Treewidth Inapproximability and Tight ETH Lower Bound
Résumé
Despite the (algorithmic) importance of treewidth, both its complexity and approximability present large knowledge gaps. While the best polynomial-time approximation algorithm has ratio O( √ log OPT), no approximation factor could be ruled under P = NP alone. There are 2 O(n) -time algorithms to compute the treewidth of n-vertex graphs, but the Exponential-Time Hypothesis (ETH) was only known to imply that 2 Ω( √ n) 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 Ω(n) time, unless the ETH fails. We further derive, under the latter assumption, that there is some δ > 1 such that δ-approximating Treewidth requires time 2 n 1-o(1) .
1 See their definition in Section 2.2.
2 Some additive inapproximability is known for treewidth [6].
3 Assuming the so-called Small Subset Expansion conjecture (that the edge expansion of sublinear vertex subsets is hard to approximate), it can be showed that any constant-approximation of Treewidth is NP-complete [20].
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