Approximating Pathwidth for Graphs of Small Treewidth

Abstract

We describe a polynomial-time algorithm which, given a graphGwith treewidtht, approximates the pathwidth ofGto within a ratio of\(O(t\sqrt {\log t})\). This is the first algorithm to achieve anf(t)-approximation for some functionf.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 leastth+2 has treewidth at leasttor contains a subdivision of a complete binary tree of heighth+1. The boundth+2 is best possible up to a multiplicative constant. This result was motivated by, and implies (withc=2), the following conjecture of Kawarabayashi and Rossman (SODA’18): there exists a universal constantcsuch that every graph with pathwidth Ω(kc) has treewidth at leastkor contains a subdivision of a complete binary tree of heightk.Our main technical algorithm takes a graphGand some (not necessarily optimal) tree decomposition ofGof widtht′ in the input, and it computes in polynomial time an integerh, a certificate thatGhas pathwidth at leasth, and a path decomposition ofGof 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 heighth. 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.