A Single-Exponential Time 2-Approximation Algorithm for Treewidth

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. In particular, our algorithm improves upon both the previous best approximation ratio of 5 in time 2(O(k))n and the previous best approximation ratio of 3 in time 2(O(k))n(O(1)), both given by Bodlaender et al. [FOCS 2013, SICOMP 2016]. Our algorithm is based on a local improvement method adapted from a proof of Bellenbaum and Diestel [Comb. Probab. Comput. 2002].