Faster Canonical Forms for Primitive Coherent Configurations: Extended Abstract

Abstract

Primitive coherent configurations (PCCs) are edge-colored digraphs that generalize strongly regular graphs (SRGs), a class perceived as difficult for Graph Isomorphism (GI). Moreover, PCCs arise naturally as obstacles to combinatorial divide-and-conquer approaches for general GI. In a natural sense, the isomorphism problem for PCCs is a stepping stone between SRGs and general GI. In his 1981 paper in the Annals of Math., Babai proposed a combinatorial approach to GI testing via an analysis of the standard individualization/refinement (I/R) technique and proved that I/R yields canonical forms of PCCs in time exp(~O(n1/2)). (The tilde hides polylogarithmic factors.) We improve this bound to exp(~O(n1/3)). This is faster than the current best bound, exp(~O(n1/2)), for general GI, and subsumes Spielman's exp(~O(n1/3)) bound for SRGs (STOC'96, only recently improved to exp(~O(n1/5)) by the present authors and their coauthors (FOCS'13)). Our result implies an exp(~O(n1/3)) upper bound on the number of automorphisms of PCCs with certain easily described and recognized exceptions, making the first progress in 33 years on an old conjecture of Babai. The emergence of exceptions illuminates the technical difficulties: we had to separate these cases from the rest. For the analysis we develop a new combinatorial structure theory for PCCs that in particular demonstrates the presence of "asymptotically uniform clique geometries" among the constituent graphs of PCCs in a certain range of the parameters. A corollary to Babai's 1981 result was an exp(~O(n1/2)) upper bound on the order of primitive but not doubly transitive permutation groups, solving a then 100-year old problem in group theory. An improved bound of exp(~O(n1/3)) (with known exceptions) follows from our combinatorial result. This bound was previously known (Cameron, 1981) only through the Classification of Finite Simple Groups. We note that upper bounds on the order of primitive permutation groups are central to the application of Luks's group theoretic divide-and-conquer methods to GI.