Private coins versus public coins in interactive proof systems
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Trading group theory for randomness
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Does co-NP have short interactive proofs?
Information Processing Letters
On the degree of transitivity of permutation groups: A short proof
Journal of Combinatorial Theory Series A
Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity class
Journal of Computer and System Sciences - 17th Annual ACM Symposium in the Theory of Computing, May 6-8, 1985
On the diameter of Cayley graphs of the symmetric group
Journal of Combinatorial Theory Series A
Explicit construction of linear sized tolerant networks
Discrete Mathematics - First Japan Conference on Graph Theory and Applications
Journal of the ACM (JACM)
Permutation groups without exponentially many orbits on the power set
Journal of Combinatorial Theory Series A
Faster isomorphism testing of strongly regular graphs
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Isomorphism of graphs with bounded eigenvalue multiplicity
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Graph isomorphism, general remarks
STOC '77 Proceedings of the ninth annual ACM symposium on Theory of computing
On the nlog n isomorphism technique (A Preliminary Report)
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Canonical labelling of graphs in linear average time
SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
Computational complexity and the classification of finite simple groups
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
Quasipolynomial-time canonical form for steiner designs
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Multi-stage design for quasipolynomial-time isomorphism testing of steiner 2-systems
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We derive structural constraints on the automorphism groups of strongly regular (s.r.) graphs, giving a surprisingly strong answer to a decades-old problem, with tantalizing implications to testing isomorphism of s.r. graphs, and raising new combinatorial challenges. S.r. graphs, while not believed to be Graph Isomorphism (GI) complete, have long been recognized as hard cases for GI, and, in this author's view, present some of the core difficulties of the general GI problem. Progress on the complexity of testing their isomorphism has been intermittent (Babai 1980, Spielman 1996, BW & CST (STOC'13) and BCSTW (FOCS'13)), and the current best bound is exp(Õ(n1/5)) (n is the number of vertices). Our main result is that if X is a s.r. graph then, with straightforward exceptions, the degree of the largest alternating group involved in the automorphism group Aut(X) (as a quotient of a subgroup) is O((ln n)2ln ln n). (The exceptions admit trivial linear-time GI testing.) The design of isomorphism tests for various classes of structures is intimately connected with the study of the automorphism groups of those structures. We include a brief survey of these connections, starting with an 1869 paper by Jordan on trees. In particular, our result amplifies the potential of Luks's divide-and-conquer methods (1980) to be applicable to testing isomorphism of s.r. graphs in quasipolynomial time. The challenge remains to find a hierarchy of combinatorial substructures through which this potential can be realized. We expect that the generality of our result will help in this regard; the result applies not only to s.r. graphs but to all graphs with strong spectral expansion and with a relatively small number of common neighbors for every pair of vertices. We state a purely mathematical conjecture that could bring us closer to finding the right kind of hierarchy. We also outline the broader GI context, and state conjectures in terms of "primitive coherent configurations." These are generalizations of s.r. graphs, relevant to the general GI problem. Another consequence of the main result is the strongest argument to date against GI-completeness of s.r. graphs: we prove that no polynomial-time categorical reduction of GI to isomorphism of s.r. graphs is possible. All known reductions between isomorphism problems of various classes of structures fit into our notion of "categorical reduction." The proof of the main result is elementary; it is based on known results in spectral graph theory and on a 1987 lemma on permutations by Ákos Seress and the author.