Novel Techniques for Automorphism Group Computation

Abstract

Abstract. Graph automorphism (GA) is a classical problem, in which the objective is to compute the automorphism group of an input graph. In this work we propose four novel techniques to speed up algorithms that solve the GA problem by exploring a search tree. They increase the performance of the algorithm by allowing to reduce the depth of the search tree, and by effectively pruning it. We formally prove that a GA algorithm that uses these techniques correctly computes the automorphism group of the input graph. We also describe how the techniques have been incorporated into the GA algorithm conauto, as conauto-2.03, with at most an additive polynomial increase in its asymptotic time complexity. We have experimentally evaluated the impact of each of the above techniques with several graph families. We have observed that each of the techniques by itself significantly reduces the number of processed nodes of the search tree in some subset of graphs, which justifies the use of each of them. Then, when they are applied together, their effect is combined, leading to reductions in the number of processed nodes in most graphs. This is also reflected in a reduction of the running time, which is substantial in some graph families.

Engineering a Fast Probabilistic Isomorphism Test

“…With the exception of conauto [12], modern practical tools have no specific mode for isomorphism testing, hence either computing entire automorphism groups or canonical labelings [6-8, 11, 15]. In their implementation, it is often the case that computing canonical labelings is significantly more expensive than automorphism group computation, since there are fewer known algorithmic techniques and tricks that can be applied (see [9,15]).…”

“…Indeed, the solvers rely on finding the entire automorphism group to then in turn prune search for the canonical form of the graph. In fact, even the state-of-the-art isomorphism test mentioned earlier [12] collects automorphisms when testing for isomorphism.…”

A Characterization of Individualization-Refinement Trees

“…These include algorithms computing automorphism groups, isomorphism solvers, canonical labeling tools used for computing normal forms, and to some extent recently also machine learning computations in convolutional neural networks [1,17]. In fact all competitive graph isomorphism/automorphism solvers, specifically nauty/Traces [15,16], bliss [11,12], saucy [8,9], conauto [13,14], and dejavu [2,3] fall within the framework. These tools alternate colorrefinement techniques (such as the 1-dimensional Weisfeiler-Leman algorithm) with backtracking steps.…”

Parallel Computation of Combinatorial Symmetries

“…The current state-of-the-art implementations of solvers computing automorphism groups are bliss [12], nauty and Traces [18], conauto [17] as well as saucy [7]. All of the mentioned algorithms follow the individualization-refinement (IR) framework.…”