- Markus Pfeiffer (School of Computer Science, St Andrews, UK)
- Christopher Jefferson (School of Computer Science, St Andrews, UK)

Backtrack search is a vital part of solving many problems in computational group theory, such as graph isomorphism, finding (non-point) stabilisers and normalisers in permutation groups, finding canonical images of objects under a group action, or short solutions to equations over a free group.

Efficient backtrack search implementations require a symbiosis of efficient algorithms, high-performance code, and sophisticated mathematical methods to prune search space. Some of the most prominent backtrack methods in computational mathematics are McKay’s graph isomorphism algorithm, Jeffrey Leon’s partition backtrack method for permutation groups, and Bernd Schmalz’ “snakes and ladders” algorithm.

Many breakthroughs in AI-search, such as learning, heuristics and parallelism, can improve performance by multiple orders of magnitude, and are applicable in computational group theory.

We would like to invite experts from AI, combinatorics, computational group theory and related areas with the aim of sharing and exchanging ideas, problems, results and implementations.

The Partition Backtracking algorithm of Jeffrey S. Leon has provided the basis for the state of the art for a large range of permutation group problems, including intersection, stabilizer and normaliser, for almost 30 years. This algorithm is however poorly understood.

This talk will give a rapid overview of the most important parts of the algorithm. In particular, I will talk about refiners, which are the central core of the algorithm. I will demonstrate some new refiners, and show how adding new refiners can provide and easy way to both improve performance on existing problems and extend existing partition backtrack systems to support new problems.

Leon claims that a method related to his celebrated Partition Backtrack Algorithm (PBA) can be used to solve canonical-representative-type problems, however speciﬁc details were never given. Although the PBA has been used to vastly speed up computation of normalisers and subgroup conjugacy in permutation groups, there has been very little progress made in methods to ﬁnd canonical subgroups up to conjugacy.

In this talk I describe how to combine aspects of the PBA and McKay’s graph isomorphism algorithm nauty to ﬁnd canonical images of groups up to conjugacy in the symmetric group. I will focus on speciﬁc diﬀerences between the two algorithms and how they relate to pruning the search tree. In addition, I will address practical concerns such as implementation and performance details.

Short abstract: Building on previous work on orbital graphs, we consider questions of theoretical interest that are also directly relevant for backtrack search algorithms. The underlying idea is to use orbital graphs as a tool for pruning the search tree in backtrack methods. Just two examples: Can we characterise the groups that can be recognised from their orbital graphs? When dealing with groups that have orbital graphs of different “quality” as a pruning tool, how can we efficiently detect the best graphs?

We consider the problem of classifying combinatorial-geometric structures by computer. A BLT-set is a set of q + 1 points in an orthogonal space Q(4,q) deﬁned over a ﬁnite ﬁeld with q elements. The points satisfy a quadratic equation such as x²₀+x₁x₂+x₃x₄=0 and there is a triple condition which says that no other point on the quadric is collinear to three of the points in the chosen set. The problem is that of classifying the possibilities of these sets up to equivalence under the orthogonal group.

After some reductions which involve the symmetry of the quadric, the problem can be phrased as a problem of ﬁnding rainbow cliques in suitable graphs. This is a diﬃcult problem in computer science and quite a few CPU-cycles have been utilized to push the classiﬁcation further. Several inﬁnite families of BLT-sets are known but there is still a lot of example which are sporadic. The present talk describes work to make the search more eﬃcient and extend the classiﬁcation to the next open case, which is when q = 73. Our computations are facilitated using the C++ package Orbiter as well as some independent implementations ustilizing multithreading.

Many problems in Combinatorics and related ﬁelds reduce to the problem of computing orbits of groups acting on ﬁnite sets. These problems are about classifying objects of a certain type. The main techniques are these: There is canonical augmentation and Snakes and Ladders. In this talk, we would like to give the theoretical framework for both techniques and compare the performance of some of the available implementations. Orbiter is a C++ package which provides an implementation of Snakes and Ladders. Another implementation exists in the computer algebra system GAP, using the Fining package for ﬁnite geometry. An implementation in Magma would be desirable. The canonical augmentation algorithm has been implemented in Nauty and more recently in Traces.

Solving equations in free (and hyperbolic) groups has been of theoretical interest in geometric and combinatorial group theory. As one central example, the known algorithm to decide isomorphism of finitely presented hyperbolic groups requires solutions to equations over free groups.

We demonstrate some recent progress in solving equations over free groups using some techniques from constraint solving, and some bespoke techniques for free groups.

In this presentation a universal strategy to solve isomorphism problems is developed. The strategy is based on the homomorphism principle for group actions, double cosets and the ”Snakes and Ladders” algorithm.

Many discrete structures can be built from simpler ones by using the fundamental homomorphism theorem applied to group actions. For a wide range of discrete structures this is not applicable, since there is no suitable unidirectional chain of homomorphisms. Following the ’snakes and ladders’ strategy by using homomorphisms in both directions, an appropriate chain of homomorphisms (ladder) can be built.

The original snakes and ladders algorithm had several drawbacks, namely the extensive demand on memory and the inadequate runtime when used for single structures. These downsides have now been overcome by extending the mathematical basis by the so-called ”Strong Paths”. This strategy is applicable for the construction of double cosets. Since many isomorphism problems can be expressed by means of double cosets they can be solved by this approach.

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