Perfect refiners for permutation group backtracking algorithms

TL;DR

This paper introduces the concept of perfect refiners to enhance backtrack search efficiency in permutation group algorithms, providing a unified framework and new formulations that improve pruning power and applicability.

Contribution

It defines perfect refiners, classifies their existence across various group settings, and implements a new backtrack search formulation in the Vole software.

Findings

Classifies groups and cosets with perfect refiners.

Provides perfect refiners for stabilisers and transporter sets.

Applies to normaliser and subgroup conjugacy problems for 2-closed groups.

Abstract

Backtrack search is a fundamental technique for computing with finite permutation groups, which has been formulated in terms of points, ordered partitions, and graphs. We provide a framework for discussing the most common forms of backtrack search in a generic way. We introduce the concept of perfect refiners to better understand and compare the pruning power available in these different settings. We also present a new formulation of backtrack search, which allows the use of graphs with additional vertices, and which is implemented in the software package Vole. For each setting, we classify the groups and cosets for which there exist perfect refiners. Moreover, we describe perfect refiners for many naturally-occurring examples of stabilisers and transporter sets, including applications to normaliser and subgroup conjugacy problems for 2-closed groups.