Graph Isomorphism in Quasipolynomial Time

We show that the Graph Isomorphism (GI) problem and the related problems of String Isomorphism (under group action) (SI) and Coset Intersection (CI) can be solved in quasipolynomial (exp ( (log n) ) ) time. The best previous bound for GI was exp(O( √ n logn)), where n is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, exp(Õ( √ n)), where n is the size of the permutation domain (Babai, 1983). The algorithm builds on Luks’s SI framework and attacks the barrier configurations for Luks’s algorithm by group theoretic “local certificates” and combinatorial canonical partitioning techniques. We show that in a well–defined sense, Johnson graphs are the only obstructions to effective canonical partitioning. Luks’s barrier situation is characterized by a homomorphism φ that maps a given permutation group G onto Sk or Ak, the symmetric or alternating group of degree k, where k is not too small. We say that an element x in the permutation domain on which G acts is affected by φ if the φ-image of the stabilizer of x does not contain Ak. The affected/unaffected dichotomy underlies the core “local certificates” routine and is the central divide-and-conquer tool of the algorithm. For a list of updates compared to the first arXiv version, see the last paragraph of the Acknowledgments. ∗ Research supported in part by NSF Grants CCF-7443327 (2014-current), CCF-1017781 (2010-2014), and CCF-0830370 (2008–2010). Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the author and do not necessarily reflect the views of the National Science Foundation (NSF).