arXiv subjects
Publications and source records attributed to Wilf A. Wilson.
We describe a new algorithm for finding a canonical image of an object under the action of a finite permutation group. This algorithm builds on previous work using Graph Backtracking, which extends Jeffrey Leon's Partition Backtrack framework. Our methods generalise both Nauty and Steve Linton's Minimal image algorithm.
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.
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.
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.
We describe and count the maximal subsemigroups of many well-known monoids of transformations and monoids of partitions. More precisely, we find the maximal subsemigroups of the full spectrum of monoids of order- or orientation-preserving transformations and partial permutations considered by V. H. Fernandes and co-authors (12 monoids in total); the partition, Brauer, Jones, and Motzkin monoids; and certain further monoids. Although descriptions of the maximal subsemigroups of some of the aforementioned classes of monoids appear in the literature, we present a unified framework for determining these maximal subsemigroups. This approach is based on a specialised version of an algorithm for determining the maximal subsemigroups of any finite semigroup, developed by the third and fourth authors. This allows us to concisely present the descriptions of the maximal subsemigroups, and to more clearly see their common features.