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Lucia Moura

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Depart ment of Cornputer Science University of Toronto

Copyright @ 1999 by Lucia Moura

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Abstract

Polyhedral Aspects of Combinatorial Designs

Lucia Moura

Doctor of Philosophy

Graduate Department of Cornputer Science

University of Toronto

1999

A t-(v, k, A) design is a collection of k-subsets (cailed blocks) of a v-set such that

every t-subset of the v-set is contained in exactly X blocks; t-(v, k, A) packhg designs

and couering deszgns are defined by replacing the word exaetly by ut most and ut le&,

respectively. The polyhedron associated with a design is the convex hull of the incidence

vectors s E ~ ( 1 ) of al1 designs of that kind. Given s > 2 and v 2 k 2 t 2 1, a family A

of 12-subsets of [l, v ] is said to be s-wise t -intersecting, if any s members Al, . . . , A, of A

are such that (AI n . . . n A,( 2 t; Let us denote by Is(v, k, t ) the set of ali such families.

The thesis is divided into three parts. In the first part, a theoretical investigation of

design polytopes is undertaken. Let TteTr,,l and denote the polyhedra associated

wit h t-(v, k, A) designs and packings, respectively. Maximal clique facets for P : V ~ X , , are

characterized as maximal (A + 1)-wise t-intersecting families of k-sets of a v-set. Sub-

packing inequaiit ies for PtTuYk,X are derived and condit ions under w hich t hese inequalit ies

define facets are given. We provide m-sparse facets for the m-sparse %(v, k, 1) packing

polytope. We also show that if the difference of two designs is a (t, k)-pod, a nul1 design

of minimum support, the designs are adjacent as vertices of the polytope.

In the second part, we investigate the following problern: YGiven s 2 2, v > k > t,

classi@ aJl maximal (with respect to set inclusion) families in IS(v, k, t)". This is a wide

open problem in extrema1 set theory. We solve the classification problem for the kst

nontrivial case, namely s = 2, k = t + 2 and arbitrary v. We dso prove that the

classification problem does not depend on v. in order to derive these results, properties

relating kernels and generating sets of maximal families are proven, and a construction

of maximal families in Is(v + 1, k + 1, t + 1) based on the ones in Is(v, k, t ) is given.

The third part of the thesis focuses on polyhedral algorithms for constructing t - (v , k, 1)

designs and packings, and m-spôrse triple systems with m = 4,5. A branch-and-cut algo-

nthm is proposed and an implementation is described. Separation algorithms for cliques

and m-sparse inequdities are presented. A partial isomorph rejection scheme is employed

to avoid processing isomorphic subproblems in the branch-and-cut tree. The effects of

various parameters on performance are analyzed through experiments. Our method is

competitive witb many other techniques for generating designs including backtracking

and randomized search. Our algorithm produces new maximal cyclic t - (v , k, 1) packings

for t = 2,3,4,5, k = t + 1, t + 2 and small v .

iii

Acknowledgement s

First of all, 1 would like to thank my supervisor Rudi Mathon for his help, support

and enthusiasm. It has been a pleasure to work with hirn.

I would also like to t hank my cornmittee members: Mi ke Carter, Derek Corneil, Eric

Mendelsohn and Mike Molloy, For several suggestions that improved this thesis; thanks

also to Allan Borodin who was part of the final examination cornmittee. Many thanks to

Charlie Colbourn, rny external examiner, for his careful reading, feedback and interesting

suggestions.

Daniel Panario and Brett Stevens proofread drafts of the thesis. Thanks to Ralf

Borndorfer and Terry Griggs who were both very kind pointing out useful references and

witb whom 1 discussed topics related to the thesis.

Friends with whom I shared this combinatorial journey and which are also great

human beings are Luis Dissett, Ali Mahmoodi, and Brett Stevens. Another friend and

source of inspiration was my master's supervisor Carlos Humes who was responsible for

getting me addicted to t his business of doing research.

My experience at the Department of Computer Science has been very nice in many

respects. First, 1 made many good friends (which 1 won't name all) ... Then, 1 met

several people that go beyond their job description, and are able to be incredibly active

and human at the same time; 1 am thinking of Martha Hendriks, Jirn Clarke and Derek

Comeil. Ot ber dear human beings 1 have met are Rudi Mathon and Eric Mendelsohn.

I would also Like to thank my parents Wanda and Alvaro Moura, my brother Pedro

and sisters Marcia and Rejane. It is hard to know who 1 wodd be now without having

had you around, so I won't try to figure out specific things to mention.

FinaIly, 1 would like to acknowledge my ceauthon in this joint enterprise of going

overseas: Natan and Daniel. During the past years 1 had the privilege of having this happy

Little boy bringing magîc and fairy tales to my life. Natan: sometirnes I think you are too

wise for an eight year old ... My other great source of happiness and encouragement was

you, Daniel. It was very lucky of me baving you there for support, advice, enthusiasm,

companionship and even technical discussions. Let us see where life will take us next.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation and thesis overview

. . . . . . . . . . . . . . . . . . . . . 1.2 A fint glance at polyhedral theory

. . . . . . . . . . 1.2.1 Basic polyhedral t heory and lioear programming

. . . . . . . 1.2.2 Combinatorial optimization and polyhedral methods

. . . . . . . . . . . . . . . . . . . . . . 1.3 bdesigns, packings and coverings

1.3.1 Balanced iocomplete block designs and Steiner systerns . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Existence resdts for t-designs

. . . . . . . . . . 1.3.3 Determination of packing and covering numbers

. . . . . . . . . 1.3.4 Steiner triple systems avoiding subconfigurations

. . . . . . . . . . . . . . . 1.3.5 A polyhedral proof for an old theorem

2 Polytopes for Designs

2.1 uiteger programming formulations for design problems and tk-matrices . . . . . . . . . . . . 2.1.1 Designs with prescribed automorphism groups

. . . . . . . . . . . . . . 2.1 -2 O ther integer programming formulations

2.2 General set partitioning. set packing and set covering polytopes . . . . . . . . . . . . . . . . 2.2.1 Independent sets and the set packing polytope

2.2.2 Independence systems and the generalized set packing polytope . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 The set covering polytope

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Design polytopes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Basic properties

. . . . . . . . . . . . . . . . . 2.3.2 Inequalities for the polytope Pt.uvk* A

2.3.3 Inequdities for the polytope of m-sparse triple systems . . . . . . 47

2.3.4 Adjacency in the polytope Ti+. kt and nul1 designs . . . . . . . . . 53

. . . . . . . . . . . . . . . . . . 2.4 Complete descriptions of some polytopes 54

2.4.1 The Fano plane polytope or Ta.ïs. . . . . . . . . . . . . . . . . . 54

. . . . . . . . . . . . . . . . . . . . . 2.4.2 Polyhedfa for mal1 packings 56

3 Clique Facets and Intersecting Set Systems 57

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Definitions and results 58

. . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Kernels and generating sets 61

3.3 Aconstructionoffamiliesin M f ' ( v + l . k + l . t + 1 ) using M P ( v . k . t ) . . 70

3.4 Al1 maximal k-uniform pairwise t-intenecting families for k 5 t + 2 . . . 74

3.4.1 Determination of al1 families in M f ( v . t + 1. t ) . . . . . . . . . . . 75

3.4.2 Determination of al1 families in M f ( v . t + 2. t ) . . . . . . . . . . . 76

4 Polyhedrd Alg~nthms for Packings and Designs 83

4.1 The polyhedrd approach and design problems . . . . . . . . . . . . . . . 84

4.1. I The branch-and-cut metbod . . . . . . . . . . . . . . . . . . . . . 84

4.1.2 Branch-and-cut and cornbinatorial design problems . . . . . . . . 87

4.2 A branch-and-cut implementation for t - (v , k. 1 ) designs and packings . . 88

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Initiahation 88

4.2.2 Separat ion aigorithms . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2.3 Criteria for abandoning the cutting-plane algorithm . . . . . . . . 94

4.2.4 Paztial isomorph rejection . . . . . . . . . . . . . . . . . . . . . . 94

4.2.5 Branch-and-cut tree processing . . . . . . . . . . . . . . . . . . . 96

4.2.6 Other implementation issues . . . . . . . . . . . . . . . . . . . . . 96

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Computational results 97

5 Concluding Remarks and Open Problems 117

Chapter 1

Introduction

1.1 Motivation and t hesis overview

This thesis is devoted to a polyhedral study of combinatorial designs. In particular, we

investigate properties of polytopes associated with design problems, with an aim of using

t his knowledge to design efficient algorithms for constmcting designs.

Combinatorial design theory comprises the study of set systerns with some 'balancen

properties. Early research in this area includes the "36 officers problemn introduced by

Euler in 1782, and the works of Kirkman, Steiner and Cayley on triple systems and

Room squares, in the 19th century. Ln this century, statistical applications led to a

renewed interest in these objects ( s e (981). More recently, applications in many other

areas have been explored (see [24, Part VI). Applications to computer science are found

in such diverse problems as file organizatîon, distributed consensus, sorting in rounds,

interconnection networks and software test ing (231 [24] [Z']. Other important areas of

application are coding theory and cryptography.

Design theonsts are concerned with existence: enumeration and analysis of a wide

M&Y of combinatorial designs. Computational techniques play an important role in

many aspects of combinatorial design theory. Infinite families of designs are usually

produced by using recursive constructions that require the knowledge of a few "starter"

designs. Cornputers have been essentid in the search for these starter designs [102].

Another important role of compntational methods is in the enumeration of designs by

exhaustive techniques, producing a list of ail designs of certain types. This is particularly

important for investigat ing design propert ies and fomulat ing conjectures [71].

Polyhedral theory has been applied to algorithm for solving hard combinatorial o p

timization problerns. The traveling salesman problem is an example of a successful a p

plication of polyhedral algorithms [83] [84]. Little researcb has been done, however, in

the application of these techniques to combinatorial designs. To the best of our knowl-

edge, the first publication in this direction was the master's thesis by Wengrzik [103],

in w hich the problem of constnicting balanced incomplete block designs is approached

wit h polyhedral theory and algorithms. independeut ly, we started an investigation of

sirnilar problems. In (761, we proposed in general terms, a polyhedral approach to design

problems, using a different formulation and dealing with a more general class of designs,

namely t-designs, packing and coverings. In this thesis, we proceed wit h the polyhedral

study of these problems, and develop and implement an algorithm for finding des i .9~

using a branch-and-cut framework. Along the way, other interesting combinatorial prob-

lems emerged. The most evident instance was the classification of s-wise t-intersect ing

families of sets, which arose in the study of a class of facets for the packing design poly-

topes. The study of this problem led to resdts relevant to extrema1 set theory. Some of

these tbeoretical results had algorithmic implications, and in particular, led to an efficient

algorithm for facet separation.

In order to describe our results at a more concrete level, we need some definitions.

First, we defuie the combinatorial design problems we are interested in, then describe a

polyhedral refonndation of t hese problems, and h d l y outline the main contributions of

t his t hesis.

Let us recaii the definitions of t-designs, packings and coverings. A t-(v, k, A) design

is a pair (V ,B) where V is a u-set and B is a collection of k-subsets of V called block

such that every t-subset of V is contained in exactly h blocks of B. A t - (v, k, A) pocking

design and a t-(v, k, A) coaering design are d e h e d by replacing the condit ion "in exact ly

h blocks' in the above definition by Sn at most X blocks" and "in at least X blocks",

respectively. The packing number, denoted by Dx(u, k, t), is the maximum number of

blocks in a t-(v, k,X) packing design. The covering number, denoted by CA(v, k , f ) , is

the minimum number of blocks in a t - (v, k, A) covering design. It is weU known tbat a

t - (v, k, A ) design is both a maximum packing and a minimum covering design.

Central questions in combinatorial design theory are concerned with the existence

of t-designs and the determination of maximum packing designs and minimum covering

designs. The same questions are relevant when additional properties are required of these

designs.

We concentrate on desigos without repeated blocks, known as simple designs. Such a

design can be represented by an incidence vector, that is a 0-1 vector s E di) indexed by

the k-subsets of a v-set and such that xs = 1 if and only if S is a block of the design. The

polyhedron associated with a design is defined as the convex h d l of the incidence vectors

of al1 designs of that kind. Let us denote by and Ctr,k,A the polyhedra

associated with the t - (v , k, A) designs, packing designs and covering designs, respectively.

Let us introduce a new notation for the packing and covering numbers for simple packings

and coverings as &(u, k, t ) and CA(v, k, t ), respectively.

The main questions regarding these designs get translated into the polyhedral world as

follows. A t - (v , k, A) design exists if and only if t he polyhedron is nooempty; the in-

cidence vectors of the t - (v, k, A) designs are precisely the vertices of this polyhedron. The

ing maximum solutions x that are vertices of correspond to the maximum packings.

Analogously, the covecîng number is rewritten as &(v, k, t ) = rnin(lTx : x E Ct, ,c , .~}

and the corresponding minimum solutions x that are vertices of Ct*u,kJ correspond

to the minimum coverings. in addition, the three polyhedra are niceIy related by

Pt.~,k,\ n Ct,v,k,.l = Tt,u,k,.\-

Our study of design polyhedra was motivated by two potential applications. The fint

one was the use of facets of these polyhedra in algonthms t hat construct designs or decide

t hey do not exist . The second one was the possibility of deriving t heoreticd results using

the knowledge of these facets. ln this thesis, we maidy explore the algorithmic use of

lacets. Nevertheless, we illustrate the second motivation in Section 1.3.5, by giving a

polyhedral proof of a known result. Fax from being a simplet proof, it does, however,

point to a direction for m h e r investigation.

We end this section by presenting an ovenriew of the thesis, including a surnrnary of

the main contributions. In the remaining sections of this chapter, we give an introduc-

tion to polyhedral theory, and definitions and known results of the combinatorial design

problems dismssed in the thesis.

In Chapter 2, we investigate polytopes associated wit h design problems. We present

integer programming formulations for designs. These models use tk-matrices, and were

previously employed in other kinds of algorithms for t-designs. We extend these formu-

lations to packings and coverings, and discuss alternative formulations. Then, we survey

known results on related problems, namely set packing, set partitioning and set covering.

After that, we present the main results of t his chapter (Section 2.3). We study the design

polytopes, especially the polytopes for packing designs and m-sparse triple systems. For

general packing desigos, two classes of inequalities are studied, namely the generalized

clique and the subpacking inequalities. We give a characterization of generalized cliques

for packing designs as intersecting families of sets. We then sumrnarize the main conse-

quences of our results on these hmilies for the design problems. The proofs are invoived

and we give them in Chapter 3. One of the consequences of these resdts is an efficient

separation algorithm for designs with k = t + 1. We derive a new class of inequalities,

which we cal1 subpacking inequalities. We give some conditions under which these in-

equalities do not define facets, and other conditions regarding the extension of subpacking

facets of Pt,u,k,41 to facets of for v' 2 v. In particular, for k = t + 1, subpacking

facets are always extendible. Moreover, we show some specific instances of facet defining

subpacking inequalities for k = 3 and t = 2, which define facets of f2,u,3,1 for arbitrary v.

We also investigate t-designs and packings avoiding subconfigurations. More precisely, we

study the polytope of the rn-sparse triple systems. Avoidance properties can be oaturdy

formdated as addit ional inequalities. We show t hat t hese inequalit ies induce facets for

the polytope of the m-sparse triple systems. Finally, we relate null designs to adjacency

in the Tt,,,+i polytope. We show that if the ciifference of two designs is a (t, k)-pod, a n d

design of minimum support, the designs are adjacent as vertices of the Tt,u,r,i polytope.

We conclude Chapter 2 by showing some design polytopes for m a i l parameters. The

facet inducing inequalities of these polytopes were found using PORTA software, a public

domain software package for polyhedral manipulation [20].

Chapter 3 is devoted to the study of maximal s-wise t-intersecting families of sets.

These set systems correspond to generalized clique facets for the packing design polytope,

as previously mentioned. In addition, they are interesting objects in their own right, and

have been given a lot of attention in extremd set theory. A family A of k-subsets of a

v-set is said to be s-wise t-intersecting, if the intersection among any s of the subsets

in the family bas cardinality at least t ; such a farnily is said to be maximal if it is not

properly contained in any ot her family wit h the same parameters (s, t , k, v) . The famous

Erdos-KeRado theorem [33] is concerned with the maximum size of such a family for

s = 2. Several ot her ext remal propert ies of t hese families have been studied (see [42]).

The extrema1 problem we are concerned with is the classification of maximal (with respect

to set inclusion) s-wise t-intenecting families of k-subsets of a u-set. This turns out to

b e a wide open problem in extremal set theory. Our main results in this chapter are

as follows: (1) we solve the classification problem for the first nontrivial case, namely

s = 2, k = t + 2 and axbitrary v ; (2) we prove that the classification problem does

not depend on v . We study properties relating generating sets, kernels and essential

elements of maximal set systems. We use t hese properties to show that given ( ~ 7 t, k)7

the classification problem is independent on v . Then, we give a constniction for maximal

families with parameten ( s , t + 1, k + 1,v + 1) using the ones with ( s , t , k , v ) . We show

that any family with s = 2, k = t + 2, t 2 3 can be generated by this construction from

a smaller family. Combining al1 previous results we reduce the classification of maximal

families for (2, t, t + 2, v ) to the cases (2,1,3,7) and (2,2,4,8), which we enurnerate by

computer. The resitlts in the chapter appear in article [77. in Chapter 4, we propose a polyhedral algorithm for constructing t-designs and pack-

ings. The algorit hm uses the well-known branch-and-cut approach [16]. We invest igate

the application of this method to design problems and describe our implementation.

Separation algorithms are developed for clique inequalities and rn-sparse inequalities for

rn = 4,5. The diffidty of having several nodes in the branch-and-cut tree corresponding

to isomorphîc problems is addressed by a partial isomorph rejection algorithm. Corn-

putational experiments showing the interplay of various pammeters are andyzed. Our

experiments address issues such as a cornparison between branch-and-cut and branch-

and-bound techniques, the effectiveness of both the specialized clique separation and the

partial isomorph rejection, as well as the effect of other parameters on the algotithm's

performance. We compare our results to the ones obtained by a general purpose integer

programrning package, as a reference point. The experiments shed some light on the

potential and difficulties of applying these methods to design problems, and provide a

starting point for further research in this area. Besides reproducing known results, our

algorithm constructs some m h a cyclic packings, which to the best of our knowledge

have not been previously investigated. Finally, in Chapter 5, we discuss open problems

and indicate directions for further research.

1.2 A first glance at polyhedral theory

In polyhedral combinatorics, polyhedra are usually given implicitly, and our objective is

to find their explicit representaticns. In the situation we are interested in, a polyhedron

is implicitly defined by having designs as its vertices. The goal is to determine the linear

inequalities that describe this polyhedron, so that we can compute its vertices, i.e., find

the corresponding designs.

in this section, we list some classical results in polyhedral theory and outline their

relation to linear prograrnming and combinatorial opt irnizat ion. Complete treatments of

polyhedral theory can be found in the survey papers by Bachem and Grotschel [6] and

Pulleyblank [86], and in the book by Nemhauser and Wolsey [80].

1.2.1 Basic polyhedral theory and linear programming

The proofs for the results contained in this subsection can be found in [80]. Throughout

this subsection, let n and m b e positive integers. A polyhedron P E Rn is the set of points

sati+ing a finite set of Linear inequaiities; that is, for any matrix A E RmXn and vector

6 E Rn, P = {z E Rn : A z 5 b ) is a polyhedron. A polytope is a bounded polyhedron,

i-e., a poiyhedron P Rn is a polytope if and only if there exist vectors 1, u E Rn such

that 15 t % u for ail x E P.

in a linear progmmrning problem, we are interested in maximizing (rninimizing) a

linear function over a polyhedron. Let c E IRn be the profit (cost) vector and P =

( x E Rn : Ax 5 6 ) be the associated polyhedron. We can write the linear programming

problem as

Minimizatioo problems, as well as problems with different inequality signs (=,z), can be

easily brought to the form given above by simple algebraic manipulations. The study of

polyhedra is fundamental for linear programming.

First, we review some basic definitions from linear algebra; thea, we define the con-

cepts of dimension, faces, and facets of a polyhedron; finally, we state classical results on

minimal representations of polyhedra.

A point z is called a linear combination of z', .. . ,xk E Rn if there exist a = k (al,. . . ,ak) E Rk ssuch that z = Ci=, CQX'. If a 2 O then x is a conic combination.

If ~ f = , ai = O then x is an a f i e combination. If a 2 O and xik=, ai = 1 then x is a

convex combination.

Given a nonempty subset S Rn, the set of all linear (conic, affine, convex) combi-

nations of points i n 3 is called the linear (conic, afine, convez) hulf of elements in S and

is denoted by lin(S) (cone(S), aff(S), conv(S)). We say that S is a cone if S = cone(S)

and that it is a conuer sel if S = cmv(S) . It is easy to prove that any polyhedron is a

convex set, and that a polyhedron of the form { x E Rn : Ax 5 O} is a cone.

A set of points xl,. . . , zk E Rn is lineurly independent if the unique solution a E Rk

to z.., aizi = O i s a i = O for i = 1,. . . , k. Analogously, a set of points zl,. . . ,zk E Rn k is a f i e l y independent if the unique solution u E I P ~ to CL, crir i = 0, xi=, ai = O is

cq = O for i = 1,. . . , k. The following proposition relates both concepts.

Proposition 1.2.1 The folloioing statements a n equàvalent:

i. The set of points zi,.. . , zk E ILn is afinely independent.

ii. The set of points x2 - xl,. . . , zk - z1 E Rn is lineurly independent.

iii. The set of points (xl, - l ), . . . , (zk, -1) E Rn+' is linearly independent.

Since affine independence of points in a polyhedron is invariant under translation, afnne

independence is more suitable than 1inea.r independence for deding with polyhedra.

Let A E I R m X n and b E Bm. It is convenient to make some distinction among inequal-

ities that define a polyhedron P = { x E Rn : Ax 6). Let I = (1,. . . , m), I= = {i E

I : aix = b; for al1 x E P) and let I' = { i E I : a'x < bi for some z E P) = I \ I = .

We indicate by (A', 6') and (As, bs) the rows of (A , b) corresponding to I= and 15,

respectively, the equality and inequality representations of P with respect to (A, b).

A polyhedron P C Rn is of dimension k , denoted by dimP = k, i f the maximum

number of affinely independent points in P is k + 1. We Say that P is full dimensional

if dimP = n. The maximum number of linear independent rows of a matrix A E IRmXn

is the rank of A, denoted by rank(A).

Proposition 1.2.2 If P C Rn then dim P + rank(A=, 6 7 = n.

Let d E Rn and do E R. An inequality Px 5 do is said to be valid for P if it is satisfied

by al1 points of P. .4 subset F E P is called a face of P if there exists a valid inequality

drx < do such that F = P n {z E Rn : drz = do) . The inequality is said to represent or

to induce the face F. It is easy to see that P and 0 are faces of P, and that a face of a

polyhedron is itself a polyhedron. A face F of P is called proper if F # P and nontrioid

if F # 0. A maximal nontrivial proper face F of P is called a facet of P. A face of

dimension zero is called a uertez and a face of dimension one is callecl an edge of P. A

poiyhedron that has at least one vertex is said t o be pointed. Let Po = {x E Rn : Az 5 0 )

be the polyhedral cone associated with P = { x E Rn : Ax 5 6) . Any point r E Po is

called a ray of P and if r belongs to an edge of Po it is c d e d an eztreme ray of P.

The study of faces is important for linear programming. If a linear programming

problem, maxcTx subject to x E P, has an optimal solution, it is easy to see that

the set of ail optimal solutions is a face of the polyhedron P. Just note that if Q =

max(cTx : x E P), then cTx 5 ~g is a valid inequality for P that induces the face

F = { X E P : cTz = a), which is the set of optimal solutions to the problem,

The main result concerning minimal inequality representation is summarized below.

Theorem 1.2.3 Let P be a polyhedron.

i. If P is full dimensional then it has a unique (up to scalar multiplication) minimal

representation by a Jinite set of linear inequalities, each one representing each of

the facets of P.

ii. i f dimP = n - k m'th k > O then a minimal representation of P is giuen by

{x E Rn : a'x = bi, i = 1,. . . , k, ais 5 bi, i = k+l,. . . k + t ) , when { (ai , bi) } ;= , ,,..,

is a rnazimal set of linearly independent rows of (A', 6') ) and for i = k+ 1, . . . k +t , a'x 5 bi are inequalities repnsenting each one of the facets f i of P.

We now give two characterizations of facets that axe very useful in deciding whether or

not an inequality is facet inducing.

Theorem 1.2.4 Let F be o nonempty proper face of P = {x E Rn : Ax 5 b ) and

(A', 6') be the equality representation of P. Then, the following statements are equiua-

lent:

ii. dimF = dimP - 1.

iii. For any pair of inepvalities drz 5 do and dTz 5 do that are ualid for P and

represent F , there e x k t cr E Bm and y > O E W such that

By now we have seen one of the representations of a polyhedron, namely the one by a

minimum number of inequalities. Another way to represent a polyheckon is by means of

its vertices and extreme rays, as stated in the foilowing theorem.

Theorem 1.2.5 (Minkowski's theorem) Let P be a pointed polyhedron, and Po be its

associated polyhedral cone. Then

i. P has afinitc set of uertices V ,

ii. there k a finite set of eztreme rays R of P svch that cme(R) = Po, and

iii. P= conv(V) fcone (R) .

Remark: Most polyhedra we wil1 deal with are polytopes, i.e. bounded polyhedra.

If P is a polytope then P = conv(V), since for polytopes cone(R) = {O).

The previous theorern is the basis for the simplex algorithm for linear programming.

The simplex method also uses the fact that if there is an optimal solution for a linear

programming problem over a pointed polyhedron P, then there is an optimal solution

t hat is a vertex, in which case the algorithm returns an optimal vertex.

1.2.2 Combinat orial opthization and polyhedral met hods

We describe a general cornbinatorial optimization problem. Let E = {el , . . . , en) be a

ground set, ci be a weight associated wit h ei E E for i = 1, . . . , n, and 3 be a famity

of feasible subsets. We want to find S E 7 such that c ( S ) = xi:c,ES is maxirnized. A

natural representation of an element S E 3 is by a 0-1 incidence vector, that is a vector

xS E Rn such that xS = 1 if and only if ei E S. This gives a polytope representation

of the problem, namely the polytope C = conv{xS : S E T}. This polytope has the

nice property of its vertices being members of 3, as we have seen for the particular

case of designs. Moreover, solving the combinatorial problern is equivalent to finding an

optimal vertex of this polytope, which can be done via Linear programming methods. The

difficdty lies in the fact that we need a description of this polytope in terms of linear

inequalit ies.

Sometimes a combinatorial optimization problem is given in the form of an integer

prograrnmjng problern. An integer programming problem is one OF the form

max cTx

I I P [ A2 5 b

x integer.

Let P = ( x E Rn : Ax < b) . The polyhedral representation of this problem is given by

Pr = ccmv{x E P : z is integer). We would like to obtain a complete representation of Pt

by means of facet inducing inequalities. A usefd technique for hding such inequalities

is given by the ChvdtaCGomory ni t generation. Let a 2 O E Rn. Then ( a T ~ ) x 5 aTb is

a valid inequality for Pr implied by Ax 5 6. If a T A is integral then any integer solution

x to the system wiLl make ( a T A ) x integral, su the inequality (ctTA)x 5 LcrTbj is valid for

PI. Such an inequality is called a ChuPol-Gomory cut derived from As 5 b.

The following theorem was proven by Chvatal [21] for polytopes, and generalized by

Schrijver [9 11 for polyhedra.

Theorem 1.2.6 Let P and Pi be the polyhedm given by an integer programming problem

as defined preuiously. Let Po 2 > Pi - 3 Pif - 3 - 3 PI be o sequence of polyhedra

ncursiuely defined by: Po = P , Pt+' is the polyhedron obtoined by the inequalities defining

pi plus ail the Chvtital-Gomory cuts derived /rom these inequalities, for i > O . Then,

there is an integer t such that Pt = PI, Le. , the sequence às finite.

The main consequence of this theorem is that any facet inducing inequality can be ob-

tained by a finite number of applications of the Chvital-Gomory procedure. Chvatal-

Gomory cuts have been used to derive and prove the validity of inequalities for polyhedra

of structured problems. They can also be seen as a technique for generating valid and

facet inducing inequalities.

1.3 t-designs, packings and coverings

in this section, we s w e y general resuits on t-(u, k, A) designs, t - (u , k, A) packings and

t-(v, k, A) coverings. First, we defuie related designs, then we review results on the

existence of t-designs and on the determination of packing and covering numben, and

finally we describe some problems on Steiner tnpk systems avoiding subconfigurations.

1.3.1 Balanced incomplete block designs and Steiner systems

Several well-studied designs are special cases of t-designs, or to be historically a c m t e ,

t-designs are generalizations of pairwise balanced designs, first introduced in statisticd

applications.

A balanced incomplete block design BIBD(v, 6, r, k, A) is a %(v, k, A) design; the

parameters b and r are determined by the other three and stand for, respectively, the

Table 1.1: Examples of t-designs.

number of blocks and the replication of points (i.e. the number of blocks that contain

any particular point). These designs are also referred to as (v, k, A ) BIBDs.

A Steiner system S(t, k, v ) is a t - (v , k, 1 ) design. A Steiner triple system STS(v)

is a 2-(v, 3 , l ) design, and a Steiner quachpie system SQS(v) is a 3-(u, 4,1) design.

Sometimes t-(v, k, A) designs are referred to as SA(t, k, v).

1.3.2 Existence results for t-designs

It can be shown by a counting argument that For any O 5 s 5 t , a t - (v , k, A ) design is an

s-(v, k, A,) design with

In particular, A, must be an integer, for all O 5 s 5 t . Let b be t h e number of blocks and

r be the replication number in a t-(v, k, A) design. Then, clearly 6 = A. and r = Xi.

Equation (1.1) gives a necessary condition for the existence of a t-(v, k, A) design

made explicit as foilows. For any t-(v, k, A) design we must have

k - 8 . \ ("- ' ) -O t - s (mod( t - s 1).

for all O 5 s 5 t. We Say that a t-(v7 k, A) design is admissible if its parameters satisfy

the previous conditions.

Another necessary condition is given by the following theorem, which is a generaliza-

tion of Fisher's inequality.

Theorem 1.3.1 (Fisher's inequality for general t - Ray-Chaudhuri & Wilson [87])

Let t < k 5 v - t . I f (V, B ) is a simple t - (v , k, A) design, then Il31 2 (&,).

Wilson showed that the necessary conditions given in (1.2) are sufficient for X large

enough.

Theorem 1.3.2 (Wilson [lO4])

Given O < t 5 k 5 v , fhen there ezist an integer Ao, such that any admissible t - (v , k ? X )

design (possibly with repeated blocks) ezists for any A 2 ho.

The following tbeorem parantees the existence of simple (i.e. with no repeated

blocks) t-designs for arbitrarily large 1. However, the values of A are extremely large.

Theorem 1.3.3 (Teirlinck [99])

Given v > t 2 O with v t (mod (t + 1)!2tC'), then a simple t - (v , t + 1, ( t + 1)!*'+')

ezists.

The two previous theorems give no light on the existence of simple t-designs with

srnall A. We summarize existence results for such cases.

Let us start with Steiner systems or t-designs with A = 1. .4 STS(v) exists if and

ody if v = L, 3 (mod 6). A SQS(v) exists if and only if u 2: 4 (mod 6 ) . Only a finite

nurnber of S(t , k , v ) with t = 4,5 are known and none are known for f 2 6.

Let us discuss the Zdesigns or BIBDs (of course, some of these results include

S(2 , k, v ) designs). For k = 3,4,5 all admissible 2-(v, k,X) designs exist with the ex-

ception of a 2-(15,5,2) design. The existence of 2-(v, 6, A) is still unsettled, with the

smallest unsolved case being 2-(46,6,1).

For t-designs with t 2 3 and small X the situation is Far from set tled. There are many

examples and some infinite families of simple t-designs with t = 4,5,6, but these infinite

families bave growing A. Recently, many 7-designs (with X = 4 and larger) aod a few

Sdesigns have been found [12].

For a more precise account of the existence situation in general and for specific param-

etea, we refer the reader to the comprehensive collection of results, pointers to results

and tables, given in the handbook edited by Colboum and Dinitz [24], in partidar

chapters [1] [25] (571 [72].

1.3.3 Determination of packing and covering numbers

In this section, we outline some general results on the determination of the size of maxi-

mum packings and minimum coverings. For a comprehensive account of results the reader

is referred to the survey papers of Mills and Mullin [75] and Stinson [95] [96].

The foilowing theorems provide upper bounds for the maximum size of a packing

DA(v, k, t ) and lower bounds for the minimum size of a covering CA ( v , k, t ) . First, we

comment on the relation of these quantities and sirnilm ones for simple designs, which

aie denoted by D A ( v , k , t ) and cA(v, k, t). Note th& that the upper bounds on DA(v , k , t )

and lower bounds on CA(v, k , t ) are also bounds for D,\(v, k, t ) and r A ( v , Ic, t ) , since

Moreover, both packings and minimum coverings for X = 1 cannot have repeated blocks,

thus D , ( v , k , t ) = D I ( v , k , t ) and C t ( u , k , t ) = G ( v , k , t ) .

Theorem 1.3.4 (Schonheim bounds)

The follouing bound holds D.!(v, k , t ) 5 If DA(v - 1, k - 1 , t - 1 )] . Iterating this bound

giues

Moreover, ifA(v - 1 ) r O (mod ( k - 1)) and Av(v - 1) s - 1 (mod k ) , then

Analogowly, zue have C,,(v, k, t ) 5 [% Cx(v - 1 , k - 1 , t - I ) ] . Iterating this bound giues

C,(v, k, t ) 2 Lx(v, k, t ) :=

Moreover, i/A(v - 1 ) O (mod (k - 1)) and Xv(v - I ) 1 (mod k ) , then

Observe that, if a t-(v, k, A) design exists it is a t-(v, k, A) packing and a t - (v , k, A)

covering. In addition, letting 6 be the number of blocks in the t-(v,k, A) design, the

divisibility conditions given in (1.2) irnply that UA(v , k , t ) = LA(u, k, t ) = b = 8, which t

implies that a t-design is both a maximum packing and a minimum covering.

Table 1.2: Example of a packing and a covering meeting the Schonheim bounds.

then Theorem 1.3.5 (Johnson bound for packings) If v < m,

We now mention some results on the determination of packing and covering numben

(see [75] [95] [96]). These numbers are obtained t hrough constructions of packings and

coverings attaining known bounds. The packing numbers DA(v, 3,2) and &(v, 4,2) have

been completely determined. The packing numben Dl ( v , 4,3) for v f 5 (mod 6) have

also been obtained. Partial results are known for &(v , 5,2). The covering numbers

CA(v, 3,2) and Ci(v, 4,2) have been completely determined. In addition, the covering

numbers Ci(v, 4,3) for v f 7 (rnod 12) have been obtained. Partial results are known

for CA(v, 5, 2).

1.3.4 Steiner triple systems avoiding subcodigurations

As we have seen in the previous two sections, the basic questions regarding triple systems

(i.e., 2-(v, 3, A) designs, packings and coverings) have been settled. However, we find

many open problems for triple systems satisSing sume additional properties. In this

section, we concentrate on Steiner triple systems (i.e. 2-(u, 3 , l ) designs) avoiding some

forbidden configurations. The same coastraints can be imposed on 2-(v,3,1) packings

and coverings, leading to relevant problems (for example, see [18] for an application of

anti-Pasch 2-(v,3, 1) packings).

The reader is directed to the s w e y by Granneil and Griggs [45] for a detailed treat-

ment of avoidance problems.

Table 1.3: Examples of STSs with and without Pasch configurations

A (p,f)-configuration in a Steiner triple system is a set of 1 blocks (of the Steiner triple

system) spanning p elements. The Pwch or puadrikateral configuration is the unique

(6,4)-configuration, which is of the form ( { a , 6, c ) , {a , d , e), { f, b, d ) , { f , c, e)). A mitre

is a (7,s)-configuration on 7 elements, Say, a, b, c, d, el f, g of the form { {a , b, e), {a, c, f ),

{a, d, g ) , (6 , c, d ) , (e, j, g ) } . A STS is anti-Pasch (anti-mitre) if it does not contain a

Pasch (mitre) as a subconfiguration.

Let rn 1 4. An STS(v) is said to be rn-sparse if it avoids every (1 + 2,l)-configuration

for 4 5 1 5 m. ErdGs (see [63]) conjectured that for al1 rn 2 4 there exists an integer v,

such that for every admissible v > v, there exist an rn-sparse S T S ( v ) .

The 4-spane STSs are the same as anti-Pasch ones, since Pasches are the only (6,4)-

configuration. A 4sparse (or anti-Pasch) STS(v) is known to exist for al1 v = 3 (mod 6)

[14]. For the remaining case, i.e. the case v = 1 (mod 6), there are constmctions and

part i d results which leads to 314 of the values being settled [14] [47] [48] [67] [88] [97].

Anti-mitre Steiner triple systems were first studied in [26]. The 5-sparse Steiner triple

systems are the systerns that are both anti-Pasch and anti-mitre. Although there are

some results on 5-sparse STSs [26] [66], the problem is far fiom settled. As for rn 2 6,

no m-spane Steiner triple system is known.

1.3.5 A polyhedral proof for an old theorem

We illustrate how valid inequalities for the design potytope can be used for proving design

properties, by giving a polyhedral proof of Theorem 1.3.4.

Theorem 1.3.6 (Schônheim bounds)

The packing number DA(u, k, t ) satisfies

Similcrly, for the covering number CA@, k, t ) we have

ProoJ We show the inequality for the packing numbers; an analogous argument can be

applied for the c m h g uumbers.

Let PL,u,k,A be the polytope for the t - ( v , k, A) packing design. The inequalities

are valid for Pt,,,,k,A. For simpkity, we will not repeat K E ([y1) in the next sumrnations.

We shall derive the upper bound on DA(v, k, t ) by taking linear combinations of

inequalities in ( 1.3). Let S( t ) be the foUowing statement: *For any L C [l, v ] , ) L) = t - t, the inequality

is valid for Pt,v,k,A.n We prove that S( I ) holds for 1 5 t 5 t by induction on 1.

Consider the case l = 1. Let L C [L, v ] with 1 LI = t - 1. For each t-subset T > L, we

add inequalities in ( 1.3), obtaining

Reorganizing the lefi- hand side and obsenring that v - ( t - 1) inequalities were added,

we eet

This implies

and rearranging sums, we obtain

The Chvatal-Gomory cut corresponding to this inequality is

which implies S(1).

Assume now that S( t ) is tme; we shall prove S(t + 1 ) . Let L C [1 , v ] wit h ( LI =

t - ( t + 1). Then, there are v - ( t - (I + 1 ) ) sets M of cardinality t - Q containing L. By

induction, inequality (1.4) is valid for any such M, and adding these inequalities gives

Using an algebraic manipulation similar to the base case, we get

Again, taking the Chvatal-Gomory cut corresponding to the previous inequality corn-

pletes the proof for.S(P + 1).

Finally, since S(t + 1) is valid for it is satisfied in particular by the incidence

vector of a maximum packing, which completes the proof. O

In the previous proof we have used valid inequalities of the form (1.3). The method

is quite general and other inequalities could be used in place of these o n e .

Chapter 2

Polytopes for Designs

Viewing designs as integer solutions to systems of equations has contributed to the study

of t-designs in several ways. I t involves tk-matrices and their associated systerns of

equations (DP) that we define in Section 2.1.

System (DP) was used by Wilson [104] to show that the necessary conditions for the

existence of t-(v, k, A) designs are sufficient for A large enough (see Theorem 1.3.2). This

was done by showing that (DP) admits nonnegative integer solutions for large A. Note

that he required the xi's to be nonnegative integers, rather than in (O, 1). We direct the

reader to the notes "Linear algebra and designs" by Godsil [44] for a detailed study of

tk-matrices, their ptoperties and use in proving results on t-designs.

Another remarkable use of system (DP) led to the discovery of several previously

unknown simple designs for t >_ 4 and md A. Kramer and Mesner [56] first observed

how system (DP) cm be transfonned into system (DPA) (see Section 2.1.1) in order to

deal with designs with presaibed automorphism groups. The transformation often leads

to a significant redudion on the problem sizes. Solving systems of the f o m (DPA) yielded

several new results, including the fiat three examples of 5-designs with an odd number of

points by Kramer [54] and Magliveras and Leavitt [69], and several other t-designs for t =

4,5,6 by Kreher and Radziszowski [58] [59] [60], including the existence of a ô-(14,7,4)

design, the srndest nontrivial admissible &design. The latter authors applied the lattice

basis redudion algorithm proposed in [64] to solve (DPA). Schmalz [90] proposed an

algorithm to compute the isomorphism types or even the designs themselves, finding

CHAPTER 2 POLYTOPES FOR DES~GNS 20

new Gdesigns, using the same system of equations. Betten, Laue and Wassermann [L2]

employed the same equations to find several 7 and gdesigns.

In this thesis we propose the study of the polytopes associated with solutions of

the systems (DP) and (DP*), and of their extension to systems for packing and cov-

ering designs (see Section 2.1). These polytopes produce "tighter", more restrictive,

systems of inequalities. The success obtained by appiying polyhedral algorithms to other

combinatorial optimization problems [84] (also see [16]) was an incentive to pursue this

investigation, which led to the algorithms in Chapter 4. Moreover, we suggest a still

unexplored potential use of design polytopes in the theory of combinatorial designs. It

is Our hope that some of the new inequalities and facets discussed in Section 2.2 and

Section 2.3 can be used to derive new results on the existence of t-designs andior new

bounds for packings and coverings. Theorem 1.3.6 illustrates how the inequalities in

problem (PDP) can be used to derive the Schiinheim bounds for packings.

The rest of t his chapter is organized as follows. In Section 2.1, design problems are

formulated as integer programming problems. This section includes the formulation ( D P )

mentioned earlier as well as its extension for dealing with packings and coverings. These

formulations are special cases of well-known combinatorial optimization problems, narnely

set packing, set partitioning, and set covering. In Section 2.2, polyhedral studies of these

generd problems are revisited, leading as corollaries to results for design polytopes. The

main contribution of this chapter is contained in Section 2.3. Ln this section, we study

the polytopes for designs. After some basic results for the three design polytopes, two

classes of inequalities for the packing design polytope are studied, other facets for m-

sparse triple systems are developed, separat ion results for the ment ioned inequaiit ies are

shown and adjacency results for the t-design polytope are included. In the Iast section of

the chapter, Section 2.4, the reader can find some samples of mail polytopes for which

the complete description by Lineu inequalities is available.

Parts of several'sections of this chapter appeared in [76]; however, most of the main

results of the chapter (Section 2.3) are Çst shown here.

In this chapter, we state when some facets axe polynomial-time separable. The size of

a design problem is measured by the number of bits needed for its integer programming

CHAPTER 2 POLYTOPES FOR DESIGNS 21

formulation. For t-(v, k, A) designs, the problem size is exactly (:) x (:) + (:) log A. Our

measure of complexity is the number of basic operations such as arithmetic operations

and comparisons.

2.1 Integer programming formulations for design prob-

lems and t k-matrices

In this section, we present integer programming formulations for t-designs and their ex-

tensions to packings and coverings. Similar formulations for designs with prescribed

automorphism groups are shown in Section 2.1.1. Ot ber integer programming formula-

tions are shown in Section 2.1.2. Although they are not used in this thesis, they offer

alternative formulations to some of the problems discussed here.

First, we define a tk-rnatrix, which encodes the incidence between t-subsets and k-

subsets of a v-set. More precisely, let W:k be the ( y ) x (:) rnatrix with rows indexed by

the t-subsets and columns by k-subsets of a v-set

1 if T c K,

O othenvise.

For a detailed study of these matrices and their role in design theory see [44].

It is easy to see that t-(u, k, A) designs correspond to the solutions s E IR(:) of

,DP, { w;k x = A l ,

z € {O, l}(X

The maximum packings correspond to solutions r E di) of

Andogously, the minimum coverings correspond to solutions I E IR(:) of


For designs with possibly repeated blocks, the condition "3 E {O, 1)(;)" sbould be re-

placed by 'x nonnegative integer", or alternatively, the matrix WCk should be replaced

by the same matrix with each column repeated A times and the points s considered to

be in {O, 1 }A(;).

Based on these integer programming formulations, we rewrite the design polytopes

defined in Chapter 1 as

For A = 1 these are special cases of set partitioning, set packing and set covering

polytopes, respectively. For A > 1, the polytopes Pt,vli,A and Ci,,,r~r can be brought to

the iorm of the polytope for independent sets of an independence system (see Section 2.2

and Section 2.3).

The following proposition relates the problems just described. It can be proven using

the Schiinheim bounds (see Section 1.34, but we give an alternative proof.

Proposition 2.1.1 Assume that a t-(v, k,A) design ezists. Let 2' E IR(:). Then the

following staternents are equivalent:

i. z' is a solution to (DP).

ii. x' is an optimal solution to (PDP).

iii. x' is an optimal solution to (CDP).

ProoJ Let 6 = A(';) / (t), the number of blocks in a t-(v, k, A) design. Let z be any feasible

solution to (PDP). Adding the inequalities of WfL z 5 X I , we get that (:) 5 ( y ) A,

that is, lTz 5 ( y ) A/ (:) = 6. Analogously, for any feasible solution y of (CDP), we see

that lTy 2 6. By the existence of a t-(v, k, A) design, we conclude that for any optimal

solution x* of either (PDP) or (CDP) we must have lTz* = b. But this equality holds if

and only if W& x* = X I . O

Proposition 2.1.1 suggests that any of the three formulations may be used to find t-

designs.

CHAPTER 2 POLYTOPES FOR DESIGNS

2.1.1 Designs wit h prescribed automorphism groups

There are two main reasons why it is desirable to search for designs with a prescribed

automorphism group. Classes of designs admit t ing certain automorphism groups are of

special interest, such as cyclic and rotational designs. In addition, assuming the action of

an automorphism group on a design allows for a reduction on the size of the corresponding

integer programming problem.

Let V be a finite set with [VI = v. The set Sv of al1 permutations on V forms a

symmetric group where composition is the group operation. If n E Sv maps the design

Dl = (V,&) to the design D2 = (V,B2) then we Say that Dl and 4 are isornorphic.

If n fixes DI so that Bi = B2 then r is called an automorphism of Di. The set of ail

automorphisms of a design D, denoted by AutD, foms a group under the composition

operat ion.

Let A be a permutation group acting on a set V. If x E V, the orbit of x under A is de-

fined by A(x) = {y E V : n ( ~ ) = y for some R E A ) . An equivalence relation on V can be

defined by x and y being equivalent if and only if y E A ( x ) ; the equivalent classes are the

different orbits of V. This definit ion is nat urally extended to orbits of s-subsets of V. The

orbit of a n s-subset S E ( y ) is defined by .4(S) = {T E ( y ) : a( S) = T for sorne n -4) . Let t , v and k be integers with O < t 5 k 5 II, V be a set with IV1 = v , and A be

a permutation group acting on V. Let T,, i = 1, . . . , 1 ( y ) / A I , be representat ives of the

orbits of the t-subsets and K j , j = 1,. . . , ( ( ! ) / A I , be representatives of the orbits of the

k-subsets. We define Wtj as the 1 ( y ) /A( x ( (1) / A I matnx dehed by

[ w ' ] , = I{K E A ( K j ) : K > Till-

Theorem 2.1.2 (Kramer and Mesner [56]) Let A be a permutation group and I =

1 AI- T h e simple t-(v, k , A) designs odmitting an automorphisrn gwup A are the

solutions x of

As previously mentioned, the formulation

[55] [58] [59] [69] [go;.

w& 2 = A l ,

z E {O, 1)'.

(DPA) was used for h d i n g t-designs in [54]


The following proposition shows similar formulations for packings and coverings.

Proposition 2.1.3 Let A be <r permutation group, 1 = 1 ( : ) / A I and 4 = IA(Ki)l, for

i = 1 , . . . , t . The simple t -(u, k, A) packârag designs odmitting outomorphism group A a n

the solutions x E R' of

maximize [cl, . . . , ci] x

(PDPA) subject to Wtk x 5 A l ,

z € {O, l}l.

Anologowly, the simple t-(u, k , A) covering designs admitting automorphism group A are

the solutions x E R' of

(CDPA) subject to Wek x 5. Al,

x E (O, 1)'.

For designs with A = 1, problems (DPA) and (PDP*) can be brought to the form of

set partitioning and set packing problems, respectively. Note that problems (DP*) and

(PDPA) differ from set partitioning and set packing in that Wfk is not necessarily a O-1

matrix. However, any column j containing an element greater than or equal to 2 can

be eliminated, since for X = 1 any solution x of (DPA) or (PDPA) must have xj = 0.

Therefore, after performing these eliminations, problems (DPA) and (PDPA) with A = 1

become special cases of set partitioning and set packing, respectively.

2.1.2 Ot her int eger programming formulations

In t his section, we discuss alternative integer programming formulations for BIBDs, Le.

2-(v, k, A) designs, based on a quadatic system of equations. Although this formulation

is not used in the rest of the thesis, we include it here for completeness. This can be

generalized for t-designs with t > 2 at the cost of having a system of equations of degree

t rather than a quaciratic one.

Let X denote the point-bock incidence matriz of a 2-(v, k, A) design wit h bblocks,

that is a v x b matrïx X = ( zpg) indexed by the points and blocks of the design and


such t hat 1 i f p € B ,

X p , B = O otherwise.

A weli known property of these matrices is given in the following theorem.

Theorem 2.1.4 Let In and JaXm denote the n x n identity matriz and the n x m mairiz

of al1 ones, respedively. A v x b 0-1 mat& X satisfies

àf and only if X is the point-block incidence matriz of o 2-(v, k , A) design with b Qlocks

and point replication equal to r .

A proof of this theorem can be found For example in [98]. Rewriting the equations

given by the theorem, we get

In fact, the first (or second) types of inequalities can be removed from the system,

since integer solutions of the remaining system will satisfy the first (second) type of

inequalit ies for any admissible parameters.

Wengnik (1 031 studied in her Master's thesis an equivalent Linear system obtained by

introducing extra variables = zilxjl and given by


Wengrzik [103] ~tudied the polytope associated with (LDP) and proposed a polybedra-

based algorithm for solving such problems. Several instances were solved, although no

new designs were produced. Although a thorough cornparison was not done, it seems

to us that this formulation may be preferabie than the formulation in (DP) for designs

with t = 2 and larger k or A. For smaller X or k as weH as for larger t, the formulation

(DP) seems more suitable. It would be interesting to investigate the reiationship between

the formulations in (DP) and (LDP), and whether facets found from one mode1 can be

translated into facets for the other one.

Another approach for solving these systems was proposed by Mathon [?'O], by consid-

ering the relaxation of (QDP) given by

Mathon employed a conjugate-gradient method to find local minima for the Least-

square function associated with the above system, which was iterated by choosing random

initial points until a design was found. He empiricdy observed that whenever a solution

of (QDPR) was found, the integrality condition was satisfied. The following theorem

guarantees that this is always the case, and that the relaxed system ( Q D P ~ ) is equivalent

to the integral one (QDP).

Theorem 2.1.5 Let v , k, r, A, b le admissible parumeters for a 2-(v, k , A) design with b

blocks and point replication r . Let x be a solution to the system of epuations ( Q D P ~ ) .

Then, v e have xii = O o t z ; i = 1, for ail 15 i 5 V , 15 15 6.

PruoJ First, we show that c!=, z:l = r, for 1 5 i 5 v. Adding the equations (2.3)

involving a fixed i , we get ~ ~ = , , j 2 i ( ~ ~ = 1 xii yl) = (v- 1 )A. Interchanging the summations,

reammging the left-hand side and using the fact that admissible parameters satisfy (v -

b l)X = r(k - l ) , we get Cl=l~iiC;=' , , , j+i~j l = r(k - l ) , for all 1 i 5 W . By (2.2), it

follows that x:=ll,j+i xji = k - zil. Substituting tù is in the previous equation, we obtain

z i1(k - zii) = r(k - 1). Therefore, kx:= , z i l - x!=, X: = r(k - l) , and by (2.1 j it

follows that kr - c:=, x i = r(k - 1). Finally, c:=, x i = r for dl 1 5 i 5 v .

Consequently, for a l l O 5 i a V , c;=, = il)^- Hence, c:=, xii(l - +il) = O

and sioce O 5 t i i 5 1 , we have zil(l - zil) = O for al1 1 5 i 5 v , 1 5 1 5 b. This impiies

that either xi1 = O or xi1 = 1, for al1 1 5 i 5 V , 1 5 1 5 b. O

2.2 General set partitioning, set packing and set cov-

ering polytopes

In this section, we summarize results on polytopes for more general problems with similar

structures to the design problems.

Let A be an m x n 0-1 matrix and c E Rn. The set packing, set partitioning and set

covering problems are, respect ively, given by

max cTx max cTz min cTx

(pK) ( A x I l (PT) ( A x - 1 ( C l { -4 z 2 1,

1. € {O, 1)" x E {O, 1)" x E {O? 1)"

Let p be an integer number, and consider the generalizations of the previous problems

by dowing a constant right-band side

The problem for which the most is known is (PK), due to its equ i~ lence to maximum

independent sets of nodes in a graph, as we will see in Section 2.2.1. Its genealization

(pKG) is in the form of maximum independent sets in an independence system, which will

be discussed in Section 2.2.2. Both covering problems (CV) and (CVG) c m be brought

to the form of maximum independent sets in an independence system by complementing

variables, as we discuss in Section 2.2.3. Less is known about the partitioning problems

(PT) and (PT') and their polytopes, but they can be transfomec! into set packing or

set covering after some adjustment of the objective function (see [9]). Moreover, in the

case of the design problems, as we saw in Proposition 2.1.1, the equivalence is trivial, so

we can concentrate on packing and covering problems.

Al1 six problems were shown to be NP-hard for general matrix A (see [65]). However,

for the special case of design problems, it is not known whether or not the problems lie

in P. For surveys on these problems see (91 [13, Part I] [17].

2.2.1 Independent sets and the set packing polytope

The set packing polytope, the one associated with (PK), is given by

P(A) = conv{x E {O, 1)" : A x 5 1).

Let IS(G) be the polytope of the independent sets of nodes for a graph G, that is

iS(G) = conv(s E {O, 1)" : x is an incidence vector of an independent set of C).

The intersection graph of a matrix A = (a,), denoted by Ga, is the g a p h whose

vertices correspond to the columns of A and an edge connects two colurnns ji,ja if and

only if there exists a row i with aij, = a , = 1.

It is well known t bat P ( A ) = IS(CA). In addition, for any graph G, the edge-node

incidence rnatrix Ac satisfies iS(G) = P(Ao). So, the set packing polytope for a matrix

is the independent set polytope for some graph. For this reason, we will use P(G) to

denote the polytope IS(G).

A great deal of research has been doue on the independent set polytope in the past

three decades. We now present t h e main results regarding this polytope.

Proposition 2.2.1 (Busic propertzes) Let A be a 0-1 matriz and let P ( A ) be the set

packàng polytope associated with A. Then

1. the polytope P(A) is full dimensional;

& the inequalities xi 2 O a n facet inducing for P(A);


9. the polytope P(A) is down-monotone, that is, if z E P ( A ) then y E P ( A ) for all

O < y ( t; in addition, any facet inducing inquafitg <rTz < ao, ezcept for si 2 0,

has all nonnegatiue coeficienkr.

The following theorem shows that if C is an induced subgraph of G, then the facet

inducing inequalities for P(Gt) can be "lifted" to facet inducing inequalities for P(G).

Theorem 2.2.2 (Lifting of facets for subgraphs (Padberg [82], for odd-holes; Nernhauser

and Trotter [79], fur general subgraphs)

Let G = (V, E ) be a graph. Let VI Ç V and C = (VI , E t ) be the subgraph of G induced by

V'. if xjEV, ajzj 5 s is a facet inducing inequaiity fo+ P ( C ) , then then ezist integers

p,, O 5 Pi 5 s S U C ~ that

C ujxj + C p j x j 5 s i W ' j€V\Vr

is a facet inducing inequality for P(G).

In the following, we list some classical classes of valid and facet inducing inequalities

for the independent set polytope.

Clique inequalities (F'ulkerson [40], Padberg [82]).

A clique of G = (V, E) is a subset of V that induces a complete subgraph of G. A clique

is maximal if it is not properly contained in any other clique. Let C be a clique of G.

Then, the inequality

is vaiid for P(G). Moreover, the above inequality induces a facet of P(G) if and only if

C is maximal (with respect to set inclusion).

Odd hole (Padberg [82]) and odd anti-hole (Nemhauser and Trotter[79]) in-

equalit ies.

An odd hole is a circuit of odd length greater than 3 without chords. An odd anti-hole is

a graph whose complement is an odd hole.


odd holc t=2, k=3, v >c7 odd anti-hole t=1&=3,v -7

Figure 2.1: Examples of subgraphs of the intersection graph of Wh.

If H C V induces an odd hole Gr of G then

is a facet inducing inequality of P ( C ) .

If A V induces an odd anti-hole G' of G then

is a facet inducing inequality of P(Gr).

Figure 2.1 shows an odd hole and an odd anti-hole for gaphs associated with packing

design problems. .

Web and anti-web inequalities (Xbotter [100]).

Webs are generalizations of cliques, holes and anti-holes. Let n and k be integers with

n 2 2and 15 k 1:. A web W(n,k) is thegraph with vertices V, = (1, ..., n) and edges dehed by: (i, j ) E E if and only if (j - i)mod n E {k, k + 1, ... , n - k). Observe

that W(n, 1) is a clique, W(2s + 1, s ) is an odd hole and W(2s + 1,2) is an odd anti-hole.

The inequaiity

is facet inducing for P( W(n, k)) if and only if n and k are relatively prime.


An anti-web is a complement of a web, denoted by W(n, k). The inequality

is facet inducing for P(W(n, k) ) if and only if n and k are relatively prime.

Wheel inequalit ies (see [68])

Ao odd wheel of center vo is a graph with vertices {vo, V I , . . . , v,) such that V I , . . . , v,,

forms an odd cycle and vo is connected by an edge to al1 of the other vertices. If W C V

induces an odd wheel C of G with center v. then

is a facet inducing inequality for P(Ct).

Generalizations of these simple wheel inequalities by subdividing edges are proposed

by Barahona and Mahjoub [I l] and Cheng and Cunningham [19].

Rank inequalities

Let G = (V, E ) be a graph and denote by a(G) the independence (or stability) number

of G, i.e. the size of a maximum independent set of G. The inequality

is clearly valid for P(G). The following theorems give sufficient or necessary conditions

for this type of inequality to define a facet for P(G).

An edge e of a graph G is said to be critical if a(G - e) = a(G) + 1.

Theorem 2.2.3 (A sufiCient condition for facetness - Chvital [22])

Let G = (V, E ) be a graph and let Eu be a set of &tical edges. if e = (V, E*) is

connected then the inequality


Civen any proper subset of V, the cutset C = (K, V \ Vl) is the (possibly empty)

set of edges joining the nodes in to those in V \ K. Let Gi and G2 be the subgraphs

induced by and V \ K , respectively. The cutset C = (h , V \ h) is called a-critical,

if a(G1) t a ( G ) 1 a(G) + 1.

Theorem 2.2.4 (A necessary condition for [acetness - Balas and Zemel [LU])

If the inequality

defines o facet of P(G), then euey cutset of G is a-critical.

Theorem 2.2.5 (A weaker suficient condition $or fucetness - Balas and Zemel [IO])

Let G = (V, E ) be a gmph with E' its set of cdical edges und such that either G' =

(V, E* ) is connected, or G has an a-cdical cutset C = (h, = V \ h) satisfying the

follomhg conditions (lor Gi the subgraph of G induced by K, i = 1,2):

1. the inequality xjrK tj < a(Gi) defies a facet of P(Gi);

2. euery mazimum-cardinality independent set of Gi is contained in some maximum-

cardinality independent node set of G.

Theorem 2.2.6 (Necessary and sufin'ent condition for eztension of rank facets - Balos

and Zemel [IO])

Let GL = ( V I , E l ) be the subpph o f C = (V, E ) induced by V' Ç V , and such that

defies u facet of P(GL). Then (2.5) defies a lacet of P(G) i f and only i f for every

k E V \ VL , the evtset ({k), V') of G[VL U {k)] (the subgruph of G induced b y V 1 U {k))

is not a-critical


Other properties and inequalities

Proposition 2.2.7 (2-connectedness of facet support - see [IS])

Let G be a groph. If crTx 5 a0 defies a facet of P(G) , then its support G[supp aT] is

2-(node)-connected.

[aequalities derived from quadratic relaxations of the set packing polytope can be found

in [68], and from ot her relaxations in [l3].

Adjacency in the set padcing polytope

Two vertices are adjacent in a polytope if they lie in a face of dimension one (also called

an edge) of the polytope.

Theorem 2.2.8 (Necessary and suficient condition for adjacency - Chvdlal[22])

Let G = (V, E ) be a p p h , let and be independent sets of G and x1,x2 their

respective incidence vectors. Then x' and x2 are adjacent in P(G) if and only if the

subgraph of G induced by (& \ Va) u (h \ \ ) is connected.

Other adjacency results, especidy for partitionhg problems, can be found in [8] [9].

2.2.2 Indepepdence systems and the generalized set packing

polytope

An independence system is a pair (1,Z) where I is an n-set and Z is a family of subsets

of I , with the property that Il C l2 E I implies Il E Z; the individuai members of Z

are cded independent sets. Sets J C I such that J $ I are said to be dependent and

minimal such sets are called circuits. An independence system is characterized by its

farnily of circuits.

Examples:

1. Let G = (V, E ) be a graph and Z& be its family of independent sets. Then (V&)

form an independence system with E being its family of circuits.

CRAPTER 2 POLYTOPES FOR DESIGNS 34

2. Let 1 = [1, n] and let A be a O-l matrix and p a positive integer. Let Za be the family

of all the 0-1 solutions x to Ax pl. Then (I,ZA) is an iodepeodence system whose

family of circuits is given by

EA = (C C I : ICI = p+ 1 and there exists a row i of A with a, = 1 for al1 j E C). (2.6)

Observe that in the two previous examples t h e independence system is r-regular, i.e.

al1 its circuits have same cardinality r ( r = 2 and r = p + 1 for the two examples,

respectively). We will focus on r-regular independence systerns, but we remark that an

independence system is not so in general.

Let Pp(A) be the polytope corresponding to (pKc), i.e.

Pp(A) = conv{x E {O, 1)" : Az 5 p l ) .

For Pp(A), the intersection hypergraph of A is given by HA = (1, EA) , where EA is given in

(2.6). Thus, similarly to ordioary set packings, the generalized set packings correspond to

regular independence systems. We will denote by P ( S ) the polytope of the independent

sets of the independeoce systern S.

Clique inequalities for regular independence systems

Let (1,1) be a pregular independence system. A subset I' I is said to be a clique

11'1 > p and ail psubsets of I I are circuits of (1,Z).

Theorem 2.2.9 (Nemhauser and Trotter [ml) Suppose I f C I is a- maaimal (uiith respect to set-inclusion) clique in a p-regular indepen-

dence system S = (41). Then

Cliques for more generai independence systems, as weU as generalized odd-holes and

anti-holes can be f o n d in [35] [%].


Lifking for independence systems

A lifting theorem, similar to the one for independent sets in gaphs, generalizes to inde-

pendence systems.

Theorem 2.2.10 (Nemhauser and Trotter [W])

Let S'= ( I ' ,Z f ) E S = ( I J ) . If

is a faeet indueing inequality for P ( S r ) , then there ezist integers pj1 O 5 pj 5 s such that

is a facet inducing inequality for P ( S ) .

2.2.3 The set covering polytope

The set covering polytope is the polytope corresponding to (CV) given by

C ( A ) = c m v { x E {O, 1)" : Az 2 1).

Similarly, the generalized set covering problem (CV') has its polytope giveo by

The set covering polytope is known to be equivalent to the independence system

polytope through complementing variables [62] [81]. Indeed, the same is true for the

generalized set covering polytope. Applying the transformation y := 1 - x, we obtain

the two equivalent problems:

Moreover, denoting by Ip(A) the polytope associated wit h (IP), the facets of the two

polytopes are related as folIows.


Corollary 2.2.11 The inequality d x 2 a0 induces a facet for Cp(A) if and oniy if the

inequality aTz 5 aï 1 - CU induces a fucet for Ip(A) .

In addition, if A hm a constant row mm s, then Ip (A) corresponds to the polytope

for uniform independence systems Pr( A), for r = s - p. The matrix W$ of the design

problem satisfies this property.

The study of the set covering polytope is way behind the one for the set packing

polytope. The equivalence of set coverings and independence systems may explain this

unbalance, since studying independent sets in hypergraphs is more complex than in

gaphs. The concept of iotersecting graphs for the set packing matrix has a parallel with

the one for matrix minors of the set covering matrix, but the latter presents a *lack of

locaiityn in the sense that inequalities valid for the set covering associated with a matrix

minor are not even guaranteed to be valid for the original polytope, and Lifting is more

involved. Neverthelas, some research has been done on the set covering polytope, which

we briefly discuss.

Generalized anti-web inequalit ies are studied in [62] [89], generalized web inequalit ies

in [89] and composition of rank facets in [al]. The inequalities with coefficients in {O, 1,2),

are characterized in [7]. Other inequalities can be found in [28]. Conditions for rank facets

of the set covering polytope to be facet defining are given in [13] [62] . In addition, as

seen in Corollary 2.2.1 1, t h e references for independence systems from section 2.2.2 are

relevant here.

2.3 Design polytopes

In this section, we show new results about the design polytopes TtVV,kJ, PtPv,>tJ and Ct,u,k,~,

which are the main contributions of this chapter.

Section 2.3.1 contains basic properties for the three polytopes. In Section 2.3.2,

we examine the polytope Pt,U,kJ. TWO classes of inequalities are studied, namely the

generalized clique and subpacking inequalities. A characterization of generalized cliques

in t e m s of intersecting set systems is given (Proposition 2.3.6 and Corollary 2-33), as well

as connections to results to be proved in Chapter 3. We derive a new class of inequalities

CHAPTER 2 POLYTOPES FOR DESIGN 37

which we cal1 subpacking inequalities (Theorem 2.3.9) and we give conditions under which

they induce facets (Proposition 2.3.11 to Proposition 2.3.14, and Table 2.1). Separation

results for both classes of inequalities are presented (Corollary 2.3.8 and Corollary 2.3.10).

In Section 2.3.3, the polytopes for anti-Pasch and m-sparse triple systems are studied and

anti-Pasch ond m-sparse facets are given (Theorem 2.3.16 and Theorern 2.3.20), as well

as separation results (Corollary 2.3.21). In Section 2.3.4, connections between adjacency

on Tt,,,,r,r and null-designs are explored (Theorem 2.3.22).

2.3.1 Basic properties

In this section, we prove some results regarding general t-designs, packings and coverings-

Let ei E IR(;), 1 5 i 5 (;), be defined by ( e i ) i = 1 and (e ' ) j = O for j # i. The

following t heorem gives information on the dimension of the design polytopes.

Proposition 2.3.1 (Dimension)

i. If k v - t then d in~T~* ,+ ,~ 5 (;) - ( y ) .

ii. Ptvv,ca is fui1 dimensional.

iii. If X < ( i ~ : ) ihen is full dimensional.

Proo f.

i. This is a direct consecpence of the fad thôt ronk(w:k) = (3, when k 5 v - t (see

[44] for a proof). Since the inequalities in W:'x < A 1 are part of the equality repre-

sentation of Say (A', b'), it FoHows that rank(d', b=) 2 rank(W:k, A l ) =

( f ) . Therefore, by Proposition L -2.2, dimTt,v,cs 5 ( i) - (:) .

ii. Since O, e l , . . . , &) E axe f i e independent, the result follows.


Proposition 2.3.2 (Non-negativity construints)

The inequokties zi > O , i = 1,. . . , (i), ore facet inducing for Pt , ,k ,A. Monover, if

A 5 (~Ic_:) - 1, then they ore fucet inducing for Ct,v,k,A.

Pmof. It is enougb to show (:) f i n e independent points in each polytope that satisfy

zi = O. For Pt,V,k,A, the points in {ej)j2i u (O) satisfy this property. If X 5 (l~i) - 1, t hen (1 \ j e i , e j ) ) j+i u (1 \ e i ) CtVVçTA. The remit follows from the affine independence

of these points. O

Proposition 2.3.3 (Relation between packing and couering numbers)

Let v > k > t , A bc integet nurnbers and A' = (i~:) - A, A" = A(:-:) - X . Then,

Moreover, y E {O, 1)" is the incidence uector of a t - (v , k, A ) couering (simple t - ( v , k , A)

couering) if and only i f 1 - y is the incidence vector of a t-(v,k,Xt') packing (simple

t -( v, k, A') packing).

ProoJ This is a direct consequence of the equivalence between the generalized set covering

problem (CP,) and the independence system problem (IP) seen in Section 2.2.3. For

simple designs take c = 1, A = WG and p = X in (CP,) and (IP), and observe that the

optimum values for these problems are equal to c A ( v , k, 1 ) and ( v , k, t ), respectiveiy.

For designs with possibly repeated blocks, assume A to have the same columns as WCk

repeated X times. This matrîx does define the correct problern for designs with repeated

blocks, since any t-(v, k, A) packing and any minimum t-(v, k, A ) covering cannot have a

block repeated more than X times. ARer some simple algebraic manipulations, the proof

is complete. O

2.3.2 Inequalities for the polytope Pt,v,k,x

We concentrate on the packing design polytope Pt,,&*. As we have seen in previous

sections, = &( WA), Le. the polytope associated with the genemlized set packing

problem whose matrix is W$ and rïght-hand side is A.


[ 3,495 1

Figure 2.2: Intersection graph of W&.


For X = 1, the problem is an ordinary set packing. The intersection graph qtk of

the matrix W:k is the graph whose vertices correspond to the k-subsets of [1, v] and

two k-subsets are connected by an edge if and only if they intersect in at least t points.

Figure 2.2 shows an example of such a graph.

For X > 1, t h e problem is the same as finding a maximum independent set in an

independence system. The circuits of this independence system are the sets of ( A + 1)

k-subsets of [ l , v ] sbaring a common t-subset, as stated by the following proposition.

Proposition 2.3.4 (Packings and independence systems)

A t-(v, k , A) pucking is an independent set of the independence system given by the

circuits

.4 clique in such a ( A + 1)-regulax independence system corresponds to a gr ou^ of

k-subsets of [I , v ] with the following intersecting properties.

Definition 2.3.5 (s-wise t-intersecting set systems)

Giuen s 2 2 and v, t 2 1 , a family A of subsets of [1 , v] is said t o be s-wise t-intersecting,

if any s mernbers Al , . ..,A, of A are such that IAi n . . . n A,I 2 t . A family A is said

to be k-uniform if euery rnernber of A has cardinality k. Let lS(vtk , t ) denote the set

of al1 k-uniform s-wise t-intersecting families of subsets of [1, v]. Let MP(v , k, t ) denote

the set of ail families in Is(v, k , t ) that are mazimal with respect to set inclusion (i.e.

A E Is(v, k, t ) such that for any B E Is(v, k, t ) , if B > A then B = A).

Proposition 2.3.6 (Characteriration of generahed cliques)

Let A be a family of k-subsets of [1, v ] . Then A is a genemlized clique for the inde-

pendence system associated with a t -(v, k , A) packing i f and only i f A E I('+ ' ) (u t k, t ) .

Moreover, a clique A is maximal if and only if A E ~ l ( ~ + ' ) ( v , k, t ) .

Pmof. B y the definition of clique in an independence system (see Section 2.2.2), A cor-

responds to a clique for the set packing independence system if and only if A is a family

of k-subsets of [l, vl such t hat all nibfamilies of A with (A + 1) elements are circuits. By


Proposition 2.3.4, this is equivalent to A E I ( ~ + ' ) ( v , k, t ) - Clearly, the clique is maximal

if and only if A E ~ l ( ~ + ' ) ( v , k, t ) . O

The following corollary characterizes the clique inequalit ies For the polytope

Corollary 2 -3.7 (Generalized clique inequalities)

Let A E I (A+ ' ) (v , k , t ) . Then, the inegvality

is valid for the polytope Monover, the cboue inequality induees a facef of PtT,c,x

if and only $ A E MP+')(v, k, t ) .

Pmof. It follows from Proposition 2.3.6 and the clique inequalities desnibed in page 34.

In iight of Proposition 2.3.6, it is natural to ask whet her it is possible to determine the

f'orm of al1 clique ioequalities for &,,k,A, leading to the problem discussed in Chapter 3:

Pmblem. Given s, n, k and t, Est al1 families of MP(v , k, t); or alternatively,

list al1 nonisomorphic families of MP (u , k , t ) .

This problem turns out to be, in general, a nontrivial problem in extrema1 set theory.

However, we obtained some results, including the classification of cliques for k - t 5 2

which will be presented in Chapter 3. We will list these results here to summarize their

main consequences on clique inequalities; the reader is referred to Chapter 3 for proofs.

Determination of generalized clique facets for some set of padcing parameters:

1. For any t 2 1, v 2 t + 3 , A = 1 and k = t + 1: there exists exactly two distinct (up

to isomorphism) clique facets for Ptv,,,t+I.ll namely


2. For any t 2 1, v > t + 6, X = 1 and k = t + 2, tbere exists exactly 15 distinct types

of clique facets for t = 1 and 17 distinct types of clique facets for t 2 2. These

cliques are given explicitly for t = 1 and t = 2, and by a construction for t 2 3 (see

Table 3.1 for their forms).

3. For any t 2 1 , v 1 t + 2, X > 2 and k = t + 1 , there exists exactly one distinct (up

to isomorphism) generalized clique, namely:

4. For arbitrary k > t and A, t here exists a vo = vo(t, A, k) such that al1 clique facets

for v 2 v, are determined by those for vo.

The knowledge of the clique structure can help us designing separation algorithms.

For example, for the case of k = t + 1 and X = I the separation of clique inequalities

t m s out to be quite simple, as shown in the following algorithm.

Algorithm: Separat ion of clique inequalities for Pt,,t+i,

Input: a fractional solution 3 to (PDP)

Output: a violated clique ioequality or "There are no violated cliquesn

let e,,+l = (([lcI1), E ) be the intersection graph of WII,+,.

for every edge ( K t , K2) E E

take L = h; U K2

i f &(,:,) 2i(> I then

return "Violated clique: ", L

return "There are no violated cliques."

Remark: In Chapter 4, we will see t hat we do not need to examine every edge, but just

the ones in the "fractional subgraphn, i.e., ody the edges co~ecting {Ki, h;), where

O < XK,,ZK~ < 1. Moreover, the algorithm can be easily adapted to find ail violated

cliques, instead of halting whenever the first violated clique is found.

RecaU from page 21 our measure of cornplexity.


Corollary 2.3.8 (Separution of cliques in Pt,U,t+I

The clique facets in Pt,,,,t+l,i can be separated in polynomial tirne.

Proof. The statement is implied by the correctness of the previous algorithm. Any frac-

tional solution to (PDP) must satisfy W:t+ll < 1. So, inequalities (2.8) are satisfied

and the only possible violated cliques are the ones in (2.9). Note that there is exactly

one clique of type (2.9) passing through each edge (Ki, K2) in the graph G&+,, for

1 Ki u K4 = t + 2 . This shows the correctness of the algorithm. The polynomiality can

be checked by noticing that every iteration takes polynomial number of steps and the

number of iterations is at most the square of the number of variables in the problem. O

It would be interesting to generalize the previous algorithm for the case k = t + 2, in

which there are 17 different kinds of cliques.

In the following, we discuss another type of inequality for the packing design polytope.

The subpacking inequalities corne from the simple fact that in a t-(v, k, A) packing (V, B)

any B' B with V' = uBtEBtBi gives rise to a t-([V'I, k, A) packing (V', 13').

Proposition 2.3.9 (S'ubpackng inequality)

Let S [l, v ] and DA(nT k, t ) be the sire of a maximum simple t -(n, k, t ) packing. Then,

the inequality given' by

As v grows, we do not expect to identify all such inequalities, since they rely on the

knowledge of the packing numiber DA(1sI, k, t ) , /SI < v , which is one of the objectives of

solving the packing design problem. Nevertheless, it is feasible to determine the packing

number for some s m d values of ISI and then solve the separation problem for subpacking

inequalities with small ISI, as stated by the following corollaxy.

Corollary 2.3.10 (Sepamtion of subpacking inequalities)

Let C 6e a constant. Subpacking ineqvalities &th 1.91 < C can 6e separated in polynomial

CHAPTER 2 POLYTOPES FOR DES~GNS 44

Proof. For C constant and s < C, the packing numbers DA (s, k, t ) can be detemined in

polynornial tirne. Moreover, there is a polynornial number, oamely zFzk+, (3, of such

i nequali t ies. O

Note that a subpacking inequality is not always facet defining, as illustrated by the

followiog proposition.

Proposition 2.3.11 If there ezLîts a simple t - (v , k, A) design, then

does not induce a facet of Pt,V,k,A.

Proof. Since there exists a simple t - (v , k, A) design, it follows that DA(v, k, t ) = b =

A($(:). But then equation (2.11) is obtained by adding the inequalities in W@ A l ,

and thus cannot induce a facet. O

A natural problem is to determine, whenever a subpacking inequality does define a

facet for Pi,zL,l, whether it also defines a facet for Pt,,,k,l, for v 2 V. [n the case of X = 1,

the su bpacking inequalities are rank inequalities (see in Section 2.2.1 ) for independent

sets in gaph G:,k. The inequality i n (2.10) is the rank inequality associated with the

subgraph of Glk induced by S. The following proposition tell us when a subpacking

inequality can be extended.

Proposition 2.3.12 (Eztension of subpacking Jacets)

Let V > k > t . Suppose the subpacking inequality

defines a facet of Pt,ù,k,l. Then, (2.12) deJines o fucet of Pt,u,k,l, u > ü if and only if

for ail L E ([y) \ (I1;4) there ezists o t -(C, k, 1) packing design ([l, ü], B) such that

([1, v ] , l3 u {L)) is a t -(v, k, 1 ) packing design.

Proof. Let v > v and let L E (['ci) \ ([ILq) . Then, the cutset ((('2) , {L)) of the subgraph

of GYvx induced by pl?) U (L) is not crîtical if and only if the condition of the proposition

is satisfied. The proof is completed by applying Theorem 2.2.6. O

In the following corollary, we show that the extension is dways possible when k = t + 1.

Corollary 2.3.13 Let ü > k > t v i th k = t + 1 . I f the ineguality

defines a facet of PtzL+L,L, then it also defines a facet for Pt,v,t+I,I, for ail v 2 z.

Proof. Let u > 5, L E (2) \ (fl+L) and LI = Ln[l,ü]. Then ( L ' I 5 t. Let P = ( [ l , ~ ] , B) -

be any t - (F, t + 1 , l ) packing design with [BI = &(Gy t + 1 , t ) . If the inequality (2.13)

defines a facet of Pt,G,t+l,l, by Proposition 2.3.11, P cannot be a t-design. Thus, there

exists a t-subset T C [ l , ~ ] not covered by P. Let rr be any permutation on [ l , ü ] such

that z ( T ) > LI . Let P' = n(P) , and denote its blocks by BI. Then ([l, v ] , B'u {L)! is

a t - (v, t + 1 , I ) design with Dl@, t + 1, t ) + 1 blocks. Therefore, by Proposition 2.3.12,

inequality (2.13) defines a facet of Pt, , t+i , i . O

Facet iaducing subpacking inequdities

We will analyze for some small ISI, k = 3 and t = 2, whether the corresponding pack-

ing inequalities define facets. The results are summarized in Table 2.1 and open cases

are indicated by question marks. Some of the table entries corne from the following

proposition.

Proposition 2.3.14 (Miscellaneous results for subpacking inequalities)

The subpacking inequalities

induce facets of P2,v,3,11 v 2 5 and v 2 10, respectiueiy. The subpacking inequality

does not induce a facet of P2,,,J,i, for any v 2 6.


Yes

Yes

No

No

?

No

Yes

?

?

No

?

No

maximal clique - Theorem 3.4.1

Proposition 2.3.14

Proposition 2.3.14

3 STS(7) - Proposition 2.3.1 1

-


Proposition 2.3.14

-

-


- 3 STS(l5) - Proposition 2.3.1 1

Table 2.1: Facet inducing subpacking inequalities for k = 3 and t = 2.

Prao/. First we prove that inequaiity ( i ) induces a facet of P2,VJ,Il for v > 5. In light

of Corollary 2.3.13, it is enough to prove it for v = 5. By Theorem 2.2.3, it is sdlicient

to prove that every edge in G& is critical. Indeed, any edge of G:, connects verticer of

the form {a, 6, c) , {a, b, d ) for distinct a, b, c, d; thus, removing such an edge, makes the

collection ({a, 6, c) , {a, b, d ) , (c, d, e)) a vaüd independent set. Hence, any edge in Gq3 is critical.

Simikdy, in order to prove that inequaiity (ii) induces a facet of P2,VJ,1, v 2 10, it

is sufncient to show that any edge of G:; is critical. Take any maximum 2-(10,3,1)

packing, i.e. a packing with 13 blocks. Then, from the (y ) = 45 possible pairs in

[l, 101, only 13 x 3 = 39 are covered. This leaves 6 uncovered pairs that cannot be ail

disjoint, since they'are taken from 10 points. Therefore, there are two uncovered pairs

{a, b), {a, c) for distinct a, b, c, and obviously (6, c) is covered, ot hemrise (a, 6, c) wotdd

belong to this maximal packing. Let d be the point that appears together with {b, c).

Removing the edge that c o ~ e c t s {a, b, c} and {b, c, d) increases the size of the maximum

CHAPTER 2 POLYTOPES FOR DESICNS 47

packing by 1, and so t his edge is critical. Since the same is tme for every edge (just apply

a permutation on [l ,101 to get a suitable isornorphic maximum packing), we conclude

that every edge is critical.

To prove that inequality (iii) does not induce a facet of Pz,,s,i, v 2 6, assume w.1.o.g.

that S3 = [ l , 61. Adding the MLid inequalities CKE(3:TCK - ZK 5 2 for al1 S E (Il:]), we

get 3 C ll.ai XK < 12 that is equivalent to inequality (iii). Heace, (iii) cannot define a ~4 3 facet. O

2.3.3 Inequalities for the polytope of m-sparse triple systems

In t his section, we study polytopes for triple systems avoiding subconfigurations.

Let us denote by Pap(v) the polyhedron associated with anti-Pasch 2-(v, 3 , l ) pack-

ings and by P,(v) the polyhedron associated wit h m-sparse 2-(v, 3 , l ) packings. As

we have seen in Section 1.3.4, anti-Pasch and Csparse triple systems are equivalent, so

PAP(V) = P4(v)-

The main contribution of this section is a class of facet inducing inequalities For Pap(v)

given by Theorem i.3.16 and its generalization for Pm(") given by Theorem 2.3.20.

Let u s start with some basic properties of these polytopes.

Proposition 2.3.15 (Dimension)

For rn 2 4, the polyhedron Pm@) C R(;) is fd l dimensional.

Proof. Since the vectors O, es, for a l l S E (r) , are incidence vectors of trivially m-sparse 2-

(v, 3,L) packing designs and are &ely independent. We conclude that dim Pm(") = (;). O

Note that although very often the inclusion P,(v) C fz,v,3,i is proper, the maximal

clique inequali ties c.ontinue to induce facets for Pm ( v ) . This is due to the fact that in the

proof of facetness of maximal clique inequalities, independent sets with at most 2 vertices

are employed; in t h e context of our polytopes, they clearly correspond to m-sparse Steiner

triple systems and so the proof remains valid.

For the sake of clarity, we b t exhibit a class of facet-defining inequalities for the

polytope PAP(v), and then generalize it for P,(v), for m 2 4.


Let us define a quasi-Pasch to be any set of 3 distinct triples contained in some Pasch

configuration. Given a valid inequality d r x < do for a polytope P, we define its equality

set as EQ(PX 5 do) = {z E P : Px = do).

Theorem 2.3.16 (Pasch-ovoiding facets)

Let T be a 6-set contained in [l, u] . Then the inequality

defines a jacet for PAP(v). We cal1 the inequality pa(T) the Pasch-avoiding inequality

relatiue to 7'.

Prooh Throughout the proof we will assume w.1.o.g. t hat T = [1 ,6].

1. The inequality pa(T) is valid.

If we use the fact t hat Pap(v) = P4(v) the argument would be sirnpler (see The-

orem 2.3.20- Part 1 .). The foilowing lines actually show t hat anti-Pasch packings

are 4sparse.

We just have to show that no more than 3 blocks from (:) can be present in

an anti-Pasch 2-(v ,3 ,1) packing design, Say (V, B). Suppose we have at least 2

distinct blocks in B From (J). Call these two blocks Bi and B2. Then, they

either intenect in zero or one point. If they intersect in zero points then no other

block of the design can be in (z) , for such a block would necessarily bave a pair

in common witb either Bi or Elz. The last case to examine is lBl n B2J = 1.

We assume w.1.o.g. that BI = (1,2,3) and & = {l ,4, 5). From the packing

design property, the only candidates from (:) that can be in B are precisely in

C = {{2,4,6), {2,5,6), {3,4,6), {3,5,6)). Findy, we see that if more than one

set from C would be in 8, either the packing or the anti-Pasch properties would be

violated. Therefore, Il3 n (3) 1 5 3.

2. The inequality pa(T) defines a facet.

It is enough to prove that if aTx a is a valid inequality for PM(v) with EQ(aTx 5

a) > EQ(po(T)), then aTx 5 a is a scalar multiple of pa(T) (see Theorem 1.2.4).


Let S and Si be arbitrary sets in (:) such that ISn S'I = 2. Let us assume w.1.o.g.

that S = {1,2,3) and Sr = {1,2,4). There exists a pair of sets SI, 4 E (i) such

that both Bi = {Si, 4, S ) and B2 = {SI, S2, S') are quasi-Pasches, e.g., Si =

{3,4,5) and 4 = {1,5,6). Clearly (V ,Bi ) and (V,B2) are anti-Pasch 2-(v,3,1)

packings. Letting x and 2' be their corresponding incidence vecton, we observe

< a). that r = x' - est + es. Moreover, we have 2 , ~ ' E EQ(pa(T)) C EQ(a 2 - Thus, a = aTz' = aTz = aT(x' - est + es) , which implies as@ = as. We daim that

there exists a constant y such that as = y, for al1 S E (3). Indeed, this follows

trivially From the connectivity of the graph whose vertices correspond to the sets

in (T) and such t hat two sets are joined by an edge whenever t heir intersection bas

size 2.

Next we wiU.show that o~ = 0, for al1 R (3) \ (3). Let R E (3) \ (3). W e daim that there exists a quasi-Pasch in ('f) that together with R forms an

anti-Pasch 2-(v,3,1) packing. if IR n TI 5 1, then any quasi-Pasch with sets in

(1) satisfy this property. So, let us analyze the remaining case of IR f l TI = 2.

In this case, we assume w.1.o.g. that R = {l,2,7). Thus, the following quasi-

Pasch BI = {(1,3,4), {1,5,6), {2,3,5}} has the desired property. Therefore, both

BI u {R} and Bi are anti-Pasch 2-(v, 3,1) packings. Moreover, their corresponding

incidence vectors, Say x and x', are in EQ(pa(T)) C EQ(U% a). Therefore, we

conclude that a = oTz' = aTz = aT(x' + eR) = aTx' + a R , and so a~ = 0.

W e have shown that aTx 5 a can be rewritten as 7 T xs 5 P. Letting z be any SE( 3

quasi-Pasch with sets in 0, we know that T E EQ(pa(T)) C EQ(7 Es(:) zs 5

a), and one can easily derive that a = 37. Therefore, p a ( T ) and aTx a only

differ by the multiplication of t h e scalar y.

In order to generalize Theorem 2.3.16 br m-sparse 2-(v, 3 , l ) packings, we need some

definîtions and two auxiliary lemmas.

Following the definition in [46], aii Eidos conJigvration of order n, n 2 1, in a Steiner

triple system is any (a + 2, n)-configuration, which contains no ( 1 + 2,l)-configuration,


1 < 1 < n. In fact, this is equivalent to requiring that 4 5 1 < n, since there cannot be

any (4,2)- or (5,3)-configurations in a STS.

Lemma 2.3.17 (Lefmann et ab[6S, Lemma 2.31)

Let 1, r be positive integers, 1 2 1 . Then any ( 1 + 2,1+ r)-configumtion in a Steiner triple

system contains a (1 + 2, 1)-conftguration.

Lemma 2.3.18 (Construction of an Etdos configuration, for al1 n 2 4)

Consider the following ncursiue definition:

Then, for al1 n 2 4, En is an Erdôs configuration of order n.

Proof. E4 is the Pasch configuration, which is the o d y (6,4)-configurat ion. Es is the mitre

configuration, which is the only (7,5)-configuration not containing a Pasch. Assume that

E, is an Erd6s configuration of order n. We will prove by induction on n that is

an Erdos configuration of order n + 1. &l is clearly an (n + 3, n + 1)-configuration. It

remains to show that it does not contain any (lf2,l)-configuration for 1 < n. Suppose to

the contrary that there exists an ( I + 2,l)-configuration B in &+i, with 4 < 1 5 n. First,

we observe that 8 must contain one of the 2 blocks containing the last point n + 3, for

otherwise B would be contained in En. In fact, B must contain both blocks containing

n + 3, for if contains only one, cal1 it E, B \ ( E ) 1, would be an (p, 1 - 1)-configuration

with 1 - 1 5 p 5 1 + 1, and by Lemma 2.3.17, it wodd contain a (p, p - 2)-configuration,

which is contrary to our assumption.

Let x be the maximum number for which the block Ex E &+l does not appear in B.

Since x 5 n - 1 and i 5 n, we must have x > 1. So, E z + I , - . - , E B and Ez 6 B,

for some I 5 x < n - 1. By construction, Er+2,. . . , En+1 are the o d y blocks

containing { x + 3 , x + 4 ,..., n + 3 ) . Let 8' = B\ {Ez+l, Ex+2 ,..., E,+l). Then, B' C E,


has 1 - (n - x) - 1 blocks and span p 5 1 + 2 - (n - x ) - 1 points. We observe that p 2 3,

since we aie dealing with at least one triple. Then, by Lemma 2.3.17, Bt E E, contains

an (p, p - 2) configuration which is a contradiction. O

Lemma 2.3.19 L& v 2 1 2 4 and let T be an ( 1 + 2)-subset of [l,v]. Let R E (Il$),

R T. Then, there ezists an Eldos configvration S of order 1 on the points of T and a

tr iple S E S, such that S \ {S) U { R ) is an I-sparse 2-(u,3,1) packing.

Proof. Let S be an Erd6s configuration of order 1 on the points of 7' (Lemma 2.3.18

guarantees its existence). If IRf17'1 5 1, taking any S E S, the set S' = S \ {S ) U { R )

is a 2-(v, 3 , l ) packing. Otherwise, if 1 R fi TI = 2, we will choose S such that the pair

R n T appears in a block, Say S. Then, Sr = S \ { S ) ü { R ) is a 2-(v, 3,1) packing. In

either case, we claim S' does not contain an (n + 2, n)-configuration, 4 5 n < 1. Lndeed,

if that was the case, the configuration, Say B, would contain R. Moreover, since R g T

and 1 RI = 3, B \ {R) would be a (pl n - 1)-configuration with n - 1 5 p 5 n + 1. Thus,

by Lemma 2.3.17, B Ç S would contain a (p, p - 2)-configuration for p 2 3, which is a

contradiction. O

Theorem 2.3.20 (m-sparse facets)

For any 4 5 1 6 h and any (1 + 2)-subset T of [ l , v 11, the inequality

defines a facet for P&).

1. The inequality s (T) is d d .

Let ( V B ) be an m-sparse 2-(v,3,1) packing. Let BT be B n c) and let Li be the

union of ail the triples in &. Let p = 1UI. Clearly p 5 ITl = 1 + 2. Since (IL, 8) is

m-sparse, then 1 BT 1 5 p - 1. Thus, 1 BT 1 5 (1 + 2) - 1 = 1 - 1, and so s(T) is valid.

2. The inequality s(T) defines a facet.


Let oTx 5 a be a vaiid inequality for P,(v) with EQ(aTz 5 a) 2 EQ(s(T)). We

must show t hat aTx 5 a and s (T) are scalar multiples of each ot her.

Let S be an Erdiis configuration of order I . There must be two triples in S whose

intersection is a single point, cal1 those triples Si and S2. We daim S \ {Si) and

S \ {S2} are m-sparse 2-(v, 3 , l ) packings. Indeed, IS \ {Si)l = ISI - 1 = 1 - 1,

and sioce S r a s (1 - 1 )-sparse, so is S \ {Si), i = 1,2. Letting xi be the incidence

vector of S \ {Si), for i = 1,2. Then, x' = x2 - es, + es,. In addition, zl,xZ E

EQ(s(T)) Ç E Q ( ~ * Z 5 a). Thus, a = aTzz = a T d = uT(x2 - e h + es,), which

implies as, = as. Moreover, for ôny Ri, R2 E (i) with 1 RI f7 R21 = 1, we can show

that aR, = a& by the same argument as before, just by considering an appropriate

permutation on T that takes R, to Si, i = 1,2. Consider the graph whose vertices

are the triples in T and two triples are joined by an edge whenever their intersection

has size 1. Since this graph is connected, we cooclude that there exists a constant

-, such that as = 7, for al1 S E (3). Next we show that a R = 0, for all R E (:) \ (3). Let R E (3) \ (3). By Lemma 2.3.19, there exists an Erdôs configuration of order

1 , Say S and a triple S E S, such that S \ {S) U { R ) is an rn sparse 2-(v,3,1)

packing. Let x and XR be the incidence vectors of S \ {S ) and S \ {S) U {R),

respectively. Then t, ZR E EQ(s(T)) 2 ~ Q ( a ' z 5 a), and thus a = aTx =

aTzA = aT(x + eR) = aTx + a ~ , and so U R = 0.

So, similarly to the proof of Theorem 2.3.16, we conclude s (T) is facet defining.

Corollary 2.3.21 (Separation oJm-sparse facets)

For constant m 2 4, m-sparse facets can be separated in polynomial time.

Proof. The n-b& of such lacets is CE, ( y ) which, for constant rn, is bounded by a

polynomial on v . O

The proof of Corollary 2.3.21 impiies the use of the aigorithm that checks whether

each of the possible inequalities is violated. In Chapter 4, more efficient aigorithms are


given for the sepmation of m-sparse facets for some srnall m's.

2.3.4 Adjacency in the polytope and nul1 designs

A nul1 (t , v , k)-design is a (O, 1 , - 1)-vector satisfying Wtkx = O. Let xL , t2 be incidence

vectors of t - (v, k , X ) designs. Then, Wtk(xl - x 2 ) = X - A = 0, so d - z2 corresponds to

a nul1 design.

Let let k 2 t + 1, R E ( [ ' ~ l ) and T E (!;y) with K n T = 0, and 4 be an injection

from T to K. Define y E Id;) such that y~ = (- l)IKnTI whenever K C K U T and K

contains exactly one of each pair ( i , + ( i ) ) for al1 i E T, and y~ = 0, otherwise. Such y

is called a (t, Q p o d and it is known to be a nul1 (t, v , k)ilesign with minimum support

(see [44]).

For example, consider a (2,3)-pod given by R = (1,2, 3) T = {4,5,6) with 4(6) = 1,

4(5) = 2 and 4(4) = 3. Consider the following triples:

The (2,3)-pod is given by y~ = 1 for K E A, y~ = -1 for K E B and y~ = O

otherwise. The four triples in A and B both cover all pairs from [1,6] except the pairs

{l,6), {2,5) and (3,4); they form a pair of Pasches. Thus, if a STS, Say xl, contains the

Pasch B, then sa = z1 + y is a STS with B replaced by A. The next theorem shows that

if two t-(v, k, 1 ) designs x1 and z2 differ by a ( t , k)-pod, then they are adjacent vertices

of the polytope

Theorem 2.3.22 Let z1 and z2 be incidence vectors of t-(v, k, 1) designs for which

2' - zZ is a ( t , k)-pod. Then x1 and z2 o n adjacent uertices in the polytope Tt,v,kpi.


Proof. By definition x1 and x2 ôre vertices of and Pt9vs,i Using Theorem 2.2.8,

x1 and z2 are adjacent if and only q , k [ s ~ p p ( z l - z2)] is connected. Let Ki, K2 E

supp[xl - x2]. We prove by induction on 1 = 1 KI \ K2 1 = 1 K2 \ KI 1 t hat there exist a path

connecting KI and K2. If 1 = 1, Ki and K2 differ by one element, so 1 Kt n Ka 1 = k- 1 2 t,

and there is an edge connecting Ki and K2 in qk. If 1 > 1, then take s E KI \ K2. By

definition, 4 ( s ) E Ka. Take K; = Ka \ {#(s)) U (s), and observe that by the definition

of a (t,k)-pod K; E supp(xl - x2) . Then IKl \ KJ = 1 - 1 and IK2 \ KJ = I and by

induction, there exists a path in ~ , k [ s u p p ( x l - x 2 ) ] connecting Kt and K; as well as K2

and K;. Therefore, t here exists a path in q,k[supp(xl - x2)] connecting Kt and K2. O

The condition in the previous theorem is not necessary, as will be seen in the example

of Section 2.4.1.

2.4 Complete descript ions of some polytopes

We conclude this chapter by studying in more detail the Fano plane polytope (i.e. the

polytope T2,îs.t) and showing ail the facets b r the packing design polytopes with small

parameters (t = 2, k = 3 and v < 5 ) .

2.4.1 The Fano plane polytope or T2,7,3,1

The vertices of the polytope T2,1J,1 are the incidence vectors of the 2 - (7,3,1) designs.

These designs are symmetric since the number of blocks is equal to the number of points.

They are also known to correspond to projective planes of order 2. Table 2.2 shows these

designs. One can tlassifi each pair of such designs as one of 3 types: the pairs that

intersect in 3 blocks, in 1 block, and in no block. Figure 2.3 displays these intersections.

GI is the union of two K15 with edges connecting designs that intersect in 1 block; G2

is a bipartite graph with edges connecting designs with 2 blocks in cornmon; G3 is the

complement of G1 U G2 (the missing edges in the bipartite graph) and its edges connect

designs with no co&m blocks.

It t m s out that every pair of the designYs incidence vectors are linked by an edge, Le.,

a face of dimension 1, in the polytope T2,7J,1. This can be verified by exarnining every


Figure 2.3: The three types of intersection among 2 - (7,3,1) designs

pair of distinct designs ( D i , Di). If Bi,& are their respective set of blocks, we observe

that (Bi \ B j ) U (Bj \ Bi) induces a connected subgraph of W& and by Theorem 2.2.8,

there is on edge between Di and Dj in Since T2,7,3,L is a face of P2,713,1, there is

an edge connecting Di and Dj in T2+7,311

Other properties of this poiytope were determined using the Porta software by Thomas

Christof and Andreas Loebel (201. For the 30 vertices of T2,7,3,i the software computed

its equality and inequdty representation. The information obtained is summarized in

the following:

O T2T713,L E W35 has dimension 14 (its dimension achieves the upper bound of The*

rem 2.3.1). Thus, its equality system is given solely by Wi'x = 1.

0 Its inequality system contains 155 facet-inducing inequalities of two types. Thirty

five of these inequaiities axe equivdent to zi 2 0, for i = 1,. . . ,35. Each of these

ioequalities passes through exactly 24 vertices. Each of the others 120 inequalities

passes through exactly 14 vertices.

0 There are 84 inequalities passing through each vertex.


- - - - --

Table 2.2: The 30 distinct 2-(7,3,1) design~.

2.4.2 Polyhedra for small packings

Next we list the complete facet representation of polytopes for some srnaIl packings,

obtained using Porta software [20].

Chapter 3

Clique Facets and Intersecting Set

Systems

As shown in Chapter 2, the generalized clique facets correspond to set systems with

prescribed intersection properties (see Proposition 2.3.6 and Corollary 2.3.7). In this

chapter, we take a closer look at these intersecting set systems. Our main goal is the

classification of al1 the clique facets of Pt,,,t,x, which are precisely the facets with right-

hand side being equal to 1.

We recall some definitions from Chapter 2. Given s 2 2 and v, t 2 1, a family

A of subsets of [I,v] is said to be s-luise t-intersecting, if any s members A l , . . . , A,

of A are such that IAi n . . . n AsI > t. A family A is said to be k-uniforna if every

rnember of A has cardinality k. We denote by IS(v, k, t ) the set of all k-uniforrn s-wise

t-intersecting families of subsets of [1, v] . We are interested in the families in Ia(v, k , t )

that are maximal with respect to set inclusion, Le., the families A E Is(v, k, t) such that

for any B E P ( v , k , t ) , if B > A then B = A. We denote by MF(v,k,t) the set of ail

families in ts(v, k, t ) that are maximal in this sense. The problem that motivates this

chapter is the following.

Problem 1. Given A 2 1 and v > k > t, List a l l generalized maximal clique

facet s of

By Proposition 2.3.6, this problem c m be rewritten in t e m s of intersecting families of

CHAPTER 3 CLIQUE FACETS AND INTERSECTING SET SYSTEMS

sets, taking s = X + 1, as follows.

Problem 2. Given s 3 2, v > k > t, list al1 families in M P ( v , k, t); or

alternatively, list al1 nonisomorphic families in M+, k, t).

In addition to the above ment ioned polyhedral consequences, Problem 2 is of independent

interest in extrema1 set theory and in the theory of hypergraphs. We briefly comment

on related works. The famous Erd6s-KwRado theorem [33] determined the size of the

largest family in MP(v, k, t) for u large enough, and it was a seminal paper in the study

of intersecting families. The best possible u for which this theorem remains true was

shown by Frankl [36], for t 2 15, and by Wilson [105], for general t . Other extensions

of the Erd&Kc+Rado t heorern includes works of Hilton and Milner [491 and Frankl [38].

Recently, the problem of determining the largest family in MP ( v , k, t ) was final'y settled

For the whole spectrum of v by Ahlswede and Khachatrian [2] [3].

Problem 2 involves determining all the families in MP(v , k, t), not just the largest

ones, which in general turns out to be a hard problem. Nevertheless, some partial results

on the classification of the families in iW(v , k, t ) are presented in this chapter; the most

important ones are:

1. the classification problem is solved for the first nontrivial case, namely s = 2 and

k = t + 2 ;

2. for any s, k, t , the classification problem does not depend on v.

Other intermediate and interesting r d t s are shown along the way and described in

more detail in the next section.

The results in this chapter are contained in paper [77]. In ônother work, we use

collections of intersecting set systems for studying transversal covers [94].

3.1 Definitions and results

In this chapter, we make use of two important concepts in the study of families in

MP(v , k, t), namely kernels and generating sets. First, we define the notion of a gener-

ating set of a family of subsets, as introduced by Ahlswede and Khachatnm [2] [3]. Let

CHAPTER 3 CLIQUE FACETS AND INTERSECTING SET SYSTEMS 59

us fix sorne notation. For any B C [ l , ~ ] , let U$(B) = {C C [ I , v ] : C 2 B, ICI = k),

and for a family B of subsets of [1, v ] , let U,L(B) = U B ~ ~ U:(B). Let A be a k-uniform

farnily of subsets of [l, v ] ; a family g of subsets of [1, v ] is called a generating se t of A, if

Ut(g) = A. The set A itself is a generating set of A, although not a very interesting one.

We would Iike to be able to find generating sets g that are "minimaln in some sense, for

example, the ones for which / UsE, GI or Igl are minimal. Such generating sets provide

us with a compact way to represent families of subsets, which proves to be useful for

studying Problern 2.

We turn our attention to t h e concept of a kernel of an s-wise t-intersecting family of

subsets. For a family A in Is(v , k, t), a set XC C [ l , v ] is a kernel of A if any s sets in A

meet in at least t elements of K, i.e., for any Al,. . . , As E A we have IAin.. .nA,nXCl 2 t .

Denote by n(k,s,t) the smallest integer for which every family A in U, Is(u, k, t) has a

kernel K(A) with )K(A)I 5 n(k, s, t ) (the standard definition of n(k, s, t) dues not require

the Families A to be k-uniform, but rather having members with cardinality at most k;

both definitions turn out to be equivalent). It is not obvious that n ( k , s, t ) is finite for

al1 k, s and t . Calczyxiska-Karlowitz [15] proved that n(k, 2 , l ) is finite. The first explicit

upper bound on n(k ,2 ,1 ) was given by Ehrenfeucht and Mycielski [32]. Lower bounds

and further improvements on upper bounds were given by Erdôs and Lovisz [34] and

Tuza [101]. The finiteness of n(k , s, t ) was independently proven by Frankl (371, in an

implicit fom, Kahn and Seymour [51], and Frankl and Fiiredi (391. These results imply

that at least t elements of each of the s-wise intersections of a farnily A E Is(v, k, t ) are

included in a small subset (with cardinality independent of v ) of the underlying set [I, V I . General bounds on'n(k,s, t) are given by Alon and Fiiredi [4], who proved that

They give an explicit construction for the lower bound, and also show that the upper

bound folIows fr0m.a theorem of Füredi [41]. Kohayakawa [53] shaqened the lower bound

in (3.1).

We note that the upper bound in (3.1) can be improved by a simple observation on

the last step of Alon and Fluedi's proof. tndeed, in that proof [4, Theorern 2.21, they


obtain an upper bound on the maximum size of a kernel given by the size of the union k - t + l + l U

of m sets of size k, with rn = ( ) This leads to the upper bound in (3.1). 1 % ~

However, the rn sets form an s-wise t-intersecting family, which implies t bat every set

has at least t elements in common with a fixed set, Say the first one. Therefore, the upper

bound can be improved to:

Before summarizing the contents of the next sections on this chapter, we give some

definitions and fix some notation.

Definition 3.1.1 We say that two families Al and d2 of subsets of [1, v] are isomorphic,

denoted by Al - da, if there ezist a permutation r of the set [l, v] , such that n sends A'

to A*. We also mi t e A2 = r(A1).

Notation 3.1.2 Let C be a collection of families of subsets. We denote 6y D(C) the

number of nonisomorphic members of C , that is, the number of equivalent classes indveed

by the equivalent relation given by -.

Definition 3.1.3 Let CL, C2 be collections of families of subsets. We say that collections

C1 and C2 are isomorphism-equivalent, if for every family A' E C1 there exists a family

d2 E CZ such that d2 - A', and vice-versa.

In Section 3.2, we study the relationship between kernels and generating sets of fam-

ilies in Ml"(v, k, t). We introduce the notion of a set of essential elements E ( A ) C [1, v]

for a family A E IS(v, k, t ) . We show t hat for maximal families the concepts of ker-

nels, generating sets, and essential elements are closely related (Theorem 3.2.4). In

particular. for v > 2k - t and A E MP(v, k , t ) , we exhibit generating sets g(A) and

g'(A) of A such that E(A) = U G ~ ( A ) G = U ~ ~ t ( s ( ) G is the unique kernel of A

with minimum cardinality. We also show that for v large enough, more precisely, for

v 2 no(k, s, t ) = max(2k - t + 1, n(k, s, t )), the set formed by the generating sets g'(A)

for al1 A E MF(v, k, t ) is isomorphism-equivalent to the set formed by the generating

sets g'(B) for d B E M16(no(k, s, t), k, t ). Therefore, given k, s and t , in order to solve


Problem 2 for v 2 no(k, s, t ), it is enough to list the generating sets g' for the solutions

obtained for v = no(k, S, t ) .

In Section 3.3, we give a general construction for families in Mr(v + 1, k + 1 , t + 1)

based on families in M P ( v , k, t). This construction produces nonisomorphic families

when applied to nonisomorphic ones. This implies that D ( M P ( v + I , k + 1, t + 1)) 2

D(MP(v, k , t)) , for al1 1 2 0.

In Section 3.4, we solve Problem 2 for s = 2 and k 5 t + 2. The case of k = t + 1

is simple, but is included for the sake of completeness. The solution of Problem 2 for

k = t + 2 is more interesting. We show that the construction given in Section 3.3, when

applied to a11 sets in M f ( v , t + 2, t), gives al1 the sets in M ~ ( V + 1, t + 3, t + 1), for

ail v and t 2 2. This reduces the case k = t + 2 to the determination of families in

MP (no(4, 2,2), 4,2) and ~ k ? ( n * ( 3 , 2 , l), 3 , l ) , which were obtained by cornputer. The

upper bound in (3.2) gives n(t + 1,2, t ) 5 t + 3 and n(t + 2,2,1) 5 t + 20, but we prove

t hat n(t + 1,2, t ) = t +2 and n(t +2,2, t ) = t + 6, for any t 2 1. More precisely, the results

regarding enumeratioo of nonisomorphic sets for the case of k - t < 2 are as follows. For

any

The

The

t >_ 1 we have

form of such sets is given in Theorem 3.4.1. For any v 2 t + 6, we have

generating sets for the correspondhg families are given in Table 3.1.

3,2 Kernels and generating sets

We begin this section by introducing the new concept of a number in [il v ] beiog essential

for a family in Is(v, k, t). Then, in Theorem 3.2.4, we show some properties relating ker-

nels, generating sets, and essentid numbers for families in MF(v , k, t). Ln CoroUary 3.2.5,

we rewrite n(k, s7 t) based on essential numbers for families in MP(v, k, t ) with v > 2k - t.

Then, we exhibit some specid generating sets (given in (3.5)) for families in MPfv, k, t )

CHAPTER 3 CLIQUE FACETS AND INTERSECTINC SET SYSTEMS 62

with v > no(k,s,t) = max(2k - t + 1, n(k,s, t)}, that allows us to show that a solution

for Problem 2 for k, s, t and v 2 no(k, s, t ) can be obtained from the solution for Problem

2 for k, s, t and v = no(k, s, t) (Corollary 3.2.10).

Definition 3.2.1 Let A E Ia(v, k, t ) . We Say that e E [1, v ] is essential for A, if them

ezist sets A l , . . . ,A. with e E Al n . . . n A. such that IAl n . . . n A.1 = t . The s-tuple

(Al, . . . , A,) is called an essential s-tuple for e in A. We also say that e is essential for

Aj relatively to A, for any 1 < j 5 S . The essential set of A is defined by E(A) = {e E

[l , v ] : e is essential for A).

It is clear from the above definition that every kernel of a family A E Is(v, k, t)

must contain E(A). In Theorem 3.2.4, we show that for a family A E MP(v, k, t) with

v > 2k - t, the set E(A) is the unique kernel of A with minimum cardinality. We also

exhibit a generat ing set g( A) of A t hat is formed by elements in E(A) , and t hat is largely

used in the rest of the chapter. For proving Theorem 3.2.4, we need the following two

lemmas. The first one generalizes a known result For s = 2 (see [3]).

Lemma 3.2.2 Let v > 2k - t. lfd is a fatnily in I s ( v , k, t) and g is a genemting set of

A, then g is an s-utise t-intersecting fumily.

Proof. Suppose that g is not s-wise t-intersecting. Then there exist sets Gl, . . . : Gs E g

such that IGi n . . .-n GsI < t . W e will show k-extensions 4,. . . , S, of Gi , . ..,Cs such

that ISl n . . . n S, 1 c t, which is the desired contradiction. Let us caii I = Gl n . . . fi Gs- l .

We observe that O < 111 < k, and that II n GJ 5 t - 1. Let S, be a k-extension of G,

such that IS, n II 5 t - 1. There exists such an extension, since t he numbers available

to be in theset S, amounts t o v - III+t - I z 2 k - t +1 - k + t - 1 = k. Let F C S,,

IF1 = k - t + 1, F n I = 0. Make k-extensions Si,. . .,Ss-l of Ci,.. .,G,-1 such that if

Si = G; U Li, then Li n F = 0. This is always possible, since IF1 = k - t + 1, and so

v-IF1 2 2k- t+ l - (k - t+ i ) = k. Wefirstobserve,since L i n F = @for 1 si 5s-1, that Si n .. .Ti n F = G, n .. . n G,-, n F = I n F = 0. Therefore,

This is the desired contradiction. O


Lemma 3.2.3 Let A be a farnily in MP(u, k , t ) , and for any A E A, let SA = (AnE(A)) .

i f A E A and B [1 ,v ] is svch that B 2 SA with IBJ = k, then B E A.

Proof. We will show that B E A by induction on 1 B \ Al. If 1 B \ Al = 0, trivially

B = A E A. If IB\Al 2 1, then choosex E B\A and y E A\B. Let B' = ( B \ { x ) ) ~ { y ) .

Since B' > SA and IB' \ Al < 1 B \ Al, by induction hypothesis B' E A. Since y is not

essential, B' \ {y) is t-intersecting with every (s - 1)-tuple of sets in A, and so is B.

Therefore, since A is maximal, it follows that B E A. O

The following theorem establishes connections between kemels, generating sets and

essent ial nurnbers for a family A in Ml"(v, k, t ). In particular, we introduce a generating

set formed by essential elements of A, and we show t hat when v > 2k - t the essential

set of A is the unique kernel of A with minimum cardinality. We observe that the

assumption v > 2k - t is not so restrictive. For example, when s = 2, if v 5 2k - t, then

the family formed by every k-subset of [1, v ] is pairwise t-intersecting, and therefore is

the only farnily in MP(v, k, t).

Theorem 3.2.4 Let A Qe a farnify in MP(v , kt 1 ) .

I . Let us denote bg g(A), the farnily gzven by

g(A) = { ~ n E(A) : A E A).

Then, g ( A ) is a genemting set for A.

2 For any kernel X: of A we have E ( d ) C K.

(a) for any generating set of A, the set Uce G is a kemel of A;

(6) the set E(A) is a kernel of A; monover, E(A) is the unique kemel of A with

minimum cardinalit y.

Proof. Part 1 . Trividy A C U,f(g(A)), so it remains to prove that Ut(g(A)) C A. Take

B E u ~ ( ~ ( A ) ) . Then B = S U L with S E g(A) and L C [1,v] \S. Since S E g(A) , there

exists A E A with S = A n E(A). Therefore, since B _> S = An E(A), by Lemmo 3.2.3,

B E A*

Part 2. Let e E E(A) and IC any kernel of A. Since e is essential for A t here exist

sets Al , . . . ,A, E A such that e E Ai n.. .O A. and (Al n.. .n A.1 = t. In addition, since

K i s a kernelofd, IX:nAln ... nA.12 t a n d s o e ~ n .

Part 3a. Let be a generating set of A. Since v 2 2k - t, it follows by Lemma 3.2.2

that g is an s-wise t-intersecting family. Let A: = Ucc G. Let A l , . . . ,A, be arbitrary

sets in A. Since g is a generating set, there exist sets Gi, . . . , Gs E 3, with Gi C Ai for

1 5 i 5 S. So, we have

I ICnAin ... nA.1 2 IKnG, n...nG,I = IGln ... nG.1 ( by construction of K)

2 t (since g is s-wise t-intersecting).

Therefore, the set K is a kernel of A.

Part ab. In Part 1, we showed that g(A) is a generating set of A. By constructioo,

we see that E(A) = UoEg(a) G. Since v 2 2k - t , it follows by Part 3a that E(A) is a

kemel of A. By Part 2, for any X3 kernel of A, E(A) C K, so E(A) is the unique kernel

of A with minimum cardinality. 0

A consequence of this theorem is that we can rewrite n(k, s, t ) in terms of essential

sets of families in M f ( v . k, t ) with v > 2k - t .

Corollary 3.2.5 Let k 2 t >_ 1, s 2 2, and let Emw(k, s, t ) be giuen by

Then, n(k, s, t ) = E,,(k, s, t ) .

Proof. The observations below will be used in the proof:

1. Is(v', k , t ) E f s (v ,k , t), for any v 2 ut.

2. Let A, Ar E P(v, k, t ), Ar C A. If );: is a kernel of A, then K is a kernel of A'.

3. Let A, A' E Is(v, k, t ) , A' C A. If K' is a kernel of A', then there exist X: > K' such that AG is a kernel of A,


Usiog the above observations, dong with part 3b on Theorem 3.2.4, we have:

n(k , s , t ) = rnax min lKl, (by definition of n(k, s, t ) ) u'k,AEls(u,k,t) K kcmcl of A

- - max min /ICI, (by observation 1) v>ak-t.&P(~,k,t) K kernel of A

- - max min 1 , (by observations 2 and 3) u>2k-t,AEMP(u,k,t) K kernel of A

- - max v>2k-t& MP (u,k,t)

1 E(A)I, (by part 3b in Theorem 3.2.4)

O

The rest of this section deals with finding a special generating set for families in

MP(v , k, t ) t hat is also a generating set for families in MP(v', k, t), for any v' 2 v. The

following is an example of a family A E MP(8,4,1) for which g(A) is not such a special

generating set. For v' 2 9, the family U:,(g(A)) is not maximal.

Example 3.2.6 Let g1 = {{1,2,3,4),(1,5,6,7), {3,5,6,7), {4,5,6,7), (2,5),{2,6),

(2 ,711 . The set A = U:(gl) i& in ~ P ( 8 , 4 , 1 ) , and E(A) = [l, 71. The set g(A) as

defined in (3.3) is the jollowing:

The set A' = U:(g(A)) is not maximal in f2(9,4, l), since, for example, the set B =

{2,5,8,9) intersects euery other set in A', but B $! A'. Howeuer, there exists o generating

set of A, nurnely g', that has the speeiul property that UU(g) E 12(v, 4,1) is mazimul for

al1 v 2 8.

We now introduce a new generating set.

Proposition 3-23 Let A be a fomily in MP(v, k , t ) , and g(A) be the generating set

introduced in (3.9). Denote byg'(A) the family @en by

- - gr(A) = {G E G : G E g(A), and (G) u g(d) is s-wise t-intersecting}. (3-5)

Then, i f v > 2k - i, the family g'(A) LP a genemting set for A.

CHAPTER 3 CLQUE FACETS AND ENTERSECTING SET SYSTEMS 66

ProoJ Since g(A) g g'(A), we have A = u ~ ( ~ ( A ) ) 2 ~ t ( g ' ( A ) ) . We have to show

Uy(g'(A)) c A. Let B E ~t(g ' (A)) . If Al,. . . , A.-1 E A, there exist GI,.. . , E

g(A), such that Ai > G i , . . . , A,-l > G,+ Also, B > G, for G such that {G) u g(A)

is Y-wise t-intersecting. So IAl n . . . n A,-l n BI > IG1 n . . . n GS-! n GI 2 t . Therefore

{B) u A is in I s ( v , k, t), and since A is maximal, we conclude that B E A. O

In the next theorem, we show that the generating set g'(B) can be used to produce

maximal families b s larger v's.

Theorem 3.2.8 Let v > vo 2 2k - t + 1, and let B be a jamily in Mls(vo, k, t ) . D e f i e

A to be a family in Is(v, k, t ) giuen by A = U,k(g'(B)). Then

2. the fomily A is rnuximal in I s ( v , k, t ) ; and

Pmof. Part 1. Given that B C A, we see that E ( B ) Ç E(d). So, we only need to

show that E ( A ) Ç E(B). Let e E E(A). Then, there exist Ai, ..*?As E A with

e E Ai n . . . n As and /Al n . . . n AsI= t. By construction, Ai 2 G; for some Gi E g'(B),

for 1 5 i 5 S. Using the fact that g'(B) is a generating set for B and Lemma 3.2.2, we

pet that IGl n . . . n GsI 1 t . Therefore, we must have e E Gl n . . . Cs C E ( B ) .

Part 2. Clearly A E Is(v, k, t). It is left to show that A is maximal. Let C E [l, v ] ,

with (CI = k and such that A u {C) is s-wise t-intersecting. Considering that g'(B) is a

generating set for A, we conclude that g'(l3) U {C} is a generating set for AU (C). Thus,

by Lemma 3.2.2, g'(B) U {C) is s-wise t-intersecting. Given that UCEP#(BIG = E(B) ,

it follows that g'($) u {C n E ( B ) ) is s-wise t-intersecting. It is easy to prove that

CnE(B) E g'(B). Indeed, let B be any k-extension of Cn E(B) contained in [l, vol. Thus,

B u {B) is s-wise t-intersecting, and since B is maximal, we have B E B. Considering

that C n E ( B ) c .B n E(B) and that g'(B) U (C n E ( B ) } is s-wise t-intersecting we

conclude that C n E(B) E g'(B). Therefore, any k-extension of C n E(B) contained in

[l, VI belongs to A, by definition; in particular, the set C E A.


Part 3. First, we show that g'(B) C g'(A). Let G E $(B). Let A be any k-extension

of G contained in (1, v ] ; since A is maximal, it fotlows that A E A. Observing that

G Ç E(B) = E(A), we conclude that G C A n E(A). Considering that {G) U g ( A ) is a

generating set for A, by Lemma 3.2.2, (G) U g(A) must be s-wise t-intersecting. Thus,

by the definition of g'(A), we conclude t hat G E g'(A).

Finally, we show tbat g'(A) S g'(B). Let Cr E gl(A). Then, by definition, Gr E A n E(A), for some A E A, and {C) u g(A) is s-wise t-intemeeting. Considering that

B C A, we see that g(B) C g(A). This implies that (G1) U g(B) is s-wise t-intenecting.

We just have to show that G E B n g(B) for some B E B. Let B be any k-extension

of A n E(A) contained in [l,vo]. As G B, we know that B E A. This implies B is

a k-extension of some G E g'(B), by constructioo of A. In addition, B C [1, vol, which

implies B E B. We conclude the proof by observing t hat C = Gr n E ( B ) Ç B n E ( B ) . 0

We show that the generating set g'(A) for a family A is the same as a generating set

g l ( B ) for some family B with smaller v , provided that v is large enough.

Theorem 3.2.9 Let vo 2 mw(2k-t + 1, n(k, s, t ) ) , and let A be a family in M P ( q k, t ) ,

&th v 3 va. Assume w.1.o.g. that E(A) = [l, 1 E(A)I].

Let B = 246 ((T'(A)). Then

2. the family B is maximal in IS(vo, k , t ) ; and

Proof. Part 1. Since B C A, it is clear that E(B) E(A). We have to show that

E(A) E ( B ) . Let e E E(A), then thereexîst sets Al, . . . ,A, E A with e E A i n . . .nAs

and IA i n . . . n A s I = t . Let I = A i n ... nA,. Consider Gi = A i f i E(A),for 15 i s s .

Then Gi, . . . , Gs E g(A), and 1 C Gl n . . . n G,. Thus (Gil 2 111 = t. Moreover,

Gi, . . ., Gs E g'(A), so U&({GI,. . ., (2,)) Ç B . We choose the foilowing k-extensions of

Gi, . . . , Gs contained in [1, q]:

BI = Gl U Li, for some LI E [1,vO] \ (Gi u (G2n ... n G,));


Bi = Gi u Li, for some Li C [I,vo] \ (Gi u Bi), i = 2,.. . ,S.

Thereexists always such an Li, since 1[1,vo]\(Gl~(G2n.. .nG,))I > vo-(IGl(+/4n.. .n

Gsl-I1l) 5 2k-t+l - (IGII+k-t ) 2 k-IG1l. For 2 5 i 5 s, thereexists always such an

Li, since I [ I , v ~ ] \ ( G ~ u B ~ ) ~ = vo-(IGil+IBiI-lGinB1l) >r 2k-t+l-lGil-k+t 2 k-1G;I.

Therefore BI, . . . , Bs E B and

= (Gi u Li) nG2 ... n G, (by the choiceof Lz,.. ., L,)

= ( ~ ' n c ~ n ... n ~ , ) u ( ~ ~ n ~ ~ n . . . n ~ . )

= 1.

So e E BI n . . . n B, and IBl n . . . n BsI = 111 = t . Therefore e is essential for B.

Part 2. One can easily check from B = U . (g'(A)) and U O ~ ~ ~ ~ A , G C [l , 1 E(A) I] 2 [l , vol,

that B E IS(uo, k, t). We only have to prove that B is maximal. Let C E [l , vol such

that 8' = B U (C}. is s-wise t-intersecting. Theo, g = g'(A) U {C) is a generating set

of 8'. By Lemma 3.2.2, the set 3 is s-wise t-inteaecting, which implies that for any

GI,. . . ,G,-1 E g'(A), we have IGI n .. . n G,-i n Cl 2 t. So, A u (C) is s-wise t-

intersecting, and since A is maximal we must have C e A. Therefore, Cn E(A) e g'(A),

which implies C E B.

Part 3. Let us first'prove that g'(A) C g'(B). Let a E g'(A). Then, G 2 A n E(A) , for

some .4 E A, and g(A) U {G} is s-wise t-intersecting. Since B C A and E(B) = E(A),

it foUows that g ( B ) C g(A). Thus, g ( B ) u {G) is s-wise t-intersecting. in addition,

B = CUL E B, forany L C [i,vo]\G. Then, BnE(B) = ( & L ) ~ E ( A ) = G u ( L ~ E ( A ) ) . So, G C B n E ( B ) for some B E B, and g ( B ) U {G} is s-wise t-intersecting. Therefore, - G E dB)*

We must show that g'(l3) E $(A). Let G E g'(B). Thus, G Ç B n E ( B ) , for some

B E B, and g ( B ) U {G) is s-wise t-intersecting. Given that B C A and E(B) = E(A),

we conclude that G B n E(B) = B n E(A) , with B E A. Let Gi, . . . , E g(A).

Considering that gtd) C g'(d) C g'(B), we see that G1,. . . , G,-i E g'(B). Using the fact

that g f ( B ) is a generating set of B and vo > Zk - t , by Lernma 3.2.2, we conclude that


g'(B) is s-wise t-intenecting, and in particular that IGn Gi n . . . n GS-* 1 2 t . Therefore,

g ( A ) u {G) is s-wise t-intersecting, which implies G E g'(A). O

The following corollary implies that the solution of Problem 2 for k, s, t and v 2

no(k, s, t ) can be obtained from the solution of Problem 2 for k, s, t and u = no(k, s, t ).

Corollary 3.2.10

Let G(v) = {gt(A) : A E MP(vT k, t ) ) and no(k, s , t ) = max{n(k, s, t ) , 2k - t + 1). If

VI, v2 2 no(k, s, t ) , then G(vl) is isornorphisrn-equiualent to G(v2).

Proof. Assume w.1.o.g. that vz 2 V I . Theorem 3.2.9 establishes that for every fam-

ily A E MP(v2, k, t), there exists a family B E M P ( v I , k, t ) such that g'(B) -- g'(A).

Theorem 3.2.8 shows that for every farniiy B E MP(viT k, t), there exists a family

A E M P ( v 2 , k, t ) such that g'(A) = g'(8). O

Remark 3.2.11 For A E M P ( v , k , t ), the generating set g'(A) is not necessarily minimal

with respect to set inclusion. For representing A, we rnay ako use the folioroihg generating

set:

s'(A) = {G E gt (A) : for any Gi E g f ( A ) , Gi 2 G implies Gi = G}.

It is easy to check that #(A) is a generating set of A, and that gt'(A) sat&$es the same

properties as g'( A), giuen in Theorem 3.2.9 and Corollary 3.2 IO.


3.3 A construction of families in MP(v + 1, k + 1, t + 1)

using MF(v , k, t )

In this section, we show how a farnily in MP(v , k, t ) can be used to construct a family

in MP(v + 1, k + 1,t + l) , for any s, v , k and t. We also show that if the construction is

applied to two nonisomorphic families, then the resulting families are also nonisornorphic

(Theorem 3.3.5). in addition, for v > 2k - t, this construction increases the number of

essential numben by exactly one ( Proposition 3.3.8).

The idea behind the construction is as follows. Let A E MP(v,k,t). The family

A(,,+1) obtained by adding the element v + 1 to every set in A is obviously a family in

Is(v + 1, k + 1, t + 1 ), but not necessarily maximal. The construction adds appropriate

sets to A(,+I) in order to obtain a family B that is maximal. In fact, we show that B is

the only maximal family in M P ( v + 1, k + 1, t + 1) that contains

Construction 3.3.1 Let d be a farnily ni M P ( v , k , t ) . Let = { A U {V + 1) : A E

A). For any A E A, denote £(A) = {(Ci,.. .,CI,-l) E A : ICI fl .. . n Cs-l n Al = t),

""d L(A) = (n,C,,**.,c,-,),,(,,(cl n ---n CS-,)) \ A* Let, for any A 4 E A?

. . ( { A U {i) : i E [l,v] \ A) if E(A) = a. '

and A* = U a E ~ A'( A). Dejïne B to be the Jamily A(,+il U A'.

Theorem 3.3.2 Let A be a farnily in MQv, k, t ) . Then the family B = A(,+l) U A'

given b y Construction 3.3.1 k a family in Mls(u + 1. k + 1, t + 1). itloreover, family B is

the only family in MP(v + 1, k + 1, t + 1) that contains the subfamily d(u+l).

Proof. First, w e will prove that B E Is(v + 1, k + 1, t + 1). It is clear that B is (k + 1)-

unifom, so it remains to prove that 8 is s-wise (t + 1)-intersecting. Let Bi, . . . , B, E 8,

we need to show t hat 1 BI n . . . n Bs 1 2 t + 1. We split the analysis into two cases.

1. If Bi,. . ., Bs E A~v+l), then it is trivial that lBi n . . .n BsI 2 t + 1.


2. If for some i, 1 5 i 5 s, Bi E A', we assume w.1.o.g. that Bi E A*. We

know that there exist Al,. . . , A, E A such that Bi > Ai, for 1 < i 5 S. If

IAl n . . .n A.1 2 t + 1, then we are done. This includes the cases when &(Al) = 0,

for in such case IAl n CI n ... n Cs-ll > t + 1, for any Cl ,..., Cs-1 E A. When

IAl n .. . n A,I = t, we have E(Al) # 0, and by the construction of &(AI ) , we

must have BI = Al U { 1 1 ) , with II E t(Al). But, by the construction of L(Ai), we

must haveil E ( A z n ... nAs) \Al then IBin ... nB,I 2 IAIn ... nA,I+l = t + l .

W e must show that B is a maximal family in I (v+ 1, k+ 1, t + l), and B is the only maximal

family in I(v + 1, k + 1, t + 1 ) t hat contains the subfamily A("+,). Let S C [l , v + 11 be an

arbitrary (k + L )-set t hat is s-wise (t + I )-intersecting wit h any s - 1 sets BI, . . . , E

A(u+ll. We prove both of the above statements by showing that S E B. If v + 1 E S,

then S\ { v + 1) must be s-wise t-intersecting with any s - 1 sets Ai , . . . , A,-1 E A. Since

A is maximal, S \ { v + 1) E A, and so S E B. Lf v + 1 $ S, then let a € S and cal1

S. = S \ {a}. Thus { S ) U A is s-wise (t + 1)-intenecting, which implies that {Sa) U A

is s-wise t-intersecting, and then Sa E A. If &(Sa) = 0, by construction of &(S.), we

have S = S. U { a ) E 8. Otherwise, if £(S.) # 0, for any (Cl, ..., Cs-1) E E(S.), we

have (Cl n . . . n Cs-I n S.1 = t , but ICl n . . . Cs-i n SI 2 t + 1, and so we must have

o E Cl n . . . n Cs-,. Therefore a E C(S.), and then S = Sa U { a } E B. O

Example 3.3.3 Let A E ~ f ( 6 , 3 , 1 ) giuen by the /irst colvrnn of the following table.

The set B = A(i) U &(A) is in hlP(7,4,2).

We introduce some notation that wiU be used in the next theorem, as weU as in the

sections to fouow.

Notation 3.3.4 For an y family B of subsets of [l , v ] , and for any x , y, z E [I, v] , we

deno te:


The following theorem shows that if Construction 3.3.1 is applied to nonisornorphic

families, then the resulting families are also oonisornorphic.

Theorem 3.3.5 Let d',A2 E MP(v, k , t ) , and let BL and B2 be the families in Mf ( u + I , k + 1, t + 1) given by the application of Construction 3.3. i to the families A L and A*,

respectively. Then B' - B2 implies A' -- A*.

Proof. Let R be the permutation on [l, v + 11 such that Ba = n(B1). I f n(v + 1 ) = u + 1,

then trivially d2 = n ( A 1 ) . Otherwise, we can assume w.1.o.g. that n(v + 1 ) = o. Let us

caii B = Ba = rr(B1.). For a faIILily C we denote C \ {x) = {C \ { x ) : C E C); we c m then

wri t e:

W e want to show r (A1) - d2. It is clear that Bv,u+l \ {v) - \ {v + 1). It is enough

to prove that Bu,- \ { v ) = &,,+I \ { v + 1).

We claim that for any A E Bu,= \ { v ) , the set {A) u A* is s-wise t-intersecting. By

definition, d2 is S-wise t-intersecting. Let Al,. . . , As-I E d2 and let Bi = U {V + 11,


- for 1 5 i 5 s - 1, B = A U { v ) . Observe that BI ,..., B.+ B E B. Let us cal1 IA =

Ain...nA,-in&tnd ig = B i n . . . n B . - l n ~ . First, weobserve that iA = i g \ { ~ , u + I ) ,

and s inceË E Bv,=, we know that v + 1 Ig. So llal = Ile \ {v)l > ( t + 1) - 1 = t.

So, {A) U d2 is s-wise t-intersecting.

Then, since da is maximal it follows that A E A*. Therefore, B,;, \ { v ) E A*,

which rneans \ {v) C &,v+i \ { v + 1). By an analogous argument, we show that

%,,+1\ ( V + 1) C Bu,;« \ { v ) . Therefore, B+++i \ {v + 1) = B,,;F~ \ {v ), which completes

the proof. O

An immediate consequence of the previous t heorem is that , for any s 2 2, the number

of nonisomorphic families is nondecreasing, when parameters v , k and t are increased by

one. Recall the meaning of D(*) from Notation 3.1.2.

Corollary 3.3.6 For any s, v , k and t 2 1, we have

The following l&ma is a well knowo result for s = 2 (see, eg. , [33]), saying basicaily

that, for v 2 2k - t , if A E MP(v, k, t ) , then A $ MP(v , k, t + 1).

Lemma 3.3.7 Let v > 2k - t and A E hiP(v, k, t ) . Then E(A) # 0.

Proof. Suppose to the contrary that E( A) = 0. Then, for every s sets Al, . . . , A, A, we have !Al fi ...fi As[ 1 t + 1. Let BI ,..., Bs E A such that q = IBi n ... n BsI is

minimum. Let I = Bi n . ..n We know that q = IIn B.1 2 t + 1. Let a E I n B,,

and take = (B. \ { a ) ) U ( 1 ) for some 1 E [l, v] \ (B. U 1). There exist such an 1, since

v - IL?, U II 2 2k - t - (2k - ( t + 1)) = 1. We daim B is s-wise t-intersecting with any

other s - 1 sets Ai,:. . , A,-1 E A, since B is (t + 1)-intersecting with Ai,. . . , A,+ Since

Ais maximal, thenBe A- But IBl n ... n BSdl nBI = II dl = IIn Bsl - 1 = q - 1,

which contradicts the fact that q is a minimum intersection size among any s sets in A.

O

We show that when we apply the previous construction for v 2 2k - t , the number

of essential numbers inmeases by exactly one.


Proposition 3.3.8 Let v 2 2k - t . Let A be a fumily in M f ( v , k, t ), and let the farnily

B E MP(v + 1, k + 1, t + 1 ) 6e the family given by Construction 3.9.1. Then E ( B ) =

E(A) u {v + 1).

Proof. It is easy to prove that E(A) U { V + 1) E ( B ) . Indeed, for any sets Si,. . . , S, E

A, with ISi n ... n SsI = t, the sets (Si U {v + 1)) ,...,(S. U { v + 1)) E B, have

I(Si u ( v + 1))n.. .n (S.U{V+ 1))l = t + 1. This shows E ( A ) C E(B). By Lemma 3.3.7,

t here exist such sets, and so v + 1 is essential for B.

We prove that E ( B ) S E(A) U { v + 1). Let e E E(B), e + v + 1. We must show

e E E(A). Since e E E(B), there exist Bi ,..., B. E B with e E Bi n ... n B., and

I B ~ ~ ... n s , l = t + i 2 2 .

If v + 1 E BI n . . . n B., then the sets Bi E A(,+i) for al1 1 5 i 5 S. Let Ci =

Bi \ {v + 1 }, for I 5 i 5 s; t hen, Ci, . . . Cs E A. Therefore, we have Cl, . . . , Cs E A, wit h

e ~ C ~ n . . . n C , and IC ln. . .nC,I = IB l n . . . n BJ- 1 = t , whichimpliese~ E(A).

If v + 1 # Bi n ... n B,, then let f E Bi n. .. n Bs, such that f # e, and define the

sets Ci for 1 5 i 5 s, as follows:

( Bi \ {f 1 1 ot herwise.

We daim that Ci E A, for al1 1 i 5 S. Indeed, if v + 1 E Bi then Bi E d(v+i), and so

Ci = Bi \ { v + 1) E A. Otherwise, if v + 1 $ Bi, the fact that { B i ) U A(v+l) is s-wise

( t + 1)-intersecting impiies { B i ) U A is s-wise (t + 1)-intersecting. This implies that

{C* = Bi \ {/}} U A is s-wise t-intersecting, and since A is maximal, we conclude that

Ci E A. We know that ICln...nC.I = I(Bln ... n Bs)\{f)l = t, and e E Cln...nCs,

which impiies e E E(A). CII

3.4 Al1 maximal k-uniform pairwise t-intersect ing fam-

ilies for k 5 t + 2

Ln this section, we solve Problem 2 for the case of s = 2 and k = t + 1, t + 2. The case

of kt = t + I is simple, but we include it here for the sake of completeness.

CHAPTER 3 CLIQUE FACETS AND INTERSECTINC SET SYSTEMS

3.4.1 Determination of ail families in M ~ ( v , t + 1, t )

In this subsection, we determine al1 families A E ~f ( v , t + 1, t ) for arbitrary v > t + 2,

and show that n(t + 1,2, t ) = t + 2, for al1 t 2 1.

We will use the following notation, for any k and v: (L) = { B [ l , v ] : 1 BI = k).

Proposition 3.4.1 Let t 2 1 and v 2 t + 2, and A E ~ l . ' ( v , t + 1, t). The following

hold:

Pm06 Part 1. The set ([:::J) containhg every possible subset is t-intersecting.

Part 2. Let A E @ ( v , t + 1, t ) , with v 2 t + 3. First, we observe that any distinct sets

Al, .A2 E A sat isfy IAi fi A21 = t . We analyze the following two cases.

Case a: Suppose that there exist distinct sets Ai, A*, A3 E A with IAl n Al n A3/ = f.

Assume w.1.o.g. that At = [l, t] u {t + I ) , A2 = [l, t ] u { t + 2) and A3 = [1,t] u {t + 3).

It is easy t o conclude that any A E A that is pairwise t-intersecting with At, A2 and A3

must contain [l, t], so A C B'. Since A is maximal then A = 8'.

Case b: Suppose t hat 1.4i n A2 n A3( < t , for al1 distinct sets At, A2, & E A. We can

assume w.1.o.g. that A = [l, t ] U { t + 11, B = [l,t] U { t + 2) E A. For every C E A,

C # A, B, we must have ICn [l, t ] l = t - 1 and C 2 { t + l , t +2). Therefore, A Ç ([::il), and since A is maximal, then A = ([::$. O

Remark 3.4.2 The generoting sets g'(A) for any set A E MP(v, t + 1, t ) and v 2 t + 3,

are isomorphic to one of the families:


Corollary 3.4.3 For t 2 1, we have n(t + 1,2, t) = t + 2.

Proof. From Corollary 3.2.5, we have n(t + 1,2, t ) = E,,,,(t + 1,2, t ), and by Proposi-

tion 3.4.1 we get &,(t + 1,2, t) = t $2. O

3.4.2 Determination of al l families in MP(v, t + 2, t )

In this subsection, we show tha t Construction 3.3.1 is enough to generate aii the sets

in M f ( v + 1,t + 3,t + 1) when applied to al1 the sets in ~ P ( v , t + 2 4 , for any v

and t 2 2 (Theorem 3.4.6). We also show that n(t + 2,2, t ) = t + 6 (Theorem 3.4.8).

Using these two results combined with results frorn previous sections, we manage to

reduce Problem 2 (for any s = 2, v, t and k = t + 2) to the solution of Problem 2

for (s, u, k, t ) E {(2,7,3, l ) , (2,8,4,2)} (Theorem 3.4.9). This solution is obtained by

cornputer and shown in Table 3.1.

Thtoughoue this section, we will largely use Notation 3.3.4. In addition, for any

A E Is(v, k, t), and a E [1, v], we denote by na the number of sets in A containing a, that

is na = Idal.

Lemma 3.4.4 Let. A be a family ni M f ( v , t + 2, t ) , for t 2 3. Let x E [l , v] such that

n, = ma%e[l,ul ni Then the fornily At is p o i h e ( t + 1)-intersecting.

Proof. It is clear t hat & is pairwise t-intersecting, since & C A. In order tc prove that

& is, in addition, pairwise (t + 1)-intersecting, we suppose to the contrary that there

exist Ai , Az E A-, such t hat 1 Ai A2 1 = t . Then Ai and A2 must be of the form

for distinct y*, y2,. . . ,y*, al ,& a*, Pz, different from x.

For any i, since ngi 5 n, we must have

lkgl 2 I&ZI 2 2-

W e claim that cannot be pairwise ( t + 1)-intersecting. Indeed, if it was so, it

would irnply that the family obtained by replacing x by yi in A-, denoted by 4- -


{ x ) + {yi), would t-intenect al1 other sets in A. Then, since A is maximal, we would

have 4T, - ( 2 ) + {yi} E A. Consequently, we would get (4, > 1 - (z} + {yi)) U

{ A I ) U {A2) ( = JAy7J + 2 > J.AyTPI, contradicting (3.6). Therefore, for i = 1,. . . , t ,

Since B;, Bi E &, and must also t-intersect both Al and A*, they must be of the

fo rrn

with Z1,Zi E {{al, a2), {aI, PZ), {&, al), (Pi, ,û2)). Condition (3.7) translates into Zf n

2: = 0, For al1 i = 1,. . . , t, which implies {Z:, 2:) equals either {{rri,a2), {Pi,P2)) or

{{crl,&), {/?l,m}}. Since t 2 3, by the pigeon-hole principle for some il # i2, we have

(@, @) = {p, Q). So, w.l.o.g., Z;' = z?, which implies 1 B:' n ~ $ 1 = t - 1 < t,

i.e., a contradiction. O

Lemrna 3.4.5 Let A be a farnily in MP(v, t + 2, t ) , for r 2 3. Let x E [Il v ] such thal

n, = mq~ti,,,j ni. Then for any B E A, and for any y E B we have ( B \ { y ) ~ { x } ) E A.

Prooi. Let B € A, and y E B. Let A E A. If A E A, then [ A n ( B \ { y ) ~ { x } ) ( 2

IA ri BI 2 t . Othenvise, if A E 4, by Lemma 3.4.4, we have IA n BI 2 t + 1, which

implies IA n ( B \ {y) u { x ) ) l 2 t . Since A is maximal, we must have ( B \ {y} u (3)) E A.

O

The following theorem implies that, for any t 2 3, the sets in ~W(V, t + 2: t ) are the

ones obtained from maximal sets of M f ( v - 1, t f 1 , t - 1) using Construction 3.3.2.

Theorern 3.4.6 ~ è t A be a farnily in LW&, t + 2, t ) , for t > 3. Let z E [I, v] such that

n, = rnaq,,,, ni. Then, the family Ar= A,\ (x) LP a fumiiy in MP(V - 1,t + ~ , t - 1).

Proof. By definition, Ar E 12(v - 1, t + 1, t - 1)- It remains to prove that A' is maximal in

12(v-I , t+l , t -1)- Let L'E [1,v]\{x),with ILf[ = t+1, andsuch that IL'nAl 2 t - 1 ,

for ail A E A'. Denote L = Lr U (2). We must prove that L' E A' or equivalently that


L E A. By the maximality of A , its enough to prove that for any B E A, we have

IB n L I 2 t .

Let B E A. [f B E A, then 8 \ { x ) E A', which implies ( B \ {x} n L'( = t - 1

and trivially (B n LI 2 t . So, assume 8 E dZ. We claim that B n L' # 0. Indeed, let

Ar E A'. Then IA'I = t + 1, IA 'n BI 2 t 2 3 and IA' n L'I 2 t - 1 . Then,

Let y E B n L'. By Lemma 3.4.5, (B \ (y) U {x)) E A and so (B \ (y}) E A'. By

definition of L', we have I(B \ {y)) n L'I 2 t - 1. Since y E B n L', we get IB n L'I 2 t ,

and finally IB n L 1 > t . O

We examine the consequences of the previous theorem on the size of kernels for s = 2

and t = k - 2 2 2 .

Corollary 3.4.7 Let t = k - 2 2 2. Then n(t + 2 , 2 , t ) = n(4,2 ,2) + t -2 .

Proof. Let v 2 t + 5 and let B E ~ l l ( ~ , t + 2 , t ) . By Theorem 3.4.6, there exist A E

M12(v - ( t - 2 ) , 4,2) such that each l3 is obtained by t-2 applications of Construction 3.3.1

to A. Thus, by Proposition 3.3.8, we have IE(B)I = IE(A)I + t - 2. Finally, using

Corollary 3.2.5, we get n(t + Z,2, t ) = n(4,2,2) + t - 2. O

Finally, we just need to determine n(3,2,1) and n(4,2,2) in order to obtain n(t+2,2, t )

for any t 2 1.

Theorem 3.4.8 For any t 2 1 , we have n(t + 2,2, t ) = t + 6 .

Proof. First, we observe that n(3 ,2 ,1 ) > 7 and n(4,2,2) 2 8, since families Ga E

MI2(?, 3.1) and Hg E M12(8? 4.2) (see Table 3.1) are such that E(Gs) = 7 and E ( H s ) =

8. In addition, if n(4,2,2) 5 8, using the fact that n(k - I,s, t - 1 ) 5 n(k , s , t ) - 1 , we

get n(3,2,1) 5 7 . This implies n(3,2,1) = 7 and n(4,2,2) = 8, which combined with

CoroUary 3.4.7, leads to n(t + 2,2, t ) = t + 6, for t 2 1. Thus, it is enough to show that

n(4,2,2) 5 8.

Let A E MP(v , 4,2) for arbitrary v 2 8, and let A, B E A be an essential pair for A

(there exit such a pair by Lemma 3.3.7). We can assume w.1.o.g. that A = {1,2,3,4)


Figure 3. L : Coloured edges of graphs Ç1, Ç2 and Ç3.

and B = (1,2,5,6}. We can also assume that A has no set of the form {1,2, a, b ) with

a, b E [I, v ] \ [1,6], for this would imply that {l, 2) must be containeci in every other set

in A and t hen 1 E(A) 1 = 2. Therefore, the remaining sets that are possibly coatained in

A and that include an elernent x 4 [1,6] are of the fom:

Therefore, any essential element x E (7, v ] must have an essential pair chosen from sets in

(3.8) and (3.9). We will show there are at most 2 such essential elements, which implies

I E ( 4 18.

Define a graph Ç with vertices corresponding to the possible triples accornpanying an

essential element x E 17, v], given by (3.8) and (3.9), and edges placed whenever the triples

corresponding to its endings intersect. Colour the edges whose t n p l e ~ comsponding to

its endings intersect in exactly 1 element. For each essentiai element x E [7, v] for A, let

e(z) be an edge of Ç connecting some triples Tl and Ta such that Tl U { x ) and T2 U ( 2 )

is an essential pair-for A. Let Er = { e ( x ) : x E E(A) \ [1,6]). Let Ç' be the subgraph

of induced by the set of aiI vertices that are endings for some edge in E'. Since A is

>intersecting, Ç' is a complete graph whose coloured edges fonn a matching. We will

show that such an Er must satisfy (E'I 5 2, which implies IE(A)I 5 8.

We reduce the size of the subgraphs we have to look at, by observing that another


I Cenerat ing sets for dl nonisomorphic families in M 1 ' ( v , 3, L ), for v 2 7

1 Generating sets for al1 nonisomorphic families in M12(v, 4,2), for v 2 8 1

1 L.15 11 one application of Construction 3.3.1 to C 1

1 1 ..17 11 ( t - 2) applications of Construction 3.3.1 to Hi 1

17

Table 3.1: AH generating sets for s = 2 and k = t + 2.

-

(1,2,3,7) {1,4,5,7) (2,4,6,7) {3,5,6,7) {1,2,5,6) (1,3,4,6) {2,3,4,5)

Cenerat ing sets for nonisomorphic famiLes in M ~ ( v , t + 2, t ) , for t 2 3, v > t + 6


Figure 3.2: Determination of al1 maximal families for s = 2, k = t + 2, for dl t 2 1

(see Theorem 3.4.9). p p p p p p p p p p p p - - - - - - - - - - - - - - - -

- - - - - -

essential pair for 2 E [7, VI, must be equivalent to one of the following cases: Ci =

(1,3,5,+), Di = {1,2,4,~); Cz = {1,3,5,2), 4 = {1,4,6,2); or C3 = {1,3,5,2),

4 = {2,3,6,2). For i = l,?, 3, we analyze the case of e(z) joining WC, and q, the

vertices corresponding to Ci and Di, respectively. Let Gj be the subgraph of Ç that is

induced by WC,, wpi and every vertex w that have noncoloured edges incident to both

WC, and WD,. The edges in Er are coloured and are contained in 6 for some 1 5 i 5 3.

In Figure 3.1, we show graphs &, G2 and Ç3 with thick edges representing the coloured

edges. By inspection, we conclude that 1 E'I 5 2. O

The following theorem is a combination of many of the previous results.

Theorem 3.4.9 The nonisomorphic families in ~ f ( 8 , 4 , 2 ) and in ~ P ( 7 , 3 , 1 ) corn-


pietefy detennine of1 nonisomorphic families in Mp(v, t + 2 , t ) , for ai l v 2 t + 6 and

t 2 1.

Proof. Suppose we have listed al1 nonisomorphic families in MP (8,4,2) and MP (7,3,1).

We outline how to obtain all nonisornorphic sets in Mla(v , t +2, t ) for rvbitrary v 2 t +6,

t 3 r . If t = 1 , by our assumptions, we have listed al1 nonisomorphic families in M P ( ~ +

6,t + 2, t ) = ~ P ( 7 , 3 , 1 ) . If t 2 2, perform t - 2 applications of Construction 3.3.1 to

each of the nonisomorphic sets in ~f (8,4,2). By Theorem 3.3.5 and Theorem 3.4.6, this

gives al1 nonisomorphic sets in MfL(8+( t -2 ) ,4+( t - 2 ) , 2 + ( t - 2 ) ) = M t 2 ( t +6 , t+2 , t ) .

B y Theorem 3.4.8, we see that no(t + 2, t ) = t + 6. Corollary 3.2.10 implies that

computing generating sets g'(A), a s defined in (3.5), for al1 nonisomorphic sets in M P ( t + 6, t + 2, t ) gives generating sets for all nonisomorphic sets in MP(v, t + 2, t ), for v 2 t + 6.

O

A complete list of nonisomorphic sets in MP(8, 4,2) and in MP(? , 3 , l ) is given in Ta-

ble 3.1.

Chapter 4

Polyhedral Algorithms for Packings

and Designs

Cornputational methods have been important in combinatorial design theory. Fint,

they are useful for constmcting "startern designs to be used in recursive constructions

of infinite families. Second, they have been employed for constructing complete lists of

designs with certain parameters. Such lists can be used by researchers in order to discover

properties, test hypot heses, formulate conjectures, etc. [71]. Third, exhaustive searches

can b e used for proving that some designs do not exist. A remarkable example of the

latter is the computational proof that a projective plane of order 10, Le. a 2-(L11,11,1)

design, does not ex$ [61].

Techniques that have been widely used inclnde backtracking, severd local search

methods (such as hill-climbing, simulated annealing, genetic algorithms) as well as sev-

eral algorit hrns using tk-matrices ([54] [56] 1581 [59] [60] [69] (901). For a collection of

articles on computational methods see the book edited by Wallis [IO21 and the s w e y s

by Gibbons [43] and Mathon [71]. Recently, the first polyhedral algorithm for 2-designs

was implemented using the point-block incidence matrix formulation [103]. In [76], we

proposed a branch-and-cut approach for finding t-designs, packings and coverings using

the tk-matrix formulation. In this chapter, we describe an irnplementation of this ap-

proach for h d i n g t-designs and packings. This algorithm c m be employed for the k t

and third applications desccïbed in the previous paragraph, namely searching for a design

CHAPTER 4 POLYHEDRAL ALGORITHMS FOR PACKINGS AND DESIGNS 84

with given parameters or proving that such a design does not exist.

The main contribution of this chapter is the design, implementation and experimental

analysis of a branch-and-cut algorithrn for t-(v, k, 1) designs and packings. The chap

ter is organized as follows. In Section 4.1, we describe the branch-and-cut algorithrn,

which is the most popular polybedral method for solving integer programming problems;

we also discuss briefly how some common requirements on the designs can be incorpo-

rated into this framework. In Section 4.2, we give details specific to our branch-and-cut

met hod, including separat ion algorit hms, initial fixings, partial isomorph rejection and

other implementation issues. In Section 4.3, we exhibit and discuss our computational

findings, analyzing how various parameters and variations affect the performance of the

algorithm. We also include tables of cyclic packings found by our algorithm.

4.1 The polyhedral approach and design problems

4.1.1 The branch-and-cut met hod

Branch-and-cut is a general technique for solving integer programming problerns, which

has become quite popular. See the paper by Caprara and Fischetti [I 61 for an anootated

bibliography. Remarkable success in solving large scale symmetrical traveling salesman

problems is described in [5] [84].

The method coinbines branch-and-bound and cutting-plane algotithms. in short, a

mtting-plane algorithm is employed at every node of the branch-and-bound tree. This

either leads to a solution of the subproblem or to a so caiied tailing-off phenornenon

(many iterations wit h lit tle improvement towards integrality ). In the latter case, the

subproblem can be substituted by two new subproblems, by creating a new branch in

the tree.

Branch-and-bound and cutting planes have been combined previously to the intr*

duction, by Padberg and Rinaldi [83], of the term "bmch-and-nit". The distinctive

feature of branch-and-nit is that cuts (Le. valid inequalities for the problem's polytope)

are not only added to the original problem, but at every node of the branch-and-bound

CHAPTER 4 POLYHEDRAL ALGORITHMS FOR PACK~NGS AND DESIGNS 85

t ree.

First , we review the cutt ing-plane met hod, t hen we describe the geneal branch-and-

cut algorithm in more detail. The reader who is farniliar with the approach may go

directly to the next sections, in which we discuss issues on our specific application.

Suppose we have a poiyhedral description of a combinatorial optimization probiem

by means of a large number of facet inducing inequalities, so that it would be impractical

to solve the linear programming problem using al1 inequalities. Cutting-plane algorithms

add inequalities one by one and often obtain an optimal solution before generating al1 of

t hem. A cutting-plane algorit hm is outlined below.

Algorithm: a general cutting-plane method

Let Pi be the polyhedron for the problem we want to solve. For any polyhedron Q > Pr,

let LP(Q) be the problem of finding nax{cTz : z E Q) and xqt (Q) be any optimal

vertex solution of L P(Q).

Input: a polytope PR containing PI.

Output:

(SUCC~SS, goPt): xqr.= xWt( P I ) , 0i

(failure, X T , PT): PT is s.t. PR 2 PT > PI, 21 = I ~ ~ ( P T ) .

Po = PR; k = O;

repeat

compute zk = x W r ( P k ) ;

if xk E Pr then "optimum was found";

else

solve the separat ion problem:

find d E Rn, do E R such that dTz 5 do is valid for Pr and dTxk > do;

if "separat ion was successfu.in then

p k + ~ = pkn (5 E R": PZ 5 do);

k = k + l ;

until "optimum was foundn or "separation was not succesduln

if "optimum was found"

CHAPTER 4 POLYHEDRAL ALGORITHMS FOR PACKINGS AND DESIGNS

Remark: If an integer programming formulation is used with P = {x E Rn : Ar 5 6)

and Pr = conv{z E P : x is integer) then PR may be set as P and the condition 'xk E Prn

is equivalent to "xk is integer". Moreover, if each inequdity d r x induces a facet of

Pr then the algorithm halts in a b i t e number of steps.

Cuttiog-plane algorithms alone may not be enough to efficiently solve a combinatorial

optimization problem. [t rnay happen that a complete description of the polyhedron is

not available, that the separation problem camot be efficiently solved for a class of

inequalities or t hat the above ment ioned taiiing-O ff phenomena arises. So, the cutting-

plane algorithm is cornbined with the branch-and-bound method.

Branch-and-bound is an implicit enurneration method. Let us focus on 0-1 integer

programming problems. A binary tree is associated with the solution of the problem. The

root of t h e tree represents the original reiaxed problem (i.e. the problem obtained from

the integer programming problem by dropping the integrality requirements) . The left

child and right chiid of a node N correspond to subproblems of the problem associated

with N obtained by fixing a free variable at O or 1, respectively. A list L of curent nodes

to be processed is kept. The List L is initialized with the root node. A t each step of the

algorithm, a node is removed from L and processed. The processing of a node N starts

by solving the relaxed subproblem corresponding to N . The node N is pruned, i.e., the

subtree rooted at N is neglected, if one of the following cases occur: the subproblem is

infeasible, the solution for the subproblem is integral, or the solution of the subproblem

is not integral but the optimal value is l a s than or equal to the objective value of the

best solution found. In al1 other cases, the children of node N are added to L. The

algonthm terminate when L is empty, and retums the best integral solution found as

the optimal solution.

The branch-and-cut algorithm differs fiom pure branch-and-bound in that at each

node it applies a cutting-plane algorithm rather than simply solving the relaued sub-

problem. In addition, the cuts generated in one node of the tree can be reused in another


node by keeping a pool of inequalities.

The design of a branch-and-cut algorithm involves many algorithmic and implemen-

tation decisions, several of which depend on the problem in consideration. We describe

our implementation in Section 4.2.

4.1.2 Branch-and-cut and combinat orial design problems

Besides being an alternative to tackle combinatorial design problems, the branch-and-cut

approacb offers ot ber advantages. We outlioe how to adapt the branch-and-eut algori t hm

to deal wit h specific design questions:

1. Extending designs. By extending a design, we mean the well-known process in

which a point is added to every block of a t-(v, k, A) design and extra blocks are

added in order to form a (t + 1)-(v + 1, k + 1, A) design. In many situations, design

theorists are coocerned whether a design can be extended to a larger design in this

fashion. Little modification in a branch-and-cut algorithm is required for producing

extensions: just fix to 1 the variables corresponding to the blocks arising from the

srnaDer design.

2. Fixing subdesigns. This is a generaiization of the previous feature. Here, any

number of blocks can be selected to be in the design.

3. Forbidding subconfigurations. As exemplified by the anti-Pasch designs, for-

bidding subconfigurat ions can be accomplished by adding extra inequaiit ies. Al-

though the number of such inequalities might be too large to be incorporated into

the original problem, it is possible to generate the violated ones as part of the

cutting-plane algorit hm.

4. Assuming the action of an automorphism group. This is accomplished,

as in other tk-matrix methods, by using a different onginal matrix (as seen in

Section 2.1.1).

5. Proving nonexistence results. Branch-and-cut is an implicit enumeratioo tech-

nique, and it is able to prove nonexistence of t-designs, whenever the upper bound

CHAPTER 4 POLYHEDRAL ALGORITHMS FOR PACKINCS AND DESIGNS 88

on al1 the active subproblems drops below b - E , where b is the number of blocks

in the design and E is large enough to account for approximation errors.

4.2 A branch-and-cut implementation for t-(v, k, 1)

designs and packings

Our implementation handles the following types of designs and employs the following

classes of inequali t ies:

The general framework of a branch-and-cut algonthm is given in Section 4.1. In this

section, we describe subalgorithms and other issues specific to our irnplementation.

t-(v, k, 1) designs and packings

general

general with k = t + 1

Csparse ( t = 2, k = 3)

.5-sparse ( t = 2, k = 3)

with cyclic automorphism

Some variable fixing c m be done in the original problem of finding t-designs, before

ninning the branch-and-cut algori th, as we describe now.

For a t-(v, k, 1) design all the k-subsets of [I, v ] passiog through a fixed (t - 1 )-set

can have their vaxiables fixed either to zero or one. For example, for a 2-(7,3,1) design,

since (1) has to appear exactly once together with every other element in some block,

we can assume the following blocks are in the design

Table 4.1: Types of designs and inequalities used in our implementations.

(separation of)

clique inequalities

general

general and speciaiized

general and specialized

general and specialized

general

m-sparse

inequalities

N I A

N/ A

m = 4

m = 5

N / A L


and al1 the other 3- subset of [1,7] containing (1) are not present in the design. Lo addi-

tion, a block not containing {l} must contain at most one element of each of BI, 8, B3,

so we c m assume w.1.o.g. that E = {2,4,6) is a block in the design.

In general, for any t-(u, k, 1) design and any given (t - 1)-subset S of [l, v ] , the blocks

of the design that contain S are unique up to permutations. Therefore, similarly to the

previous example, we can assume w.1.o.g. that

1. the following blocks are present in the design:

2. al1 the other subsets of (1, v] containing [1, t] are not present in the design.

4.2.2 Separat ion algorit hms

At each iteration of the cutting-plane, a separation problem must be solved for each

of the classes of facets being considered. An inequolity 8 2 < do is considered to be

violated whenever dTx - do is larger than a parameter cailed VIOLATfON-TOLERANCE.

This parameter has a great impact on the algorithm performance as we will discuss in

Section 4.3,

Next, we discuss the separation algorithms and related issues used in our implemen-

tation,

The fractional intersecting graph

The separation of clique facets relies on h d i n g violated cliques in the intersection graph

of the original matrix. For o u W:' rnatrix, this graph has (1) nodes and it would

be extremely time consuming to search for cliques or other subgraphs in such a large

graph. However, it is weiI known [50] that in any set packing problem, we can restnct

our attention to the ç e t i o n a l intersection graph, Le., the subgraph of the intersection

graph induced by the fractional &ables. More precisely, in any set packing problem,

the support of a violated inequality cannot contain variables that are equd to 1, and


the variables that are equal to O do not contribute to the violation; therefore, the v i e

lated inequalities can be found just by analyzing the fractional variables and lifting the

inequality including possible nul1 variables. In our experiments, the number of fractional

variables is roughly three times the expected number 6 of blocks, Le., around 3 Itif J y in

contrast wit h t be total number of variables being (i) . This reduces the size of the graph

we have to deal with from several tbousand to a few hundred nodes.

in our problem each variable corresponds to a k-subset of [1, v]. A canonical ordering

on the subsets is used, so that each subset corresponds to an integer number that is

the index of the variable. The fractional intersection graph of W:' is constructed in the

following way: each fractional variable corresponds to a node, and two nodes are adjacent

if their corresponding sets intenect in at least t points.

Hoffman and Padberg [50] report that decomposing the fractional graph into con-

nected components resulted in large savings for t heir set partit ioning problems, which

arise in an airline crew scheduling application. We did implement the decomposition of

the fractional graph into connected components. However, it turned out that our design

theoretical problems have fractional graphs, in most of the cases, formeci by a single

connected component .

Separation of clique facets

Clique facets are used for ail types of designs in Table 4.1. Given 4 we must find violated

cliques in the fractional graph Pk,, (corresponding to I). A clique C is considered violated

if xiic q - 1 > VIOLATION-TOLERANCE.

GENERAL-CL1 QUE-SEPARATION:

The general clique detection employed by our algonthm works for a general graph. We

borrowed several ideas from H o h a n and Padberg [50] and Nemhauser and Sigismondi

[78]. For every node v , we search for a violated clique containing v. Let N ( v ) denote the

neighborhood of v. Since every clique containing u must be contained in u U N ( v ) we

concentrate in this subgraph. We look at the neighborhood of a node v. If the neighbor-

hood is srnall, Say, under 20 nodes, we enurnerate every clique in the neighborhood, select

CHAPTER 4 POLYHEDRAL ALGORITHMS FOR PACKINGS AND DESKNS 91

the violated ones, and lift them to the larger problem (including nul1 variables). If the

neighborhood is larger we use two greedy heuristics proposed by Nemhauser and Sigis-

mondi [78]. Both heuristics select, among the candidate nodes to join a partial clique,

the one satisfying a given criteria. The first heuristic chooses the fractional nodes with

larger xi, and the second one, the fractional node closest to 0.5. However, we observed

that in most iterations the fractional variables are al1 under 0.5, so these two heuristics

turned out to be the same for our problems.

SPECIAL-CLIQUE-SEPAR (for k=t+ 1):

We implemented the special clique separation algorithm for designs with k = t + 1, given

on page 42. Recall t hat t his algorit hm uses the knowledge of the clique structure for t hese

problems and examines each edge of the graph exactly once, since there is at most one

violated clique passing through each edge. Note that we use the fractional intersection

gaph in place of Gi,,, as mentioned previously.

Sepration of 4-sparse and 5-sparse inequalities

Recall t hat m-sparse inequalities only apply to %(v, 3 , l ) designs and packings. Alt hough

the m-sparse inequalit ies are part of the original problem formulation, we do not add them

to the original problern due CO the large number of inequalities of this type; there are (p) Csparse inequalïties and (i) + (1) Csparse inequaiîties. instead, we only add rn-sparse

inequalities t hat are violated during the cut ting-plane algorit hm.

We implemented separation algorithms for Psparse and 5-sparse inequalities. Recall

that the Csparse inequalities are of the form in (4.1) and 5-sparse inequalities are of the

form in (4.1) or (4.2):

A separation algorithm could go over a l l 6-subsets or ?-subsets of [ l ,v ] and check

whether the corresponding inequalities (4.1) and (4.2) are violated. However, this wodd


be very inefficient due to the large number of possible inequalities most of which would

not be violated.

Our separation algorithm employs the foilowing idea. Fint, we detemine every set

with 3 t n p l e ~ (corresponding to nonzero variables) whose union spans a 6-subset and such

that every pair of t n p l e ~ intersects in exactly one point. Then, we check whether the

corresponding inequalities of type (4.1) are violated. Similarly, we determine al1 sets of 3

triples (corresponding to nonzero variables) spanning a 7-set and such that every pair of

t n p l e ~ intersects in exactly one point, and check whether the corresponding inequalities

of type (4.2) are violated. These conditions are not yaranteed to find all the violated

inequalities for fractional points. For example, consider a fiactional point with X K = 0.33

for al1 triples K [1,5], and X K = O for al1 ot her triples in [ l ,6j; t hen an inequality of

type (4.1) is violated for T = [1,6], but there are no 3 triples (with positive values)

contaioed in [1,6] that span [l, 61. However, when the variables are integer, inequality

(4.1) is violated if and only if a Pasch is present, in which case we know any 3 sets

in the Pasch span the &set and are pairwise intersecting. Moreover, as we progress

towards integrality, it is more likely that this condition will suEce to detect violated

bsparse inequalities. Similarly, .Fsparse inequalities are not guaranteed to be found by

this method, but wheo integrality is attained, these conditions guarantee the detection

of violated 5-sparse inequalities.

Let us now describe the method used for detecting the above mentioned conditions,

Le. for detecting the groups of 3 triples with every pair intersecting in a point and

spanning eit her a 6-subset or a 7-subset of [l , v ] .

First, we construct a 1-intersecting graph of nonzero variables, i.e. a graph whose

vertices correspond to nonzero variables and such that two nodes are connected if the

correspooding t n p l e ~ intersect in exactly one point. Then, we go over each bcycle in the

graph and check whether the union of the corresponding triples spans 6 or 7 points. In

order to avoid finding each bcycle more than once, the edges of the graph are directed

from the s m d e r triple to the Iarger one (smaller here, with respect to the canonical

ordering of triples). The algorithm searches for 3 edges of the type ( K I , K2), (K2, K3)

(h;, K3). The algorithm is outiined below.

CHAPTER 4 POL~HEDRAL ALGORITHMS FOR PACKINGS AND DESIGNS

Algorithm: separation of Csparse and bsparse inequalities.

Constnict the direct 1-intersecting grapb of nonzero variables:

Let I E IR(;) be the point to be separateci.

create a node for each K with 2~ > 0.

for every pair of nodes Ki and K2 do

if IKl (in K21 = 1 then

create a directed edge frorn min(K1, Kz ) to max(K1, Kz).

for every directed edge ( Ki, K2)

for every node K3 such that ( K2, K3) and (Ki, K3) are edges do

take S = Ki U K2U K3.

if ISJ = 6 then

if &(:) 3~ - 3 > VIOLATION-TOLEXANCE, then store (S, 3)

else

if ISI = 7 then

if ZK - 4 > VIOLATION-TOISRANCE, then store (S, 4)

Pool of cuts

Previously generated inequalities are stored in a pool of cuts in order to reuse them in

other nodes of the tree. This is a common feature in a branch-and-cut algorithm (see

1161 ==

Any cut that is generated is stored in the pool of cuts. Before Nining the separation

algorithm for various facets we check whether some inequalities from the pool are violated.

If the number of violated inequalities found in the pool is larger t han a certain value,

there is no need of calling the separation routines.

The data structure for the pool of cuts is a hash table and a linked List of inequalities.

The entries of the hash table are pointers to inequalities and collisions are resolved by

chainiog. The hash f ~ i o n is given by the sum of the square of the Mnable indices

corresponding to nonzero coefficients in the inequalities modulo the table size. Before

inserting a new constraint on the List, we check if it is already present through the pointer

in the hash table. To check if there are violated inequalities in the pool, the algorithm

CHAPTER 4 POLYHEDRAL ALGORITHMS FOR PACKINCS AND DESIGNS

searches linearly along the linked list.

4.2.3 Crit eria for abandoning the cutt ing-plane algorit hm

Our algorithm stops cutting-plane iterations if any one of the following conditions is

satisfied: the optimal solution to the LP relaxation is integrai (i.e. the subproblem

rooted at the node was solved), the subproblem is infeasible, or the addition of the cuts

is "not producing much improvementn. The last condition is measured by the number

of cuts and the "qualityn of cuts at the previous iteration. The cutting-plane algorithm

is abandoned whenever the number of cuts in the previous iteration is smaller than a

parameter MIN-NUMBER-OF-CUTS or the maximum violation is smailer t han a parameter

MIN-WORTHWHILE-VIOLATION.

4.2.4 Partial. isomorph rejection

For combinatorial design problerns, several subproblerns in the branch-and-cut tree may

be equivalent. Recall that a node in the branch-and-cut tree corresponds to the sub-

problem in which the variables in the path from the root to the node have their values

fixed either to zero or one. Let N be aoy node of the tree and denote by 78(N) and

F1(N) the collection of blocks correspanding to the variables fixed to O and 1, respec-

tively, in the path from the mot of the tree to N. If N and bf are nodes in the tree with

(FO( N), FI (N)) isomorphic to (FO(M), FI (M)) then equivalent problems are going to be

unnecessarily solved. The partial isonorph rejection we describe in this section aims at

reducing the numbo of such equivalent subproblems.

Let N be a node of the branch-and-cut tree with two children No and iVl. The original

branching scheme would make No correspond to XK = O and NI correspond to ZK = 1,

for some variable K. This branching scheme is modified so that the number of nodes in

the tree is reduced by avoiding some subproblems that are equivalent to others already

considered. as we discuss next.

Let A be the permutation group acting on [I, v ] that fixes FO(N) and Fi(N), and

let A ( K ) be the orbit of K under A. The new bmching scheme for partial isomorph

CHAPTER 4 POLYHEDRAL ALGOR~THMS FOR PACKINGS AND DESIGNS 95

rejection lets No correspond to UxL = O for al1 L E A(K)" and NI correspond to "xL = 1

for some L E A( K)". The tree reduction cornes from letting Nl correspond, w.l.o.g., to U X K = In instead. Thus the new branching scheme implies

This new branch-and-cut tree has at most as many problems as the regular one, since

when IA(K)I > 1 not only K but other variables in A ( K ) are being simultaneously fixed

at O in No.

AU we need is an algorithm that, given collections F0 and 3, of k-subsets of [l, v ]

and a k-subset K of [l , v ] , computes

1. the permutation group A acting on [l, v ] that fixes Fo and 3 1 ;

2. the orbit A(K) of K under A.

The fint problem is equivalent to finding the permutation group acting on the vertices

of a special graph that fixes some subsets of the vertices. Consider the bipartite graph

Gll,:[i,ulFouFl wbose vertex partition corresponds to points in [l, v] and sets in ü Fi, and

such that p E [ I , v] is connected to F E 75 u FI if and only if p E F. Thus, our problem

is equivalent to finding the permutation group acting on t h e vertices of the graph that

fixes vertices in [1, v], in F0 and in 3 1 . This can be computed using the package Nauty,

by Brendan McKay, the "mat powerfui generd purpose graph isomorphism program

currently availablen (quoted from [52]). For a description of the algorithms used by

Nauty see [73].

The second problem can be solved by a simple algorithm that we describe now. A

collection S of k-sets is initialized with K. At every step, a different set L in S is

considered and, for all r E A, the set n(L) is added to S. The algorithm halts when aU

sets in S have been considered, and thus, A ( K ) = S.

A srnail variation of the two previous methods can dramaticdy improve eEciency

when IF0 U 3 1 1 << v. Let R = UsEFoUr,B. Consider the graph GvouF, iinstead of

Gp,,,IFouF, and apply the method desmbed above to compute an automorphism group


A'. The points in [i, v ] \ R are isotated vertices in Gli,v13,uF,, and therefore form a cycle

in any permutation in A. Thus A' is the restriction of A to points in R. Ln order to

compute A(K) , we use the method descRbed above and compute A'(K n R), and then

compute A( K) by taking al1 the k-subsets of [l, v ] that contain sorne set in Ar(K n R).

Two kinds of improvements in efficiency are observed. In the fust part, the original graph

gets reduced by [1, v ] \ R nodes. In the second part, if 1 K n RI < k the size of the set S

in the second problern is reduced by a factor of (C($!!().

4.2.5 Branch-and-cut tree processing

The main strategies to be defined in the braoch-and-cut node processiog are: the selection

of the variable involved in the branching at a node and the selection of the next node to

be processed.

We implemented two strategies for the selection of the branching mriable in a node.

The first one selects the variable with largest fractional value, and the second one, the

variable closest to 0.5. Both strategies turned out to be equivalent, since in our problams

most of the fract ional variables are smaller t han 0.5.

Typically, the objective value of the relaxed problem and of the integer problem are

the same or quite close in our design problems. This suggests that a tree search guided

by objective value is often meaningless. Our selection of the next node to process is done

as a depth-first search, giving priority to nodes with variables fixed to 1. The ratiouale

behind this heuristic can be explained as follows. In general, fixing a variable to 1 makes

a packing problem tighter than fixing it to O. Depth-first search is employed in order to

fix as many variables to 1 as possible in an attempt to quickly find a design.

4.2.6 O t her implementation issues

Preprocessing is a technique that d o w s for a reduction on the size of a linear progmm-

ming problem. It involves several rules for variable and constraint elimination through

the testing of simple conditions. Preprocessing has proved to have a great impact on

efficient ly solving integer programming problems [SOI. No preprocasing d e , besides

the initial fixing mentioned in Section 4.2.1, was implemented. However, most of the

conditions that could allow for problem reduction do not hold initially in our problems.

For example, our problems have no empty rows or colwnns, the problems are given by

a matrix of cliques, they contain no dominated rows or colurnns, etc. On the other

hand, such conditions might hold deeper in the tree, and preprocessing could become

advantageous there. This is a direction for further research. For general discussions on

preprocessing techniques see the article by Crowder, Johnson and Padberg [30] or t h e

book by Nemhauser and Wolsey [80, pages 17-20], and for such a study in the context

of set partitioning problerns see the t hesis by Borndorfer [13] and the references therein.

Objective funct ion perturbation is a technique based on adding small ( randomly cho-

sen) quantities to each of the objective function coefficients. This is done in such a way

that the optimal solution for the perturbed problem remains the same as the original

one. This technique sometimes works well for problems with many optimal solutions, by

making some of them more attractive, leading the linear programming algorithm directly

towards thern. We experimented wit h this technique and, although in some cases we got

a reduction on the number of nodes in the tree, the linear programming solution at each

node was more expensive, which led to a greater overall time.

4.3 Computational results

In t his section, we report on computational experiments with the branch-and-cut imple-

mentation described in the previous section. The following aspects are investigated.

1. The kinds and sizes of problems o u . implementation can handle.

2. The effect of isomorphism rejection for t-designs and packings, including the cases

in which the designs do not exist.

3. The impact of cutting: a compaxison between branch-and-cut and branch-and-

bound.

4. A c o m p ~ s o o between the specialized and the general clique separation dgorithms

for t-designs and packings with k = t + 1.


5. The influence.of parameters that affect the trade-off between branching and cutting,

namely VIOLATION-TOLERAWCE and MIN-WORTAWHfLE-VIOLATION.

6. Changes on the algorithm's performance given by additional properties such as

m-sparse condit ions.

7. A comparisod of our branch-and-cut with a leading general purpose branch-and-

botmd software.

The tests are run on a Sun Ultra 2 Mode1 2170 workstation with 245 MB main memory

and 1.2 GB virtual memory, operating system SunOS 5.5.1. Our branch-and-cut consists

of about 10,000 Lines of code written in C++ ianguage and cornpiled with g++ compiler.

The following packages are linked with our code: LEDA Library version 3.2.3 [93] for

basic data structures such as Lsts and graphs, CPLEX package version 4.0.8 [29] for

solving linear programming subproblems, and Nauty ~ackage' version 2.0 (731 for finding

automorp hism groups of g a p hs.

In the following.we discuss our experimental results. We ran the algorit hm for several

designs and packing parameters which are described in Tables 4.2 and 4.5.

We explain the columns in the tables describing the experiments. The first few

columns indicate problem parameters, and they Vary from table to table. The other

columns are described as foilows. Columns (BB) and (BT) give the number of explored

branch-and-bound tree nodes and the number of tirnes the algorithm backtracks, respec-

tive.~. Columns (IE), (IR), (MA) and (IT) encode information on the isomorp h rejection

algorithm. Column (IE) gives the number of times the algorithm is executed; (IR) gives

the number of times the aigorithm returned a nontrivial orbit (one with at least two

sets); (Md) is the largest depth of a node in which a nontrivial orbit was found; (IT)

is the total time spent in isomorph rejection. The next two columns (LPT) and (LPs)

report on total time solving linear programming problems and the number of such prob-

lems, respectively. Column (ST) denotes total separation time; (Cl) and (Sp) indicate

the number of times clique and sparse inequalities are added to a subproblem. Findy,

%th an adaptation by Luis Dissett for having the automorphism group as an output parameter rat ber than a printing.


(TotT) contains the total tirne. Al1 times are measured in seconds.

la al1 tables, u and b denote the number of points and blocks in the design, respectively.

Remember that the optimal objective value is equal to 6. In the tables for packings,

column (Sch) shows the Schonheim upper bound for the corresponding packing. This

bound is used in the algorithm as an initial upper bound for the objective value. When

applicable, column (exist.) indicates whether a certain design exists. Tables 4.8 and 4.9

have a column (no#) which correspond to the parameter combination given in Table 4.7.

In Tables 4.4 and 4.5 we report on isomorph rejection statistics. Table 4.4 compares

the same instances of packjngs with and without isomorph rejection (specified by column

(IRej)). Only the packings with v = 5,11,12,14 requires a cal1 to the algorithm. From

t hese parameters, only v = 5,11 benefit from nontrivial orbits, and only v = 1 1 profits

from the isomorph rejection. We observe that the time spent in the isomorph rejection

algorithm is very small compared to the total time. The packing for v = 11 could not be

found without the isomorph rejection. The difficulty encountered for v = 11 is that the

Schônheim bound is not met by the packing size. Therefore, the program rnight find a

solution of size 17, but has to go over most of the branches to conclude it is optimal. The

isomorph rejection reduces the amount of branches to be searched. We conclude that the

isomorph rejection algorithm is effective since it spends little extra time and adds the

benefit of tackling the hardest problems.

In Tables 4.5 and 4.6 we report on the effectiveness of using cuts, as opposed to

simple branch-and-bound, for 2-(v, 3 , l ) designs and packings. For designs there is no

clear winner in terms of total time (Table 4.5). For packings, the total time using

cuts is either comparable or substantidy smaller than without cuts, especially for the

larger instances (Table 4.6). Ln all cases, using cuts reduces the number of explored

tree nodes and the number of times the algorithm backtracks. This is reflected in the

often smaller number of solved Linear programming problems and time spend on solving

them. However, the time spent on cut separation makes the cutting version worst for

some instances. Larger problems should benefit from the cutting version, since linear

programming tends to dominate the running time.


Tables 4.8 and 4.9 summarize the r e d t s of 8 nuis corresponding to a combination of

parameters describèd in Table 4.7. Even and odd oumbered mns correspond to special-

ized and general separation algorit hms, respect ively. From t hese tables we observe t hat

the specialized separation is done much faster than the general one (see column (ST)).

In al1 runs the specialized separation produced savings in the total running time of up

to 50%. These tables also show t h e influence of the parameters MIN-WORTHWHILE

VIOLATION and VIOLATION-TOLERANCE. Let us denote them by MWV and V,

respectively. An interval [II,,,,, v,,] is assigned to each of these parameters. The alg*

rithm initially sets the value of the parameter to v,,; as the number of fractional vari-

ables decreases, the parameter is continuously reduced towards v,,,,. The 4 combinations

shown in Table 4.7 are tried. The best runs involve (MWV, V) = [(0.3,0.3), (0.3,0.3)]

and [(0.3,0.6), (0.3,0.6)]. The main conclusion is that the performance is positively of-

fected by requiring stronger cuts. In al1 other tables, these parameters are set as M W

= V = [(O.3,0.6)].

In Tables 4.8 and 4.9 our nuis are also compared to the general purpose integer

progamming solvei oRered by CPLEX. As problem sizes grow, our algorithm runs much

fa te r than CPLEX (2 to 10 tirnes faster in the two largest problems). This cornparison

is only included as a reference. Other problems such as m-sparse cannot be solved by

such a general package since the number of inequalities defining the integer programming

problem is very large. We only manage to solve some instances of these problems, by

adding ody the viilated m-sparse inequalit ies in the cut ting-plane algorit hm.

Tables 4.10 and 4.1 1 show the solution of Psparse and 5-sparse Steiner t n p k systems.

For Csparse systerns we observe that v = 13 is harder than v = 15. In fact, for v = 13

the program has to establish its nonexistence, which is time consuming. However, the

same instance v =. 13, for which a %pane system is also nonexistent, can be solved

more eficiently due to the addition of 5-sparse facets. This makes the algorithm detect

its nonexistence earlier. The algonthm encounters diffidties in solving 4-sparse and

.5-sparse problems much earlier than when solving ordinary Steiner triple systems. For

Csparse and bsparse problems the number of tree nodes grows imrnensely starting on

v = 19 and v = 15, respectively. The second part of both tables indicates problems

that did not finish after 2 hours of running time. Our conclusion is that the m-sparse

restriction imposes .extra difficulties for this method. Further research should be done in

order to solve larger 4,Ssparse problems.

Steiner quadruple systems are shown in Table 4.12. The case v = 14 is already a hard

instance. In eariier versions of this implementation it took about 6 houn to solve this

problem. Currently, it takes about 40 minutes to solve it. The next instance v = 16 is

still a challenge for this implementation. Although the design is known to exist, other

computational met hods also fail to find t his design.

Tables 4.13-4.21 report on cyclic designs and packings. Recall t hat t hese problems are

solved by assuming a cyclic automorphism group action on the design, which produces

reductioas of the problem size, as shown in Tables 4.2 and 4.3. Tables 4.14-4.17 compare

the size of maximal t - (v, k, 1) packings to the size of regular packings for 1 = 2,3,4,5,

k = t + 1, t + 2, and small v. Columns (B1) and (C) indicate the number of base blocks

and the total number of blocks in the cyclic packiags, respectively. To the best of our

knowledge, this is the first time this quantities are computed. in columns (D) and (S)

we include known values for the size of a maximal ordinary packing and Schooheim

upper bounds, respectively. Thus, we must have C 5 D 5 S. Observe that in most

cases C is not much smaller than D (or not much smaller than S in the cases that D

is unknown, see Tables 4.16 and 4.17). In Tables 4.18 and 4.20, we show a sample of

statistics on some runs for cyclic packings; their corresponding base blocks are shown in

Tables 4.19 and 4.21. The packings can be obtained by developing the blocks modulo

v. Our experiments show t hat the sizes of cyclic packings are close to maximal ordinary

packings (compare (C) and (D)), and they are much easier to compute and compact to

store. Therefore, these objects should be very attractive for applications. The size of a

maximal cyclic packing also offers a good lower bound for the packing number. Cyclic

Steiner triple systetns are relateri objects, which have b e n given some attention (see for

instance [85]).

Our previous analysis outlines the most efficient strategies and variations. The results

show that the speciaiized clique separation greatly improves the ninning time. The partial

isomorpb rejection algorithm is also effective given the Lttle time it spends and the great


deal of savings it offers for hard problems. In addition, the influence of other parameten

is discussed. The branch-and-cut algorithm can be appiied to a wide range of designs,

as our experiments have shown. Cyclic packings seem to be interesting objects which

deserve furt her investigation.

design

ordinary designs

before fixing -- --

after fixing

cyclic designs

Table 4.2: Problem sizes and statistics for t-designs

t(v,k, 1) packings 2-(5,W)

cyclic packings frows #cols - -

- -

h

Table 4.3: Problem sizes and statistics for packings

CHAPTER 4 POLYHEDRAL ALGORITHMS FOR PACKINGS AND DESIGNS 1 04

v b Sch lRej II BB BT 1 IE IR Mx. IT 1 LPT LPs

no 2 O O 0 - O 0.02 3 yes 11 O O O - O 0.08 14

- .

yes 13 O O O - O 0.12 14- no 1 3 0 0 0 - O 0.13 14 yes 77 27 27 O - 0.05 1.82 121 no 77 27 O O - O 1.81 121

(') the algorithm lailed to find the designs even after exploring 400,000 branches.

Table 4.4: The effect of partial isomorph rejection on 2-(v ,3 ,1) packings

b:

Sch:

IRej :

IE:

IR:

Md:

IT:

number of blocks

isomorph rejection algorit hm

number of c d s ta IRej

number of successf' IRej's

Max. depth of node in successfull IRej

Total tirnein IRej

BB: number of explored B&B nades

BT: number of backtracks

LPT: total time solving LP problems

LPs: number of LP problems

ST: total time in separation alg.

Cl: total number of added cliques

TotT: total time


Table 4.5: Banch-and-eut versus branch-and-bound for 2-(v, 3 , l ) designs

v b 7 7 7 7 9 12 9 12 13 26 13 26 15 35 15 35 19 57 19 57 21 70 21 70 25 100 25 LOO

Cuts

Yes no

yes no

yes no

yes no

yes no

yes no

yes no

- 6.28 70.83 290

BB BT 1 1 O 1 O 1 O 6 O 40 17 1 O 1 O 35 1 112 42 58 8 57 12 60 O

31 155 31 155 33 176 33 176

IE IR Mx1 IT O 0 0 * O

O 0 - O O 0 - O O 0 - O O 0 O 17 O - 0.14 O 0 - O O 0 - O 1 O - 0.02 42 O - 0-78 8 O - 0.23

12 O - 0.34 O O O

7 4 6 6 0 - 0.29

yes no

YS no

LPT LPs O 1 O 1

0.01 1 0.02 1 0.07 8 0.38 40 0.06 1 0.06 1 1.45 39 5.2 112 3.86 63 3.02 57 9.96 69 16.19 74

l13 7 358 128 174 26 267 71

ST Cl O O O O O O O O

0.1 4 O O O O O O

2.66 5 O O

6.17 7 O O

17.27 12 O O

7 O - 0.73 128 O - 13-95 26 O - 3.31 71 O - 10.46

TotT O

0.02 0.01 0.01 0.18 0.57 0.08 0.08 4.17 6.68 10.29 3.81 27.44 16.99

87.61 127 277.69 358 309.52 198 209-06 267

76.35 25 O O

114.61 53 O O

165.56 311.52 430.36 235.84

Table 4.6: Branch-and-cut versus branch-and-bound for 2-(u, 3 , l ) packings

v b 5 2 5 2 6 4 6 4 7 7 7 7 8 8 8 8 9 12

10 13 10 13 11 II 1 1 17 12 20 12 20 13 26 13 26 14 28 14 28

Cuts yes no yes no

yes no

yes no

yes 9 1 2 n o . 4

yes no

yes no

yes no

yes no

yes no

BB BT 3 3 3 O 3 O 1 O 1 O 5 O 8 2 O

O

13 O 649 324 1063 531

848 414

44 14 77 27 314 145

lE IR MA IT 1 1 1 O O 1 1 1 O O

O 0 - O O 0 - O O 0 - O O 0 - O O 0 - O

1 1 0 O O 0 O O 0 O

1 1 0 0 0 - O O O - O

324 11 10 0.26 531 17 10 0.33

1 7 1 1 0 - O 414 O - 1.75

1 3 0 0 0 d O 14 O - O 27 O - 0.05 145 O - 0.15

LPT LPs O 3 O 3 O 3

0-01 3 0.01 1 0.01 1 0.02 8 0.04 8 0.02 3 0.02 4 0.08 14 0.06 13 6.41 829 7.91 1063 0.22 24 5.69 848 0.12 14 0.34 44 1.82 121 3.57 314

ST CI O O O O

0.01 O O O O O O O

0.02 6 O O O 1 O O

0.05 4 O O

2-46 319 O O

0-25 11 O O

0.34 1 O O

0.85 86 O O

TotT 0.01 0.01 0-01 0.01 0.02 0.02 0.06 0.05 0.04 0.02 0.14 0.08 9.94 9.72 0.53 8.97 0.38 0.46 2.91 4.71

CHAPTER 4 POLYHEDWL ALGORITHMS FOR PACKINGS AND DESIGNS 107

1 1 clique separation aigorithm I

Table 4.7: Parameter combination for several runs.

1 V MWV [O. 1,O. 11 [0.1,0.3]

BB BT 1 LPT LPs 1 ST Cl I TotT

general spefialiaed m# 1 rua# 2

Table 4.8: CLique separation and parameter variations.


25 100 25 100 25 100 25 100 25 100 25 100 25 LOO 25 100 25 100 27 117 27 117 27 117 27 117 27 117 27 117 27 117 27 117 27 117 31 155 31 155 31 155 31 155 31 155 31 155 31 155 31 155 31 155 33 176 33 176 33 176 33 176 33 176 33 176 33 176 33 176 33 176

- run# - - I 3 5 7 - 2 4 6 8 - Cplex - - 1 3 5 7 2 4 6 8 - Cplex - 1 3 5 7 - 2 4 6 8 - Cplex - 1 3 5 7 - 2 4 6 8 Cplex -

BB BT LPT LPs 71 4 23.1 110 71 4 23.5 110 64 O 11.62 81 57 O 12.49 65 60 1 26.37 98 60 1 25.75 98 66 O 17.11 91 60 O 9.966 69

102 74.5 144

ST Cl TotT 92.21 91 115.76 92.21 93 116.28 79.51 30 91.38 72.7 9 85.42 20.28 97 46.94 19.96 97 45.99 18.49 42 35.78 17.27 12 27.44

1 63.83

Table 4.9: Clique separat ion and parameter variations.

v b exist

7 - no

9 12 yes

13 - no

15 35 yes

19 57 yes

Table 4.10: Cspane 2-(v, 3 , l ) designs

BB BT

1 O

v b exist

7 - no

9 - no

13 - no

BB BT 1 LPT LPs 1 ST Cl Sp 1 TotT 1

LPT LPs

O 2

Table 4.1 1: 5-sparse 2-(v, 3? 1) designs

The problems with "(*) 7200" in the column TotT could not be solved within 2hs of

CPU.

ST Cl Sp

0.02 O 7

TotT

0.04


Table 4.12: Steiner Quadruple Systems: S ( v , 4, l ) designs

v b exist.

TotT 0.01 0.07

2488.7

v b 8 14 10 30 14 91

7 Y- - no

26 yes

35 yes

57 yes

70 yes

100 yes

27 117 yes

31 155 yes

LPT LPs

Table 4.13: Cyclic 2-(v, 3 , l ) designs

ST CI O O O O

35.59 57

BB BT 1 1

1700 838

IE IR Mx1 IT O 0 0 - O O 0 0 - O

O O - O

LPT LPs O 1

0 .O7 1 1974.75 1755


Table 4.14: Cyclic 2-(v, k, 1) packings

BI: base blocks in the cyclic packing C: size of maximal cyclic packing

D: size of maximal packing S: Schonheim upper bound

CHAPTER 4 POLVHEDRAL ALGORITHMS FOR PACKINGS AND DES~GNS

Table 4.15: Cyclic 3-(v, k, 1 ) packings

- indicates our algorithm did not find the cyclic packings

? iqdicates that the regular packing number is unknown


Table 4.16: Cyclic 4 ( v , k, 1) packogs

Table 4.17: Cyclic 5-(v, k, 1) packings

- indicates o& algorithm did not h d the cyclic packings

? indicates that the regular packing nurnber is unknown

(*) indicates that a (not necessarily optimal) cyclic packing was found by the algorithm


v C D Sch

6 3 3 4

7 7 7 7

8 10 14 14

9 9 18 18

10 30 30' 30

11 33 35 35

12 45 51 54

13 52 65 65

14 84 91 91

15 105 105 105

16 132 140 140

17 153 156 157

18 198 198 202

19 228 228 228

20 285 285 285

SI 315 315 315

LPT LPs

O 1

O 1

O 2

O 2

0.02 5

0.12 27

0.36 78

1.79 384

3.68 549

0.23 35

32.21 2645

257.49 16648

230.44 12029

96.89 467

203.40 6611

441.22 13433

TotT

Table 4.18: Cyclic 3-(v, 4,1) packings


Base Blocks

Table 4.19: Cyclic 3-(v,4,1) packings

LPT LPs v C D Sch

8 8 8 . 11

9 9 18 25

10 30 36 36

11 66 66 66

12 7 2 ? 84

13 117 ? 140

14 154 ? 182

TotT BB BT

1 O

1 O

1 O

1 O

65 31

1639 819

5048 2523

Table 4.20: Cyclic 4 ( v , 5,l) packings

Table 4.21: Cyclic P ( v , 5,l) packings

v Base BIocks

Chapter 5

Concluding Remarks and Open Problems

In this thesis, we have investigated the polyhedrai structure of combinatorial design

problems. This study has led to a new algorithmic technique for constructing various

types of designs such as t-designs and packings, cyclic packings, and Steiner triple systems

avoiding su bconfigurat ions.

Extrema1 problems for set systems were a common theme: maximal intersecting fam-

ilies of sets (Chapter 3) and Erd6s configurations (Section 2.3.3) are both defined based

on extrema1 properties. Indeed, extrema1 problems are in the very nature of polyhedrai

t heory and of the designs we dedt with. To study a polyhedron, one investigates its ver-

tices and facets, which are its nonempty faces of smdlest dimension and its proper faces

of largest dimension. Maximal packings are the designs with most blocks such that each

pairwise intersection of blocks does not exceed a given parameter t ; maximal m-sparse

packings are triple systems with most blocks avoiding the uconcentration" of points in a

few blocks. So, it is not surprising that many of the open problems we now describe are

related to extrema1 set theory.

Polyhedral theory and designs

Problem 1. Use valid inequalities for the design polytopes to derive results for infinite

families of designs.

In the algorithm of Chapter 4, we use these inequalities in order to determine maximal

packing designs for &ecific parameters. So, the algorithm determines the packing number

CHAPTER 5 CONCLUDING &MARKS AND OPEN PROBLEMS 118

for fixed ( t , u, k, A). Can valid inequalities or facets (of the design polytope) be used to

determine other properties, such as upper bounds on packing numbers, for an infinite

family of parameters?

Algorithma involving set systems

An aoswer to the next question would be useful not only for problems discussed in this

t hesis, but also for designing ot her algori t hms involving set systems.

Problem 2. Let Ai, A2,. . . , A, be (nonisornorphic) farnilies of k-sets of a v-set and let

B be a family isomorphic to one of them.

1. Compute j such t hat family Aj is isomorphic to 8, in an efficient way. In part icular,

what is the minimum number of sets in D that have to be examined in order to

determine Aj ?

2. If' we know that Aj and B are isomorphic, what is the minimum number of sets of

B to be examined in order to determine the permutation of [l, v] that sends B to

A,?

We will illustrate these problems through an example. Suppose {Al, A*, . . . ,&) =

M&J, t + 1, t), and B is an arbitrary family in Mf ( v , t + 1, t). Proposition 3.4.1 tells

us three things: (1) p = 2, (2) by examining any 3 sets of B we can determine which of

the two families B is isomorphic to, and (3) if B is isomorphic to ([lG2I), then 2 sets

are enough to determine the permutation that sends B to (['SI) * The clique separation

algonthm in page 42 is based on these iacts. A first step would be to extend these results

to (t + 2)-subsets of a v-set. In this case, p = 17 for t 2 2, and the solution of these

problems would give us an efficient clique separation dgorithm for designs with k = t +2.

Problem 3. Design a branching scheme that uses the structure of the set systerns for

the branch-and-cut algorithm.

Given a subconfiguration corresponding to a node in the tree, the algorithm shodd

branch according to the possible ways of extending this subcodguration.

Problem 4. Find an efficient algorithm for enmeration of distinct (up to isomorphism)

maximal iatersecting set systerns.

CHAPTER 5 CONCLUDING REMARKS AND OPEN PROBLEMS 119

The naive algorithm generates d the set systems and t ben eliminates the isornorphic

ones. An efficient algorithm should avoid the generation of isornorphic ones in the first

place. General methods for this kind of generation are discussed in [74].

Maximal intersecting set systems

We pose a conjecture that generalizes some results from Cbapter 3.

Conjecture 1. Let k > t 2 1, s 2 2. Then, there exist a to = to(k - t , s ) and

a vo = vo(k - t, s) such that any family in MP(v, k, t ) is an extension of a family in

MP(vo, to + (k - t), to), for all t 2 toi and v 2 vo + ( t - to) . In particular, the number of

families distinct up to isomorphism in M P ( v , k, t ) is completely determined by k - t.

In this thesis, we prove this is true for s = 2 and k - t = 1,2. For s = 2 and k - t = 1,

Proposition 3.4.1 implies to = 1 and vo = 3. For s = 2 and k-t = 2, Theorem 3.4.9 states

t hat to = 2 and vo = 8. In the first case, this means Mf(3,2 ,1) completely determines

MP ( v , t + 1, t ) for any t 2 1 and v 2 t + 3, and in the second case MF (8,4,2) completely

determines al1 Mf2(u,t + 2,t) for any t 2 2 and v 2 t + 6. 1s it true that the same

happens for ot her values of (k - t ) ?

The proof of this conjecture wodd imply that the classification problem (Problem 2

on page 58) for maximal s-wise t-intersecting systems of k-sets of a v-set depends ody

on s and k - t . We have already shown it does not depend on v (Coroliary 32-10).

The following conjecture concerns keniels and essential sets of intersecting families

(see Chapter 3).

Conjecture 2. Let s 2 2 and t < k and, for any v > k, let e(v) = maxAÉ~~~(u,r, , ) 1 E(A)I.

Then, if û is the smdest integer such that e(ù) < ü, then n(t , s, k) = e ( ~ ) .

This conjecture implies that whenever the maximum size among ail kernels of famiiies

in M P ( v , k, t ) is strictly smaller t han v, no other family with II' 2 v can have a kernel

of size v. If true, i ve c m design an algorithm to determine a(t , s, k), that is able to halt

whenever such a Z is found, and so has to examine up to n(t,s, k) different v's. This

conjecture is true for s = 2 and k - t = 1,2 (see Table 3.1).

CHAPTER 5 CONCLUDING REMARKS AND OPEN PROBLEMS

Other aigorithrnic problerns

There are other natural extensions of out results and algorïthms that emerge from this

thesis. For exarnple, other classes of facets could be investigated, the branch-and-cut

algorithm could be extended to deal with other kinds of designs (such as coverings, de-

signs with higher A, and admitting other automorphism groups). In addition, it would

be interesting to investigate whether the combination of techniques such as tactical de-

compositions (see [71]) wi t h the polyhedral approach leads to more efiicient algorit hms.

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