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Home , Unimodular matrix

Perfect Ideal and Balanced Matrices

Michele Conforti

y

Gerard Cornuejols

Ajai Kap o or

z

and Kristina Vuskovic

December

Abstract

In this pap er we survey results and op en problems on p erfect

ideal and balanced matrices These matrices arise in connection with

the set packing and set covering problems two programming

mo dels with a wide range of applications We concentrate on some of

the b eautiful p olyhedral results that have b een obtained in this area in

the last thirty years This survey rst app eared in Ricerca Operativa

Dipartimento di Matematica Pura ed Applicata Universitadi Padova Via Belzoni

Padova Italy

y

Graduate School of Industrial administration Carnegie Mellon University Schenley

Park Pittsburgh PA USA

z

Department of University of Kentucky Lexington Ky USA

This work was supp orted in part by NSF grant DMI and ONR grant N

Introduction

The mo dels known as set packing and set covering have

a wide range of applications As examples the set packing mo del o ccurs in

the phasing of trac lights Stoer in pattern recognition Lee Shan

and Yang and the set covering mo del in scheduling crews for railways

airlines buses Caprara Fischetti and Toth lo cation theory and vehicle

routing Sometimes due to the sp ecial structure of the constraint

the natural relaxation yields an optimal solution that

is integer thus solving the problem We investigate conditions under which

this integrality prop erty holds

Let A b e a matrix This matrix is perfect if the fractional set packing

p olytop e fx Ax g only has integral extreme p oints It is ideal if

the fractional set covering p olyhedron fx Ax g only has integral

extreme p oints It is balanced if no square submatrix of o dd order contains

exactly two s p er row and p er column The concepts of p erfection and bal

ancedness are due to Berge The concept of idealness was introduced

by Lehman under the name of widthlength prop erty Berge proved

that a matrix is balanced if and only if it and all its submatrices are p erfect

or equivalently if and only if it and all its submatrices are ideal

n

When a p olyhedron Q R only has integral extreme p oints the linear

+

n

program max fcx x Qg has an integral optimal solution x for all c R

for which it has an optimal solution Therefore p erfect ideal and balanced

matrices give rise to integer programs that can b e solved as linear programs

for all ob jective functions

Another interesting situation o ccurs when the dual linear program has

an integral optimal solution for all integral ob jective functions for which it

has an optimal solution let B b e a matrix with integral entries A linear

system B x b x is total ly dual integral TDI if the linear program

max fcx B x b x g has an integral optimal dual solution y for

n

all c Z for which it has an optimal solution Edmonds and Giles

proved that if the linear system B x b x is TDI then the p olyhedron

fx B x b x g only has integral extreme p oints

Perfect Matrices

Theorem Lovasz

For a matrix A the fol lowing statements are equivalent

i the linear system Ax x is TDI

ii the matrix A is perfect

iii max fcx Ax x g has an integral optimal solution x for al l

n

c R

iv max fcx Ax x g has an integral optimal solution x for al l

n

c f g

Clearly i implies ii implies iii implies iv where the rst implication

is the EdmondsGiles prop erty and the other two are immediate What is

surprising is that iv implies i and in fact that ii implies i

Graph theory is very relevant to the study of p erfect matrices In fact his

torically this is where the motivation for the study of p erfection originated

In Berge prop osed a conjecture ab out graphs In the same pap er

Berge also prop osed an elegant weakening of this conjecture to entice re

searchers to investigate the topic The weaker conjecture was proved in

by Lovasz and is known as the p erfect graph theorem This result implies

Theorem and in fact is equivalent to it since the key step of the pro of

amounts to showing that iv implies i ab ove Berges conjecture known

as the strong p erfect graph conjecture has b een thoroughly investigated but

is still unsettled

In a graph a clique is a set of pairwise adjacent no des The chromatic

number is the smallest number of colours needed to colour the no des so that

adjacent no des have distinct colours Since all no des of a clique must have a

distinct colour the chromatic number is always at least as large as the size

of a largest clique A graph G is perfect if for every no de induced subgraph

of G the chromatic number equals the size of a largest clique The p erfect

graph theorem states that a graph G is p erfect if and only if its complement

G is p erfect The complement of graph G is the graph G having same no de

set and complement edge set

The connection b etween p erfect graphs and p erfect matrices was stud

ied by Fulkerson See also Chapter in Schrijver The next theorem

due to Chvatal gives a crisp statement of this connection It is obtained

through the concept of cliqueno de matrix of a graph A cliquenode matrix

of a graph G is a matrix whose columns are indexed by the no des of G

and whose rows are the incidence vectors of the maximal cliques of G

Theorem Chvatal

A matrix is perfect if and only if its nonredundant rows form the

cliquenode matrix of a perfect graph

A ma jor op en question is to characterize the graphs that are not p erfect

but all their prop er no de induced subgraphs are These graphs are called

minimal ly imperfect A hole is a chordless cycle of length greater than three

and it is odd if it contains an o dd number of edges Odd holes are minimally

imp erfect since their chromatic number is three and the size of the largest

clique is two but all prop er induced subgraphs are bipartite and therefore

p erfect If G is minimally imp erfect then so is its complement by Lovaszs

p erfect graph theorem In particular complements of o dd holes are minimally

imp erfect

Conjecture Berge

The odd holes and their complements are the only minimal ly imperfect

graphs

Several classes of graphs that arise in applications such as comparability

graphs and interval graphs are p erfect In these sp ecial cases fast combina

torial algorithms have b een developed for solving the set packing and related

problems A comprehensive collection of pap ers on p erfect graphs can b e

found in the b o ok edited by Berge and Chvatal

Ideal Matrices

In a network with source s and destination t a path from s to t is an st

path and an edge set disconnecting s from t is an stcut It is easy to see

that the pro duct of the minimum number of edges in an stpath by the

minimum number of edges in an stcut is at most equal to the total number

of edges in the network This lengthwidth inequality can b e generalized

to any nonnegative edge lengths and widths w the minimum length of

e e

an stpath times the minimum width of an stcut is at most equal to the

scalar pro duct w This lengthwidth inequality was observed by Mo ore

and Shannon and Dun Construct two matrices A and B as

follows the columns are indexed by the edges of the network the rows of

A are all the incidence vectors of minimal stpaths and the rows of B are

all the incidence vectors of minimal stcuts Mo ore Shannon and Dun

show that the widthlength inequality implies that b oth A and B are ideal

Lehman showed that ideal matrices always come in pairs and that

the widthlength inequality is in fact a characterization of idealness Another

imp ortant result of Lehman ab out ideal matrices is the following

Theorem Lehman

For a matrix A the fol lowing statements are equivalent

i the matrix A is ideal

ii min fcx Ax x g has an integral optimal solution x for al l

n

c R

iii min fcx Ax x g has an integral optimal solution x for al l

n

c f g

Statements i and ii are equivalent by the denition of idealness and

ii implies iii is immediate The dicult part of Lehmans theorem is that

iii implies ii Here contrary to the situation for p erfection idealness of

A do es not imply TDIness of the linear system Ax x This can b e

seen using A Q dened as follows

6

C B

C B

C Q B

6

A

Cho osing c the unique optimal dual solution is y

1 1 1 1

Yet it is easy to check that the p olyhedron fx Ax g

2 2 2 2

only has integral extreme p oints

Therefore we are led to dene another class of matrices a matrix

A has the max ow min cut property MFMC prop erty if the linear system

Ax x is TDI Why is this prop erty called MFMC The reason is

that when A is the of stpaths versus edges min fcx Ax

x g is the classical minimum stcut problem Its dual is the maximum

ow problem The max ow min cut theorem of Ford and Fulkerson

states that for all nonnegative integral vectors c there exists a maximum

ow that is all integral By analogy Seymour coined the term MFMC

prop erty for any matrix A with such a dual integrality prop erty By the

theorem of Edmonds and Giles if a matrix has the MFMC prop erty

then it is ideal

Conjecture Conforti and Cornuejols

For a matrix A the fol lowing statements are equivalent

i the matrix A has the MFMC property

ii min fcx Ax x g has an integral optimal dual solution y for

n

al l c Z

+

iii min fcx Ax x g has an integral optimal dual solution y for

n

al l c f g

Statements i and ii are equivalent by the denition of the MFMC

prop erty and ii implies iii is immediate The dicult part of the conjec

ture is to show that iii implies ii This holds in an imp ortant sp ecial case

namely when A has the binary property This happ ens when any minimal

vector that satises Ax has an o dd intersection with every minimal row

of A Seymour shows that the matrix Q is the unique minimal violator

6

of the MFMC prop erty that has the binary prop erty It follows easily from

this deep theorem that when A has the binary prop erty iii implies ii in

Conjecture

Similarly to get a b etter understanding of idealness one might try to list

or at least describ e constructively the minimal violators of this prop erty

Sp ecically a matrix A is said to b e minimal ly nonideal if the p olyhedron

fx Ax g has a fractional extreme p oint but all p olyhedra obtained

from it by setting a variable x equal to or to only have integral vertices

j

Lehman gives three innite classes of minimally nonideal matrices But

as for minimally imp erfect graphs it is an op en problem to list all the min

imally nonideal matrices In fact the situation app ears more complicated

than for minimally imp erfect graphs since in addition to Lehmans three

innite classes there is a host of small examples that are known Cornuejols

and Novick proved that there are exactly minimally nonideal circulant

matrices with k consecutive s k a matrix is circulant if it is square

and its rows are all the cyclic shifts of the rst row A minimally nonideal

where the s are not consecutive is the Fano matrix

C B

C B

C B

C B

C B

C B

C B F

7

C B

C B

C B

C B

A

This example was already known to Lehman Recently L utolf and

Margot have enumerated all minimally nonideal matrices with up to

columns They also found quite a number of examples with and with

columns However only three matrices with the binary prop erty are

of o dd cycles known to b e minimally nonideal F the incidence matrix O

7 K

5

of versus edges in the complete graph K and the incidence matrix bO

5 K

5

complements of cuts versus edges in K

5

Conjecture Seymour

are the only and bO Up to permutation of rows and columns F O

K 7 K

5 5

minimal ly nonideal matrices with the binary property

An imp ortant sp ecial case of this conjecture was solved recently by Guenin

namely when the matrix A is the incidence matrix of o dd cycles versus

edges of a graph Another recent attempt at Seymours conjecture was made

in

Although a complete list of all minimally nonideal matrices currently

seems out of reach Lehman gives striking prop erties of minimally non

ideal matrices First he proves that their fractional set covering p olyhedron

has a unique fractional extreme p oint He also proves that except for the

matrices in one of his innite classes the unique fractional extreme p oint for

1 1 1

all other minimally nonideal matrices is of the form A similar

k k k

theorem holds for minimally imp erfect matrices see Padberg

Perfect and Ideal 0; 1 Matrices

The concepts of p erfect and of ideal matrices can b e extended to

matrices Given a matrix A denote by nA the column vector whose

th th

i comp onent is the number of s in the i row of matrix A The

matrix A is perfect if its fractional generalized set packing p olytop e fx

Ax nA x g only has integral extreme p oints Similarly

the matrix A is ideal if its fractional generalized set covering p olytop e

fx Ax nA x g only has integral extreme p oints

A matrix is total ly unimodular if every square submatrix has

equal to In particular all entries are A milestone pap er in

the study of integer p olyhedra is that of Homan and Kruskal which

characterizes totally unimo dular matrices see also Chapter in Schrijver

It follows from this characterization that a totally unimo dular matrix

is b oth p erfect and ideal

It is well known that several problems in prop ositional logic such as SAT

MAXSAT and logical inference can b e written as integer programs of the

form

n

minfcx Ax nA x f g g

These problems are NPhard in general but they can b e solved in p olytime

by linear programming when the corresp onding matrix A is ideal In

fact in this case SAT and logical inference can b e solved very fast by

resolution see Section of

Ho oker was the rst to relate idealness of a matrix to that of

a family of matrices A similar result for p erfection was obtained by

Conforti Cornuejolsand de Francesco These results were strengthened

by Guenin for p erfection and by Nobili and Sassano for idealness

We present Guenins theorem The other results have a similar avor



Given a matrix A its completion A is the matrix obtained as

follows if two rows a and a satisfy a a for some j and a a

i k ij k j il k l

for all l j then add row a a to A if it is not already present Rep eat

i k



the pro cess until no more rows can b e added By construction A is a

matrix Now construct

P N



A

I I



where P and N are matrices of the same dimensions as A dened by

 

p if and only if a and n if and only if a and I

ij ij

ij ij

is the Finally we say that A is irreducible if the p olytop e

fx Ax nA x g is not entirely contained in one of the

hyperplanes x or x

j j

Theorem Guenin

Let A be an irreducible matrix Then A is perfect if and only if the



matrix A is perfect

There is a connection b etween p erfect matrices and p erfect bidi

rected graphs a concept introduced by Johnson and Padberg In this

pap er they also prop ose a conjecture relating p erfect bidirected graphs to

p erfect graphs This conjecture has recently b een proved by Sewell

Balanced Matrices

Balanced matrices come up in various ways in the context of facility

lo cation on trees see Tamir

Berges motivation for introducing balancedness was to extend to hyper

graphs the notion of A matrix A can b e viewed as the

nodeedge matrix of a hypergraph H the no des of the hypergraph H corre

sp ond to the rows of A and the edges corresp ond to the columns with edge

j containing no de i if and only if a A hypergraph is balanced if its

ij

no deedge matrix is balanced Therefore a hypergraph is balanced if it has

no o dd cycle in which every edge contains exactly two no des of the cycle

Berge dened a hypergraph to b e bicolorable if its no des can b e partitioned

into two classes say red and blue in such a way that every edge of cardinality

two or greater contains at least one blue and at least one red no de

Theorem Berge

A hypergraph is balanced if and only if al l its node induced subhypergraphs

are bicolorable

When sp ecializing this theorem to graphs we get the well known and easy

to prove result that a graph has no o dd cycle if and only if it is bipartite

Several prop erties of bipartite graphs extend to balanced hypergraphs For

example Berge and Las Vergnas generalize Konigs theorem see for

example Theorem in West A is a set of pairwise

nonintersecting edges and a transversal is a no de set that intersects all the

edges

Theorem Berge and Las Vergnas

In a balanced hypergraph the maximum cardinality of a matching equals

the minimum cardinality of a transversal

Recently Conforti Cornuejols Kap o or and Vuskovic have shown

that the celebrated theorem of Hall Theorem in west ab out the

existence of a p erfect matching in a bipartite graph also extends to balanced

hypergraphs A matching is perfect if every no de b elongs to an edge of the

matching

Theorem Conforti Cornuejols Kapoor and Vuskovic

A balanced hypergraph has no perfect matching if and only if there exist

disjoint node sets R and B with jRj jB j and every edge contains at least

as many nodes in B as in R

Interestingly the notion of balanced matrix motivated as a gener

alization of bipartite graphs can itself b e extended to matrices while

still preserving many imp ortant prop erties

Sp ecically a matrix is balanced if in every square submatrix with

two nonzero entries p er row and p er column the sum of the entries is a

multiple of four The class of balanced matrices also prop erly includes

totally unimo dular matrices

A matrix A is bicolorable if its rows can b e partitioned into blue

rows and red rows in such a way that every column with two or more nonzero

entries either contains two entries of opp osite signs in rows of the same

colour or contains two entries of the same sign in rows of dierent colours

For a matrix this denition coincides with Berges notion of bicolorable

hypergraph

Theorem Conforti and Cornuejols

A matrix is balanced if and only if al l its row submatrices are bicol

orable

Berges theorem stated in the introduction also extends

Theorem Conforti and Cornuejols

Let A be a matrix Then the fol lowing statements are equivalent

i the matrix A is balanced

ii every submatrix of A is perfect

iii every submatrix of A is ideal

Recognition

Given a matrix A are there p olytime algorithms to recognize whether

A is p erfect ideal or balanced No such algorithm is known for p erfection

or idealness However Conforti Cornuejols and Rao give a p olytime

recognition algorithm for balancedness The situation is the same for

matrices no algorithm is known for checking that a matrix is p erfect

or ideal but Conforti CornuejolsKap o or and Vuskovic give a p olytime

algorithm for checking balancedness The algorithm is complicated and its

computational complexity although p olynomial is rather high Nevertheless

the basic idea underlying it is very simple The algorithm is based on a

theorem stating that if a matrix is balanced then either it b elongs to a basic

class or else it can b e decomp osed into two smaller matrices using a well

dened decomp osition op eration Based on this theorem the recognition

algorithm recursively decomp oses the matrix until no further decomp osition

exists Then each of the nal blo cks is checked for balancedness For this

approach to work one must b e able to recognize basic balanced matrices

in p olytime and the decomp osition op eration must have three prop erties

i the fact that a matrix can b e decomp osed must b e detected in p olytime

ii the two blo cks of the decomp osition should b e balanced if and only if the

original matrix is balanced and iii the total number of blo cks generated in

the algorithm must b e p olynomial

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