Quasipolynomial-time canonical form for steiner designs

A Steiner 2-design is a finite geometry consisting of a set of "points" together with a set of "lines" (subsets of points of uniform cardinality) such that each pair of points belongs to exactly one line. In this paper we analyse the individualization/refinement heuristic and conclude that after individualizing O(log n) points (assigning individual colors to them), the refinement process gives each point an individual color. The following consequences are immediate: (a) isomorphism of Steiner 2-designs can be tested in nO(log n) time, where n is the number of lines; (b) a canonical form of Steiner 2-designs can be computed within the same time bound; (c) all isomorphisms between two Steiner 2-designs can be listed within the same time bound; (d) the number of automorphisms of a Steiner 2-design is at most nO(log n) (a fact of interest to finite geometry and group theory.) The best previous bound in each of these four statements was moderately exponential, exp(~O(n1/4)) (Spielman, STOC'96). Our result removes an exponential bottleneck from Spielman's analysis of the Graph Isomorphism problem for strongly regular graphs. The results extend to Steiner t-designs for all t≥2. Strongly regular (s.r.) graphs have been known as hard cases for graph isomorphism testing; the best previously known bound for this case is moderately exponential, exp(~O(n1/3)) where n is the number of vertices (Spielman, STOC'96). Line graphs of Steiner 2-designs enter as a critical subclass via Neumaier's 1979 classification of s.r. graphs. Previously, nO(log n) isomorphism testing and canonical forms for Steiner 2-designs was known for the case when the lines of the Steiner 2-design have bounded length (Babai and Luks, STOC'83). That paper relied on Luks's group-theoretic divide-and-conquer algorithms and did not yield a subexponential bound on the number of automorphisms. To analyse the individualization/refinement heuristic, we develop a new structure theory of Steiner 2-designs based on the analysis of controlled growth and on an addressing scheme that produces a hierarchy of increasing sets of pairwise independent, uniformly distributed points. This scheme represents a new expression of the structural homogeneity of Steiner 2-designs that allows applications of the second moment method. We also address the problem of reconstruction of Steiner 2-designs from their line-graphs beyond the point of unique reconstructability, in a manner analogous to list-decoding, and as a consequence achieve an exp(~O(n1/6)) bound for isomorphism testing for this class of s.r. graphs. Results, essentially identical to our main results, were obtained simultaneously by Xi Chen, Xiaorui Sun, and Shang-Hua Teng, building on a different philosophy and combinatorial structure theory than the present paper. They do not claim an analysis of the individualization/refinement algorithm but of a more complex combinatorial algorithm. 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 need to deal with s.r. graphs satisfying "Neumaier's claw bound," permitting the use of a separate set of asymptotic structural tools. In joint work (in progress) with Chen, Sun, and Teng, we address that case and have already pushed the overall bound below exp(~O(n1/4)) The present paper is a methodologically distinct and stand-alone part of the overall project.