Permutation group algorithms based on directed graphs (extended version)

TL;DR

This paper introduces a novel framework for permutation group algorithms using directed graphs, enhancing existing methods like partition backtrack, and demonstrates improved efficiency through experiments.

Contribution

It generalizes the partition backtrack algorithm by organizing backtrack search around directed graphs, enabling richer problem representations and often reducing search space.

Findings

Algorithms often produce smaller search spaces than partition backtrack

Framework effectively handles various permutation group problems

Implementation available in GraphBacktracking for GAP

Abstract

We introduce a new framework for solving an important class of computational problems involving finite permutation groups, which includes calculating set stabilisers, intersections of subgroups, and isomorphisms of combinatorial structures. Our techniques generalise 'partition backtrack', which is the current state-of-the-art algorithm introduced by Jeffrey Leon in 1991, and which has inspired our work. Our backtrack search algorithms are organised around vertex- and arc-labelled directed graphs, which allow us to represent many problems more richly than do ordered partitions. We present the theory underpinning our framework, and we include the results of experiments showing that our techniques often result in smaller search spaces than does partition backtrack. An implementation of our algorithms is available as free software in the GraphBacktracking package for GAP.