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(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 t⁰ 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 (t⁰ + 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.