Treewidth Inapproximability and Tight ETH Lower Bound

TL;DR

This paper proves that approximating Treewidth within a factor of 1.00005 is NP-hard and that exact computation requires exponential time, establishing tight lower bounds under the ETH assumption.

Contribution

It introduces a simple linear reduction from 3-SAT to Treewidth, establishing new hardness and time complexity bounds for approximation and exact computation.

Findings

1.00005-approximation of Treewidth is NP-hard

Exact Treewidth computation requires exponential time

Approximation within certain factors requires subexponential time under ETH

Abstract

We present a simple, self-contained, linear reduction from 3-SAT to Treewidth. Specifically, it shows that 1.00005-approximating Treewidth is NP-hard, and solving Treewidth exactly requires $2^{\Omega(n)}$ time, unless the Exponential-Time Hypothesis fails. We further derive, under the latter assumption, that there are some constants $\delta > 1$ and $c>0$ such that $\delta$-approximating Treewidth requires time $2^{\Omega(n/\log^c n)}$.