Groups, Graphs, Algorithms: The Graph Isomorphism Problem∗

Graph Isomorphism (GI) is one of a small number of natural algorithmic problems with unsettled complexity status in the P /NP theory: not expected to be NP-complete, yet not known to be solvable in polynomial time. Arguably, the GI problem boils down to filling the gap between symmetry and regularity, the former being defined in terms of automorphisms, the latter in terms of equations satisfied by numerical parameters. Recent progress on the complexity of GI relies on a combination of the asymptotic theory of permutation groups and asymptotic properties of highly regular combinatorial structures called coherent configurations. Group theory provides the tools to infer either global symmetry or global irregularity from local information, eliminating the symmetry/regularity gap in the relevant scenario; the resulting global structure is the subject of combinatorial analysis. These structural studies are melded in a divide-and-conquer algorithmic framework pioneered in the GI context by Eugene M. Luks (1980).