A Simple Local Search Gives a PTAS for the Feedback Vertex Set Problem in Minor-Free Graphs

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 [13]. However, their algorithm is a combination of many advanced algorithmic tools such as contraction decomposition framework introduced by Demaine and Hajiaghayi [10], Courcelle's theorem [9] and the Robertson and Seymour decomposition [29]. 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 + ) factor. A polynomial-time approximation scheme for a minimization problem is an algorithm that, given a fixed constant > 0, runs in polynomial time and returns a solution within 1 + of optimal.2 A PTAS is efficient if the running time is of the form 2 poly(1/ ) n O(1) .By generalizing the contraction decomposition of Demaine and Hajiaghayi to minor-free graphs, Fomin, Lokshtanov, Raman and Sauraubh [13] obtained a PTAS for the FVS problem in this class of graphs. A graph is H-minor-free, or simply minor-free, if it excludes some fixed graph H as a minor. In this work, we assume that |V (H)| is a constant. We note that the class of minor-free graphs are vastly bigger than planar graphs and bounded-genus graphs. A typical example is the complete bipartite graph K 3,n which has unbounded genus but is K 5 -minor-free. In Section 5, we show that in some sense, minor-free graphs are the limit for which we are still able to obtain a PTAS for this problem.A common theme in all known algorithms is complication in both implementation and analysis. The algorithm of Kleinberg and Kumar [19] is obtained by recursively applying the planar separator theorem by Lipton and Tarjan [22] and analyzing several special cases. The algorithm by Demaine and Hajiaghayi [10] employs the primal-dual relationship of planar graphs to decompose the graphs into several bounded treewidth instances, then applies dynamic programming (DP) to solve the FVS problem on bounded treewidth graphs. DP on bounded treewidth graphs is a very strong algorithmic tool. However, the implementation details typically are quite complicated. Additionally, the NP-hardness complexity of finding a tree decomposition of minimum width in planar graphs is still a long standing open problem. The algorithm of Cohen-Addad et al. [7] for bounded-genus graphs is not simpler and has worst running time; however, it can work with nodeweighted graphs. Given the complicated nature of the algorithms for planar and bounded-genus graphs, it is not surprising that the technical level of the algorithm by Fomin, Lokshtanov, Raman and Sauraubh [13] for minor-free graphs is much higher. It uses advanced tools such as Courcelle's theorem [9] and the Robertson and Seymour decomposition [29]. We note that the decomposition of Robertson and Seymo...