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.

我们引入了一个新框架来解决一类涉及有限置换群的重要计算问题，其中包括计算集稳定器、子群的交集和组合结构的同构。我们的技术受到“分区回溯”的启发和概括，这是 Jeffrey Leon 在 1991 年引入的当前最先进的算法。但是，我们使用标记有向图来组织我们的回溯搜索算法，而不是有序分区，这允许更丰富地表示许多问题，同时通常会导致更小的搜索空间。在本文中，我们介绍了支撑我们框架的理论，描述了我们的算法，并展示了一些实验的结果。我们算法的实现可作为免费软件在GAP 的GraphBacktracking包。