Approximating pathwidth for graphs of small treewidth

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√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 $Ω(kc )$ 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.