Multi-stage design for quasipolynomial-time isomorphism testing of steiner 2-systems
Xi Chen, Xiaorui Sun, Shang‐Hua Teng · 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.