Permutation group algorithms based on directed graphs

Jefferson, Christopher; Pfeiffer, Markus; Wilson, W. A.; Waldecker, Rebecca

doi:10.1016/j.jalgebra.2021.06.015

Cited by 9 publications

(11 citation statements)

References 6 publications

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“…We performed our experiments using the GraphBacktracking [6] and Back-trackKit [7] packages for GAP [3]. BacktrackKit provides a simple implementation of the algorithms in [8,11,12], and provides a base for GraphBacktracking.…”

Section: Methodsmentioning

confidence: 99%

Permutation group algorithms based on directed graphs

Jefferson¹,

Pfeiffer²,

Waldecker³

et al. 2021

Preprint

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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.

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Section: Methodsmentioning

confidence: 99%

Permutation group algorithms based on directed graphs

Jefferson¹,

Pfeiffer²,

Waldecker³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The results discussed here build on work in [JPWW21], and we include elements of an earlier draft of that article [JPWW19, Section 5.1]. We therefore thank the VolkswagenStiftung (Grant no.…”

Section: Introductionmentioning

confidence: 91%

“…In this section, we give some notation and definitions, and we introduce background concepts including backtrack search in the symmetric group. Our terminology and notation closely follows that of [JPWW21].…”

Section: Background and Notationmentioning

confidence: 99%

“…In each step down into the recursion, new stacks are appended to the existing ones. This may be done by a refiner, in an attempt to remove redundant parts of the search space (Section 3), or by a splitter, in order to divide the search space into smaller parts that can be searched recursively (see [JPWW21,Section 6]).…”

Section: Classical Backtracking Partition Backtracking and Graph Back...mentioning

confidence: 99%

“…This technique was enhanced with orbital graphs in [The97,JPW19]. More recently, this use of orbital graphs inspired a reformulation in [JPWW21] around labelled digraphs.…”

Section: Introductionmentioning

confidence: 99%

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Perfect refiners for permutation group backtracking algorithms

Jefferson¹,

Waldecker²,

Wilson³

2021

Preprint

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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 2closed groups.

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