New refiners for permutation group search

Abstract

Partition backtrack is the current generic state of the art algorithm to search for subgroups of a given permutation group. We describe an improvement of partition backtrack for set stabilizers and intersections of subgroups by using orbital graphs. With extensive experiments we demonstrate that our methods improve performance of partition backtrack -in some cases by several orders of magnitude.

Permutation group algorithms based on directed graphs

“…For Leon we use the group and coset refiner described in [11]. For Orbital we use the DeepOrbital group and coset refiner from [8], although we get similar results for all the refiners described in that paper. For the Strong and Full techniques, we use refiners of the kind described in Section 8.1 for groups and cosets, using orbits and orbital graphs.…”

“…In this section, we provide experimental data comparing the behaviour of our algorithms against partition backtrack, in order to highlight the potential of our techniques. We repeat the experiments of [8,Section 6], which demonstrated improvements from using orbital graphs, and we also investigate some additional challenging problems. In many cases we observe a significant advancement with our new techniques.…”

“…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. We note that where we reproduce experiments from [8], we produce the same sized searches.…”

“…BacktrackKit provides a simple implementation of the algorithms in [8,11,12], and provides a base for GraphBacktracking. We note that where we reproduce experiments from [8], we produce the same sized searches.…”

“…By encoding information about the given problem into those partitions, it is possible to cleverly prune (i.e., omit superfluous parts of) the search space. There have already been some extensions and improvements, inspired by graph-based ideas of McKay (see for example [14] and [8]). This leads us to believe that more powerful pruning, and ultimately better performance, could be obtained by using graphs directly, at the expense of the increased computation required at each node of the remaining search.…”