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].