S*-matrices and SNS*-matrices

We first recall some basic facts from sections 1.2 and 2.1. Let A be an m by m + 1 matrix. By definition, A is an S*-matrix if and only if each submatrix of A of order m is an SNS-matrix. By (iv) of Theorem 2.1.1, A is an S*-matrix if and only if there exists A strict signing D such that AD and A(–D) are the only column signings of A each of whose rows are balanced. The matrix A is an S-matrix if and only if A is an S*-matrix and AIm+1 = A and A(–Iw+1) = –A are the only column signings of A each of whose rows are balanced.

Let v1, v2, … vm+1 be the column vectors of A. Then v1, v2,…, vm+1 are the vertices of an m -simplex whose interior contains the origin if and only if the right null space of A is spanned by A vector w each of whose entries is positive. This implies A geometric description of S-matrices [6]:

Clearly, each row of an S*-matrix must contain at least two nonzero entries. The following theorem [7] shows that an m by m + 1 matrix with at least three nonzero entries in each row is not an S*-matrix.

Abstract

A permutation group G of degree n has a natural induced action on words of length n over a finite alphabet Σ, in which the image xg of x under permutation g ∈ G is obtained by permuting the positions of symbols in x according to g. The key result in “Pólya theory” is that the number of orbits of this action is given by an evaluation of the cycleindex polynomial PG(z1, …, zn) of G at the point z1, …, zn = |Σ|. In many cases it is possible to count the number of essentially distinct instances of a combinatorial structure of a given size by evaluating the cycle-index polynomial of an appropriate symmetry group G.

We address the question “to what extent can Polya theory be mechanised?” There are compelling complexity-theoretic reasons for believing that there is no efficient, uniform procedure for computing the cycle-index polynomial exactly, but less is known about approximate evaluation, say to within a specified relative error. The known results — positive and negative — will be surveyed.

Preliminaries

This article is concerned with a topic in computational algebra, which combines aspects of combinatorics, algorithmics, and computational complexity. On the assumption that most readers will be unfamiliar with at least one of these, the first section aims to give a brief account of key facts.

Elementary group-theoretic preliminaries

Let Σ be a finite alphabet of cardinality k, and G a permutation group on [n] = {0, …,n – 1}.

Introduction

A number of graph-theoretic models, involving various kinds of diffusion processes, lead to basically one and the same issue of “global connectivity” of the graph. These models include: random walks on graphs, especially their use in sampling algorithms; the “avalanche” or “sandpile” model of catastrophic events, which is mathematically equivalent to “chip-firing” games; load balancing in distributed networks; and, somewhat more distantly but clearly related, multicommodity flows and routing in VLSI. In this paper we survey some recent results on the first two topics, as well as their connections.

Random walks. The study of random walks on finite graphs, a.k.a. finite Markov chains, is one of the classical fields of probability theory.

INTRODUCTION

This year, 1995, is the centenary of the death of the Reverend Thomas Pennington Kirkman, Rector of Croft and Fellow of the Royal Society. In mathematical circles he is known, amongst other things, for his 1847 paper [Kl] in which he showed that Steiner triple systems of order v exist for all v ≡ 1 or 3 (mod 6). Since then, hundreds of papers have been written on many different aspects of such systems. Nevertheless, there are still fundamental but challenging questions which are unsolved. In this paper we consider just one of these concerned with the intersections of families of Steiner triple systems. We survey known results, present some recent advances, and pose a number of problems which suggest possible directions for future research in this area.

We start with some basic definitions. Recall that a Steiner triple system of order v (briefly STS(v)) is a pair (V, B) where V is a v-set, and B is a collection of 3-subsets of V called triples such that each 2-subset of V is contained in exactly one triple. A family (V,B1), …,(V,Bq) of q Steiner triple systems of order v, all on the same set V, is a large set of STS(v) if every 3-subset of V is contained in at least one STS of the collection. Two STS(v), (V,B1),(V,B2) are disjoint if B1 ∩ B2 = Ø, and almost disjoint if |B1 ∩ B2| = 1. Interest in families of disjoint STS also dates back to the last century: Cayley [C] determined that the maximum number of disjoint STS(7) is two and Sylvester [S] found a large set of 7 mutually disjoint STS(9).

INTRODUCTION

The discovery by Vaughan Jones in 1984 of a new polynomial invariant of links was the starting point of spectacular advances in knot theory which suddenly brought together previously unrelated concepts from various branches of mathematics and physics. One particularly fruitful idea was to consider a link diagram as an abstraction of a physical system of elementary objects (molecules, atoms, particles…) interacting in a local fashion. These local interactions are described by a statistical mechanical model. In the context of physics, much of the relevant information is then given by the partition function. The basic facts from the point of view of knot theory are that one can define natural conditions on the parameters of the model which insure that the partition function is a link invariant, and that one can actually find models satisfying these conditions which yield non-trivial link invariants. In fact, the models which correspond to link invariants are closely related to the exactly solvable or integrable models which are of particular interest in physics.

We shall be mainly concerned here with the version of this approach to the construction of link invariants which is based on spin models. There the local interactions can be viewed as taking place between the vertices of a graph and along the edges of this graph. The simplest case is that of the Potts models, which give rise in the context of graph theory to the Tutte polynomial, and in the context of knot theory to the above mentioned Jones polynomial. Spin models for link invariants can be defined in terms of matrices satisfying certain equations.

Introduction

One important area of research in Combinatorial Mathematics is that of the existence, construction and enumeration of designs of various different sorts. Given the general conditions denning a design it is often the case that very simple necessary conditions for its existence can be derived in terms of the so-called parameters of the design, but by and large it is very difficult to prove or disprove that the conditions obtained are also sufficient. Thus, even if necessary conditions are obtained we are generally left with either an immediate non-existence result or the problem of attempting to establish the existence by an explicit construction of the design. Of course, if the design concerned does not actually exist then this can often be a lengthy process. The first example that springs to mind is a projective plane of order 10, where the necessary conditions obtained by the Bruck-Ryser-Chowla Theorem do not disprove its existence, and yet the plane does not exist [16].

If the parameter sets of the designs are ‘small’ then it has often been the case that direct computer-free methods have led to a construction, and in some cases a complete classification has been achieved. In certain sporadic instances with ‘large’ parameter values it is also possible to construct and classify the corresponding designs without using a computer, but it is generally true that the larger the parameter set the more difficult the problem of determining all non-isomorphic designs with that given set of parameters.

INTRODUCTION

Consider a game of Twenty Questions in which someone thinks of a number between one and one million. A second person is allowed to ask questions to each of which the first person is supposed to answer only yes or no. Since one million is just less than 220, it is clear that a “halving” strategy (i.e. asking “Is the number in the first half million?”, and so on) will determine the number within twenty questions. But now suppose that up to some given number e of the answers may be lies. How many questions does one now need to get the right answer?

This is Ulam's searching problem, posed by Stanislaw Ulam (1976) in his autobiography “The Adventures of a Mathematician”.

The problem has recently been solved for all values of e (for the cases of both 220 and 106 objects). We give the solution in Figure 1 and an outline of the proof in Section 3.

More generally, we may consider the problem of finding the smallest number f(M,e) of yes-no questions sufficient to determine one of M objects if up to e of the answers may be lies. In Section 4, we survey the present state of knowledge regarding this function.

In Section 5, we consider a version of Ulam's problem without feedback, where all the questions must be asked in advance of receiving any answers. This is equivalent to a problem in the theory of error-correcting codes.

The British Combinatorial Conference returns to Scotland in 1995 with the fifteenth in a series of international meetings concerned with all branches of Combinatorics. Nine distinguished researchers, representing both Mathematics and Computing Science, have been invited to deliver the principal lectures, and this volume contains the survey articles which they have submitted in advance. These contributions certainly justify the early interest shown in the conference, and should pave the way towards another successful meeting. The second essential ingredient is the series of short talks presented by delegates out with the plenary sessions, and papers contributed in conjunction with these will be considered for a special edition of Discrete Mathematicsb edited by Douglas Woodall.

The year 1995 marks the hundredth anniversary of the death of the Rev.T.P.Kirkman, and one day of the conference will be designated ‘Kirkman Day’ in his honour. The talks on this day, including those by Rosa and Spence, will be devoted to topics related to Kirkman's achievements in Combinatorics.

I am grateful to the authors and referees for their co-operation in meeting the necessary deadlines, and I am indebted to Roger Astley of Cambridge University Press for his assistance in the preparation of the text. The British Combinatorial Committee acknowledges with thanks the financial support provided by the London Mathematical Society and the Institute of Combinatorics and its Applications.