Abstract
Graph automorphism (GA) is a classical problem, in which the objective is to compute the automorphism group of an input graph. Most GA algorithms explore a search tree using the individualization-refinement procedure. Four novel techniques are proposed which increase the performance of any algorithm of this type by reducing 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 an input graph. Then, we describe how these techniques have been incorporated into the GA algorithm conauto, asconauto-2.03, with at most an additive polynomial increase in its asymptotic time complexity. Using a benchmark of different graph families, we have evaluated the impact of these techniques on the size of the search tree, observing a significant reduction both when they are applied individually and when all of them are applied together. This is also reflected in a reduction of the running time, which is substantial for some graph families. Finally, we have compared the search tree size of conauto-2.03 against those of other popular GA algorithms, observing that, in most cases, conauto explores less nodes than these algorithms.
Highlights
Graph automorphism (GA), graph isomorphism (GI), and finding of a canonical labeling (CL) are closely related classical graph problems that have applications in many fields, ranging from mathematical chemistry [1, 2] to computer vision [3]
We propose a novel approach to early automorphism detection (EAD) which allows infering an automorphism without the need to reach a leaf-node of the search tree
We compare the size of the search tree of conauto-2.03 against those of nauty, Traces, saucy, and bliss
Summary
Graph automorphism (GA), graph isomorphism (GI), and finding of a canonical labeling (CL) are closely related classical graph problems that have applications in many fields, ranging from mathematical chemistry [1, 2] to computer vision [3]. This technique helps, for example, when not all the components in a component-based graph are isomorphic Note that this backjumping only relies on the structure of the partitions at the nodes of the search tree, while the backjumping proposed in [15] relies on the conflicts found during the search for automorphisms. If the number of times a certain conflict has been found at a node (not in the first-path) exceeds the number of times it was found in the node at the same level of the first-path, no more branches need to be explored in this node This technique helps pruning the search tree in a large variety of graph families, and it is an improvement over the conflict propagation described in [7].
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