Graph Isomorphism in Quasipolynomial Time

TL;DR

This work proves GI, SI, and CI admit quasipolynomial-time algorithms, significantly advancing beyond the longstanding exp(√n) barrier. The core strategy blends Luks's divide-and-conquer framework with two novel symmetry-breaking tools: local certificates (which either build global automorphisms or certify local obstructions) and combinatorial partitioning that isolates a canonically embedded Johnson graph as the primary obstruction. Central to the approach are the Design Lemma and the Split-or-Johnson framework, which together reduce high-arity relations to binary structures and then exploit Johnson-group actions to drive recursive reductions. The analysis leverages Weisfeiler-Leman refinements, CFSG consequences (via Cameron), and a careful cost accounting that yields a final bound of exp((log n)^{O(1)}). The results establish a near-complete understanding of GI’s complexity landscape, highlighting the pivotal role of Johnson groups and local-to-global symmetry, with implications for related isomorphism problems and complexity theory.

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 φ 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 φ if the φ-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.

Theorems & Definitions (240)

• Theorem 1.1.1
• Corollary 1.1.2
• Theorem 1.1.3
• Corollary 1.1.4
• Theorem 1.3.1
• Remark 1.3.2
• Theorem 1.3.3
• Definition 2.2.2
• Theorem 2.2.3: Bochert, 1897
• Definition 2.3.1: Induced substructure