Local Search is a PTAS for Feedback Vertex Set in Minor-free Graphs

TL;DR

This paper demonstrates that a straightforward local search algorithm provides a Polynomial-Time Approximation Scheme (PTAS) for the Feedback Vertex Set problem in minor-free graphs, simplifying previous complex approaches.

Contribution

It introduces a simple, easy-to-implement local search method that achieves a PTAS for FVS in minor-free graphs, contrasting with prior complex algorithms.

Findings

Local search yields a PTAS for FVS in minor-free graphs.

The local optimum is within a (1+ε) factor of the global optimum.

The approach is simpler and more practical than previous methods.

Abstract

We show that a simple local search gives a PTAS for the Feedback Vertex Set (FVS) problem in minor-free graphs. An efficient PTAS in minor-free graphs was known for this problem by Fomin, Lokshtanov, Raman and Sauraubh. However, their algorithm is a combination of many advanced algorithmic tools such as contraction decomposition framework introduced by Demaine and Hajiaghayi, Courcelle's theorem and the Robertson and Seymour decomposition. In stark contrast, our local search algorithm is very simple and easy to implement. It keeps exchanging a constant number of vertices to improve the current solution until a local optimum is reached. Our main contribution is to show that the local optimum only differs the global optimum by $(1+\epsilon)$ factor.