Multi-stage design for quasipolynomial-time isomorphism testing of steiner 2-systems

• Authors: Xi Chen;Xiaorui Sun;Shang-Hua Teng
• Venue: Proceedings of the forty-fifth annual ACM symposium on Theory of computing
• Year: 2013

A standard heuristic for testing graph isomorphism is to first assign distinct labels to a small set of vertices of an input graph, and then propagate to create new vertex labels across the graph, aiming to assign distinct and isomorphism-invariant labels to all vertices in the graph. This is usually referred to as the individualization/refinement method for canonical labeling of graphs. We present a quasipolynomial-time algorithm for isomorphism testing of Steiner 2-systems. A Steiner 2-system consists of points and lines, where each line passes the same number of points and each pair of points uniquely determines a line. Each Steiner 2-system induces a Steiner graph, in which vertices represent lines and edges represent intersections of lines. Steiner graphs are an important subfamily of strongly regular graphs whose isomorphism testing has challenged researchers for years. Inspired by both the individualization/refinement method and the previous analyses of Babai and Spielman, we consider an extended framework for isomorphism testing of Steiner 2-systems, in which we use a small set of randomly chosen points and lines to build isomorphism-invariant multi-stage combinatorial structures that are sufficient to distinguish all pairs of points of a Steiner 2-system. Applying this framework, we show that isomorphism of Steiner 2-systems with n lines can be tested in time smash{nO(log n)}, improving the previous best bound of smash{exp(~{O}(n1/4))} by Spielman. Before our result, quasipolynomial-time isomorphism testing was only known for the case when the line size is polylogarithmic, as shown by Babai and Luks. A result essentially identical to ours was obtained simultaneously by Laszlo Babai and John Wilmes. They performed a direct analysis of the individualization/refinement method, building on a different philosophy and combinatorial structure theory. We comment on how this paper fits into the overall project of improved isomorphism testing for strongly regular graphs (the ultimate goal being subexponential exp(no(1)) time). In the remaining cases, we only need to deal with strongly regular graphs satisfying "Neumaier's claw bound," permitting the use of a separate set of asymptotic structural tools. In joint work (in progress) with Babai and Wilmes, we address that case and have already pushed the overall bound below smash{exp(~{O}(n1/4))}. The present paper is a methodologically distinct and stand-alone part of the overall project.