An improved isomorphism test for bounded-tree-width graphs

TL;DR

This paper presents a faster fixed-parameter tractable algorithm for testing isomorphism of graphs with bounded tree width, improving previous methods by refining graph decomposition and automorphism group analysis.

Contribution

It introduces an improved isomorphism test for bounded-tree-width graphs with better running time and a canonization algorithm, utilizing advanced group-theoretic techniques.

Findings

New algorithm runs in time 2^{k polylog(k)} poly(n).

Provides structural restrictions on automorphism groups of bounded-tree-width graphs.

Offers a canonization algorithm with slightly worse but practical running time.

Abstract

We give a new fpt algorithm testing isomorphism of $n$-vertex graphs of tree width $k$ in time $2^{k\operatorname{polylog} (k)}\operatorname{poly} (n)$, improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time $2^{\mathcal{O}(k^5\log k)}\operatorname{poly} (n)$. Based on an improved version of the isomorphism-invariant graph decomposition technique introduced by Lokshtanov et al., we prove restrictions on the structure of the automorphism groups of graphs of tree width $k$. Our algorithm then makes heavy use of the group theoretic techniques introduced by Luks (JCSS 1982) in his isomorphism test for bounded degree graphs and Babai (STOC 2016) in his quasipolynomial isomorphism test. In fact, we even use Babai's algorithm as a black box in one place. We also give a second algorithm which, at the price of a slightly worse running time $2^{\mathcal{O}(k^2 \log k)}\operatorname{poly} (n)$, avoids the use of Babai's algorithm and, more importantly, has the additional benefit that it can also used as a canonization algorithm.