Treewidth Inapproximability and Tight ETH Lower Bound

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 $2^{Ω(n)}$ time, unless the Exponential-Time Hypothesis fails. We further derive, under the latter assumption, that there are some constants $δ> 1$ and $c>0$ such that $δ$-approximating Treewidth requires time $2^{Ω(n/\log^c n)}$.