Approximating Pathwidth for Graphs of Small Treewidth

Abstract

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.