László Babai

We show that the Graph Isomorphism (GI) problem and the related problems of String Isomorphism[12] (under group action) (SI) and Coset Intersection (CI)[13][14] can be solved in quasipolynomial ${\displaystyle \exp \left(\left(\log n\right)^{O\left(1\right)}\right)}$ time. The best previous bound for GI was ${\displaystyle \exp \left(O\left({\sqrt {n\log n}}\right)\right),}$ where ${\displaystyle n}$ is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, ${\displaystyle \quad \qquad \exp \left({\tilde {O}}\left({\sqrt {n}}\right)\right),}$ where ${\displaystyle n}$ 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.