Graph Isomorphism in Quasipolynomial Time

TL;DR

This paper presents a quasipolynomial time algorithm for solving the Graph Isomorphism problem and related problems, significantly improving previous bounds and employing group-theoretic and combinatorial techniques.

Contribution

It introduces a quasipolynomial time algorithm for GI, building on Luks's framework and overcoming previous barriers with new group-theoretic methods.

Findings

Graph Isomorphism can be solved in quasipolynomial time.

The algorithm extends to String Isomorphism and Coset Intersection problems.

Johnson graphs are identified as the main obstructions to canonical partitioning.

Abstract

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 ($\exp((\log n)^{O(1)})$) time. The best previous bound for GI was $\exp(O(\sqrt{n\log n}))$, where $n$ is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, $\exp(\tilde{O}(\sqrt{n}))$, 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 {\phi} that maps a given permutation group $G$ onto $S_k$ or $A_k$, 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 {\phi} if the {\phi}-image of the stabilizer of $x$ does not contain $A_k$. The affected/unaffected dichotomy underlies the core "local certificates" routine and is the central divide-and-conquer tool of the algorithm.