Abstract

Let Ω be a finite set and let G be a permutation group acting on Ω. The permutation group G partitions Ω 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 (Ω, G) ϵ 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 difficulty of the problems. In particular, we would like to know for which input sets I the problems are tractable.

Introduction

Let Ω be a finite set and let G be a permutation group acting on Ω. The permutation group G partitions Ω into orbits: Two elements of Ω are in the same orbit if and only if there is a permutation in G which maps one element to the other. This survey focuses on three related computational problems, each of which is defined with respect to a particular input set I:

1. Given an input (Ω, G) ϵ I, count the orbits.

2. Given an input (Ω, G) ϵ I, approximately count the orbits.

3. Given an input (Ω, G) ϵ I, choose an orbit uniformly at random.

The goal is to quantify the computational difficulty of the problems. In particular, we would like to know for which input sets I the problems are tractable.

Many interesting orbit-counting problems come from the setting of “Polya theory”.