We describe a polynomial-time algorithm which, given a graph G with treewidth t, approximates the pathwidth of G to within a ratio of \(O(t\sqrt {\log t})\). This is the first algorithm to achieve an f(t)-approximation for some function f.

Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at least th+2 has treewidth at least t or contains a subdivision of a complete binary tree of height h+1. The bound th+2 is best possible up to a multiplicative constant. This result was motivated by, and implies (with c=2), the following conjecture of Kawarabayashi and Rossman (SODA’18): there exists a universal constant c such that every graph with pathwidth Ω(k^c) has treewidth at least k or contains a subdivision of a complete binary tree of height k.

Our main technical algorithm takes a graph G and some (not necessarily optimal) tree decomposition of G of width t′ in the input, and it computes in polynomial time an integer h, a certificate that G has pathwidth at least h, and a path decomposition of G of width at most (t′+1)h+1. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of height h. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC’05) for treewidth.

我们描述了一个多项式时间算法，给定一个树宽为t的图G ，将G的路径宽度近似为 \(O(t\sqrt {\log t})\) 的比率。这是第一个实现某个函数f的f(t)近似值的算法。

我们的方法基于以下关键见解：每个具有大路径宽度的图都具有大树宽或包含大完整二叉树的细分。具体来说，我们表明每个路径宽度至少为th +2 的图的树宽度至少为t或包含高度为h +1的完整二叉树的细分。边界th +2 最好达到乘法常数。这个结果是由以下 Kawarabayashi 和 Rossman (SODA'18) 的猜想推动并暗示（c = 2）：存在一个通用常数c使得每个路径宽度为 Ω( k ^c ) 的图的树宽度至少为k或包含高度为k的完整二叉树的细分。

我们的主要技术算法采用图G和输入中宽度为t ′ 的G的一些（不一定是最优的）树分解，并在多项式时间内计算整数h ， G 的路径宽度至少为h的证书，以及路径分解宽度至多为 ( t ′+1) h +1的G。该证书与高度为h的完全二叉树的细分的存在密切相关（并暗示）。然后通过将此算法与 Feige、Hajiaghayi 和 Lee (STOC'05) 的树宽近似算法相结合，获得路径宽度的近似算法。