On Approximating Cutwidth and Pathwidth

TL;DR

This paper presents improved approximation algorithms for cutwidth and pathwidth problems in graphs, utilizing a novel metric decomposition technique suitable for min-max objectives, advancing the state-of-the-art in graph ordering approximations.

Contribution

Introduces a new metric decomposition method that achieves near-logarithmic approximation ratios for cutwidth and pathwidth, surpassing previous recursive partitioning approaches.

Findings

Achieves $ ext{log}^{1+o(1)}(n)$ approximation for cutwidth.

Achieves $ ext{log}^{1+o(1)}(n)$ approximation for pathwidth.

Introduces a new metric decomposition technique for min-max graph problems.

Abstract

We study graph ordering problems with a min-max objective. A classical problem of this type is cutwidth, where given a graph we want to order its vertices such that the number of edges crossing any point is minimized. We give a $ \log^{1+o(1)}(n)$ approximation for the problem, substantially improving upon the previous poly-logarithmic guarantees based on the standard recursive balanced partitioning approach of Leighton and Rao (FOCS'88). Our key idea is a new metric decomposition procedure that is suitable for handling min-max objectives, which could be of independent interest. We also use this to show other results, including an improved $ \log^{1+o(1)}(n)$ approximation for computing the pathwidth of a graph.