A Single-Exponential Time 2-Approximation Algorithm for Treewidth

TL;DR

This paper presents the first faster-than-exact 2-approximation algorithm for treewidth, achieving a balance between approximation quality and computational efficiency for graphs.

Contribution

It introduces a novel 2-approximation algorithm for treewidth with a runtime of 2^{O(k)} n, improving over previous ratios and speeds.

Findings

First 2-approximation algorithm faster than exact methods

Achieves approximation ratio of 2 in time 2^{O(k)} n

Improves previous best ratio of 5 by Bodlaender et al.

Abstract

We give an algorithm that, given an $n$-vertex graph $G$ and an integer $k$, in time $2^{O(k)} n$ either outputs a tree decomposition of $G$ of width at most $2k + 1$ or determines that the treewidth of $G$ is larger than $k$. This is the first 2-approximation algorithm for treewidth that is faster than the known exact algorithms, and in particular improves upon the previous best approximation ratio of 5 in time $2^{O(k)} n$ given by Bodlaender et al. [SIAM J. Comput., 45 (2016)]. Our algorithm works by applying incremental improvement operations to a tree decomposition, using an approach inspired by a proof of Bellenbaum and Diestel [Comb. Probab. Comput., 11 (2002)].