Losing Treewidth In The Presence Of Weights

TL;DR

This paper presents a randomized polynomial-time approximation algorithm for the Weighted Treewidth-$ta$ Deletion problem, extending previous results to weighted graphs and more general planar graph problems.

Contribution

It introduces a novel random sampling technique of protrusions to achieve constant-factor approximation for weighted treewidth deletion problems.

Findings

Provides a polynomial-time constant-factor approximation algorithm for weighted treewidth deletion.

Extends unweighted graph results to weighted graphs and planar F-M-Deletion problems.

Answers open questions from prior research on weighted treewidth problems.

Abstract

In the Weighted Treewidth-$\eta$ Deletion problem we are given a node-weighted graph $G$ and we look for a vertex subset $X$ of minimum weight such that the treewidth of $G-X$ is at most $\eta$. We show that Weighted Treewidth-$\eta$ Deletion admits a randomized polynomial-time constant-factor approximation algorithm for every fixed $\eta$. Our algorithm also works for the more general Weighted Planar $F$-M-Deletion problem. This work extends the results for unweighted graphs by [Fomin, Lokshtanov, Misra, Saurabh; FOCS '12] and answers a question posed by [Agrawal, Lokshtanov, Misra, Saurabh, Zehavi; APPROX/RANDOM '18] and [Kim, Lee, Thilikos; APPROX/RANDOM '21]. The presented algorithm is based on a novel technique of random sampling of so-called protrusions.