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)].

翻译：我们给出一个算法：对于给定的$n$顶点图$G$和整数$k$，该算法在时间$2^{O(k)} n$内要么输出一个宽度不超过$2k+1$的$G$的树分解，要么判定$G$的树宽大于$k$。这是首个比已知精确算法更快的树宽2近似算法，特别地，它改进了Bodlaender等人[SIAM J. Comput., 45 (2016)]在时间$2^{O(k)} n$内给出的先前最佳近似比5。我们的算法通过对树分解应用增量改进操作来实现，其思路受Bellenbaum与Diestel[Comb. Probab. Comput., 11 (2002)]中一个证明的启发。