Treewidth Inapproximability and Tight ETH Lower Bound

Despite the (algorithmic) importance of treewidth, both its complexity and approximability present large knowledge gaps. While the best currently known polynomial-time approximation algorithm has ratio.. (v. log OPT), no approximation factor could be ruled out under P. NP alone. There are 2.. (..) -time algorithms to compute the treewidth of..-vertex graphs, but the Exponential-Time Hypothesis (ETH) was only known to imply that 2.. (v..) time is required. The reason is that all the known hardness constructions use Cutwidth or Pathwidth on bounded-degree graphs as an intermediate step in a (long) chain of reductions, for which no inapproximability nor sharp ETH lower bound is known. We present a simple, self-contained reduction from 3-SAT to Treewidth. This starts filling the former gap, and completely fills the latter gap. Namely, we show that 1.00005-approximating Treewidth is NP-hard, and solving Treewidth exactly requires 2.. (..) time, unless the ETH fails. We further derive, under the latter assumption, that there are some constants.. > 1 and.. > 0 such that.. -approximating Treewidth requires time 2.. (../log....).