-rank(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.
% P > is a representation of d = inv(B) as a product of prime factors
If d =
and p a function from NZ into N defined recursively as

n;=,

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
all ( m , n ) integer matrices ( m , n E N) with maximal row rank and invariant d,
equals the integer number K ( d ) .


A. Bachem

4

The number of nonisomorphic groups G ( B ) ,where A varies over all ( m , n ) integer
matrices ( n E N) with r a n k ( B ) E { m , n } and invariant d, equals L ( d , m ) .
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 the 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 the 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
k

el<,:= max

E~

J=I

leads to

To prove the second part of the proposition we first note that r a n k ( B ) m. Since
two groups G ( A ) and G(B) with matrices having both less than m columns,
cannot be isomorphic, the second statement follows obviously from the first one.

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 respresentation 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
Nx = N b

modB

x integer
has a solution iff S*(A)-' V, U, b is integer. In this case, all solutions are of the form
x=b

modH

x integer

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

S(S(M), uML)= ( I m , O m , n ) .
With diag(tl,. .., t k ) : = S * ( M ) ,tk+,:=O( i = 1 , . . ., m - k ) and D : = UML we get
immediately

(t)

gcd(t,,d,)= 1,

i

=

1 ,..., m ,

where d , : = g c d ( D , , / j = 1,. . ., n ) ( i = 1 , . . ., m ) .
Obviously the system
Nx=O

modB

x integer
is equivalent to the system

y integer,
and using (t) it is also equivalent t o
( S * ( M ) ,O m . s - k ) y= 0 mod WMUML

y integer.


A. Bachem

6

Let y

be a ( k , s - k ) partition of y , then we get

= ( y i , y:)'

S * ( M ) y l= O

mod WMUML.

y l , y z integer

Let K i ( i = 1 , . . ., k ) be unimodular matrices, which transform the ith row of
into ( d , , ~. ., ., 0). Using

d:=W,U,L

Ei:=K , diag(1,. . ., 1, t;', 1,. . ., 1 ) K 1
i = 1,.. ., m we define

n
1

E:=

E,.

i=k

By induction on i one can easily show that
1

diag(1,. . ., t i + l , .. ., t m ) y l = fi

fl Eiz
j=,

n
1

yz,

j=i

E,Z integer

is equivalent (for all i
(*1

=

1 , . . ., m ) t o

S * ( M ) y l= 0 m o d B

y l , Y Z integer
so that
yl

=

DEz

y 2 , Ez

integer

is equivalent t o (*).
Since E-' is an integer matrix and x

= KMy,the

equation

x = (KMVMYlf Ryz)

completes the proof.
Theorem (2.2). With the notations of theorem (2.1) we get
(i) S * ( L )= S(A)-'UA U i ' S * ( B )
(ii) s * ( H ) =
idiag(t,,-,+,, . . ., t , )
where S * ( L ) = :diag(tl,. . ., t m ) .
Proof. Because of

L = s * ( A ) - ' U A ~ ,~' B B ,

(i) follows immediately from the equation
S * ( L )= S * ( L & )

=

S*(LKBVB).


Reduction and decomposition of integer programs

7

Let

where Is-' denotes an ((s - k ) , ( s - k)) identity matrix. Because of H =
K M (W,U,L, P ) , we conclude S * ( H )= S,*(WMUML,
P), that is

where Q denotes the first k rows of U,.
From the proof of theorem (2.1) we know that

S*(L)= S*(H(U,L))
so

= diag(tl,.

. ., t,,,),

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

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) =
m. Then we get
G ( N / B )= G ( S * ( E ) ) ,

that means the group G ( N / B ) is isomorphic to the group G(S*(E)),where E : =
WM UML and L:=S*(N,B)-' U(N,B)N,M : = S * ( N , B ) - 'U(N,B)B.
Corollary (2.4).

0 : = UE S*(M)-'W,U, S*(N,B)-'U(fi,B)
is an isomorphism from G ( N / B ) to G ( S * ( E ) ) .
Corollary (2.5). The order of G ( N / B )equals

inv (B)
det (S*(N,B))
Proof of Theorem (2.3). Let K be a unimodular matrix, so that N K is up to
permutations of rows in Hermite normal form. Let N be the matrix NK without the
zero columns. Obviously we have G ( N / B )= G ( N / B ) .Let

{ N ) : = { x ~ ~ m / x = Nfor
y a

~ E Z ' }

be a subgroup of (Z"', + ). Because h, : {N}+ he ({R})is a homomorphism (Proposition 1.3) G ( N / B )is isomorphic to the factor group

A. Bachem

8

{I?}/ kern el (he)

1

where kernel(hB)= {x E {I?} x = 0 mod B}.
With Theorem (2.1) we conclude

1

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

f:= S*(M)-'W,U,BL-'.
Then

f

:(R}+Zk

is an isomorphism and f (kernel(he)) = { z E Z'

I z = 0 mod WMUML}.Thus we get

{I?}/kernel(h,) = Z k/kernel(&)

and because UE is also an isomorphism we get the isomorphism
G(N/B) = G(S*(E)).
The corollaries follow immediately from Theorem (2.3) in conjunction with
Theorem (2.2).

3. Partitioning of integer programs over cones

The computational effort to solve the problem
min c ' x
s.t. Nx + By = b
x E N', y E Z"
usually grows rapidly according to the determinant of B. It is therefore sometimes
advantageous to decompose the problem into smaller subproblems and to link the
optima of the subproblems to a solution of the masterproblem. We give now two
examples of decomposing problem (3.1) in case the matrix N is of the form

I

=IN1

N =

or

0

. . N,

A l , .. . ......, A,
N,

b=

N=

0

N,

(3.3)


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

1

SG(N, b/B):={x E N ‘ 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 bz(y)e G(N, /B,,),
minc:x,
x E SG(N,,b,(y)/B,) otherwise,
the optimal value of the subproblems.
z(b,(y)): =

[

Proposition (3.4). The programs

min c’x
x E SG(N, b/B),

(3.5)

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

i = 2 , ..., r

min c,x, = c’X,
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
=z
I

I

ciy

+ 2 .z(b,(y))l y E N )
1=2

proves one part of the proposition, however the reverse direction is trivial.
Let again N be an ( m , r ) integer matrix which has form (3.3) and define
zl(xz,.. .,x,):=minc,x,
s.t.

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.


10

A. Bachem

Proposition (3.5). The programs
min c’x
x E S G ( N ,b l B )

and

min c,x,
X,

+ z,-](x,)

E S G (Nr,
br/Br,)

are equivalent.
Proof. If we denote by

c’X:= min c’x
x E SG(N, b l B )

we obviously get
c , f l = min clxl

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. The paper has been revised substantially while
he was a visiting professor at the University of Bonn.

References
[ l ] G.H. Bradley and P.N. Wahi, Integer Programming Test Problems, Report No. 28, Yale University,
New Haven, December 1969.
[2] L. Comtet, Advances Combinatorics (Reidel, Dordrecht, 1974).
[3] R.E. Gomory, On the Relation between Integer and Non Integer Solutions to Linear Programs,
Proc. Nat. Acad. Sci. 53 (1965) 260-265.
[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
Standards-B, Vol. 78B (1974) 3-6.


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Annals of Discrete Mathematics 1 (1977) 13-47
@ North-Holland Publishing Company

SOME VALID INEQUALITIES FOR THE SET
PARTITIONING PROBLEM* .
Egon BALAS
Carnegie-Mellon University

We introduce a family of inequalities derived from the logical implications of set partitioning
constraints and investigate their properties and potential uses. W e start with a class of
homogeneous canonical inequalities that we call elementary, and discuss conditions under which
they are (a) valid, (b) cutting planes, (c) maximal, and (d) facets or improper faces of the set
partitioning polytope. We give two procedures for strengthening nonmaximal valid elementary
inequalities. Next we derive two nonhomogeneous equivalents of the elementary inequalities,
which are of the set packing and set covering types respectively. Using the first of these
equivalents, we introduce a “strong” intersection graph, a supergraph of the (common)
intersection graph, whose facet generating subgraphs (cliques, odd holes, etc.) give rise t o valid
inequalities for the set partitioning problem. These inequalities subsume or dominate the similar
inequalities that one can derive for the associated set packing problem. O n e subclass can be used
to enhance orthogonality tests in implicit enumeration or column generating algorithms. Further,
we introduce two types of composite inequalities, obtainable by combining elementary inequalities 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 importance 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


E. Balm

14

I

min{cx 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

1

P = conv{x E R" Ax

=

e, xi

=0

or 1,;E N }

the setpartitioning polytope, and denote the linear programming relaxation of P by
LP={xER"IAx=e,xsO}.
Clearly, vert P = P n (0, l}",
We will also refer to

p = conv{x E R" I Ax

=se, x, = 0 or l , j E N } ,

the sef packing polytope associated with P.
Whenever P # 0, we have
dim P =G dim LP = n - r(A )
where r(A) is the rank of A.
An inequality
7rx

s

rro

(1)

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

=

570

(1')

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 dimP 2 1, which implies n > r(A). If dim P = dimLP,


Some oalid inequalities

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
x E P such that
T:x*

+ j E N2
-{k)

TjX,

>To.

This notion is the same as that of a minimal inequality (see Gomory and Johnson
[12]; and, more recently Jeroslow [13]), except that here we find it more convenient
to consider inequalities of the form S rather than 3 , in order to have a

nonnegative righthand side.
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 to 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 to 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.


E. Balas

16

2. Elementary inequalities
We shall denote
M
k

={i E

1

M

a,k

=

I},

I

N, = { k E N atk= l},

1

N,k = { j E N, a,ak = o),

G
k

= M\

Mk,

k E N,

Is,= N , N,,i E M,
iE

G k ,

k E N.

N,, is the index set of those columns a, orthogonal to ak and such that a,, = 1.
Since alk= 0 (as a result of i E G k ) , x k = I implies that at least one of the variables
x,, j E N , k , must be one.
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. Since Q C NikC Ni for some i E M, the sum of the variables indexed by Q
cannot exceed 1.
Proposition 2.1.

For every k

EN

and i E

a,, the inequality

is satisfied by all x E P.
Proof. From the definition of Nik,for every x E vert P, XI, = 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

k

I.

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

of these inequalities may be identical.
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.