Vsep-New Heuristic and Exact Algorithms for Graph Automorphism Group Computation

TL;DR

This paper introduces one exact and two heuristic algorithms for computing graph automorphism groups, utilizing individualization and refinement, with heuristics significantly reducing runtime while maintaining near-accuracy.

Contribution

The paper presents novel heuristic strategies and a new exact algorithm for graph automorphism group computation, improving efficiency and introducing new cell selection methods.

Findings

Exact algorithm has exponential worst-case time complexity.

Heuristic algorithms run in polynomial time.

Heuristics are many times faster than the exact algorithm.

Abstract

One exact and two heuristic algorithms for determining the generators, orbits and order of the graph automorphism group are presented. A basic tool of these algorithms is the well-known individualization and refinement procedure. A search tree is used in the algorithms - each node of the tree is a partition. All nonequivalent discreet partitions derivative of the selected vertices are stored in a coded form. A new strategy is used in the exact algorithm: if during its execution some of the searched or intermediate variables obtain a wrong value then the algorithm continues from a new start point losing some of the results determined so far. The algorithms has been tested on one of the known benchmark graphs and shows lower running times for some graph families. The heuristic versions of the algorithms are based on determining some number of discreet partitions derivative of each vertex in the selected cell of the initial partition and comparing them for an automorphism - their search trees are reduced. The heuristic algorithms are almost exact and are many times faster than the exact one. The experimental tests exhibit that the worst-cases running time of the exact algorithm is exponential but it is polynomial for the heuristic algorithms. Several cell selectors are used. Some of them are new. We also use a chooser of cell selector for choosing the optimal cell selector for the manipulated graph. The proposed heuristic algorithms use two main heuristic procedures that generate two different forests of search trees.