ix Chapter 1 Introduction 1 1.1 Algorithmic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Parallel Computational Complexity . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.5 Random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.6 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 2 Parallel Uniform Generation of Unlabelled Graphs 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Sampling O...

Let be a finite set and let G be a permutation group acting on The permutation group G partitions into orbits. This survey focuses on three related computational problems, each of which is defined with respect to a particular input set I. The problems, given an input ( ; G) 2 I, are (1) count the orbits (exactly), (2) approximately count the orbits, and (3) choose an orbit uniformly at random. The goal is to quantify the computational diculty of the problems. In particular, we would like to know for which input sets I the problems are tractable.

Informally, an \unlabelled combinatorial structure" is an object such as an unlabelled graph (in which the vertices are indistinguishable) or a structural isomer in chemistry (in which dierent atoms of the same type are indistinguishable). Computational experiments such as those described in this volume often rely on random sampling to generate inputs for the experiments. This paper surveys work on the problem of eciently sampling unlabelled combinatorial structures from a uniform distribution. 1 Introduction Most of the experimental work described in this volume involves rst randomly sampling combinatorial structures and second using the randomly-chosen structures as inputs to computational experiments. In order for the experiments to be valid, the distribution from which the combinatorial structures are drawn must be precisely specied. In order for the experiments to be computationally feasible, the random-sampling algorithms must be ecient. This survey is devoted to the ...

The substitution method has proven to be an effective tool for bounding the critical probability of a variety of percolation models. Nevertheless, until recently substitution method calculations have been done by hand. This has severely restricted the size of computationally feasible substitution regions. We examine the computational problems posed by the substitution method, beginning with an analysis of the component calculations. We seek to better understand the nature of the computational problem, hoping that better understanding will lead to improvements. Our goal is a little different from that of most algorithmic investigations. Since the substitution method constitutes a proof, there is little reason to perform a particular computation more than once. We use each speed improvement to attempt a new, larger computation that will lead to tighter bounds on the critical probability. Our major results can be grouped into two categories: recognition of links between substitution method calculations and well-known results in other areas of mathematics,