  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, and in particular improves upon the previous best
approximation ratio of 5 in time $2^{O(k)} n$ given by Bodlaender et al. [SIAM
J. Comput., 45 (2016)]. Our algorithm works by applying incremental improvement
operations to a tree decomposition, using an approach inspired by a proof of
Bellenbaum and Diestel [Comb. Probab. Comput., 11 (2002)].
