Permutation group algorithms based on directed graphs

TL;DR

This paper presents a novel graph-based framework for permutation group algorithms that generalizes existing methods, leading to more efficient search processes for problems like stabilizer calculations and isomorphism testing.

Contribution

It introduces a new graph-based approach to permutation group algorithms, extending the partition backtrack method with directed graphs for improved efficiency.

Findings

Framework generalizes partition backtrack

Results show smaller search spaces in experiments

Implementation available in GAP's GraphBacktracking package

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 are inspired by and generalise 'partition backtrack', which is the current state-of-the-art algorithm introduced by Jeffrey Leon in 1991. But, instead of ordered partitions, we use labelled directed graphs to organise our backtrack search algorithms, which allows for a richer representation of many problems while often resulting in smaller search spaces. In this article we present the theory underpinning our framework, we describe our algorithms, and we show the results of some experiments. An implementation of our algorithms is available as free software in the GraphBacktracking package for GAP.