Local search is a PTAS for feedback vertex set in minor-free graphs

Abstract

We show that an extremely simple local search gives a PTAS for the Feedback Vertex Set (FVS) problem in minor-free graphs. It keeps exchanging a constant number of vertices to improve the current solution until a local optimum is reached. The previous PTAS by Fomin, Lokshtanov, Raman and Saurabh [1], despite theoretical efficiency, is much more complicated in the sense that it combines many advanced algorithmic tools such as contraction decomposition framework by Demaine and Hajiaghayi [2], Courcelle's theorem [3] and the Robertson and Seymour decomposition [4].Our main technical contribution is to show that the local optimum only differs the global optimum by a (1+ϵ) factor. We do so by introducing Steiner points, those who are not in the local and optimal solutions, to the standard analysis of local search. We believe that our technique is of independent interest.