Graph isomorphism in quasipolynomial time parameterized by treewidth

TL;DR

This paper presents a quasipolynomial-time graph isomorphism algorithm that leverages graph decompositions based on treewidth, extending Babai's earlier work to a broader class of graphs.

Contribution

It introduces a quasipolynomial-time algorithm for graph isomorphism parameterized by treewidth, generalizing Babai's quasipolynomial-time test.

Findings

Algorithm runs in time $n^{polylog(k)}$ for graphs with minimum treewidth $k$

Extends Babai's quasipolynomial-time graph isomorphism test

Utilizes graph decompositions within a group-theoretic framework

Abstract

We extend Babai's quasipolynomial-time graph isomorphism test (STOC 2016) and develop a quasipolynomial-time algorithm for the multiple-coset isomorphism problem. The algorithm for the multiple-coset isomorphism problem allows to exploit graph decompositions of the given input graphs within Babai's group-theoretic framework. We use it to develop a graph isomorphism test that runs in time $n^{\operatorname{polylog}(k)}$ where $n$ is the number of vertices and $k$ is the minimum treewidth of the given graphs and $\operatorname{polylog}(k)$ is some polynomial in $\operatorname{log}(k)$. Our result generalizes Babai's quasipolynomial-time graph isomorphism test.