Subset feedback vertex set on graphs of bounded independent set size

Abstract

The (Weighted) Subset Feedback Vertex Set problem is a generalization of the classical Feedback Vertex Set problem and asks for a vertex set of minimum (weight) size that intersects all cycles containing a vertex of a predescribed set of vertices. Although Subset Feedback Vertex Set and Feedback Vertex Set exhibit different computational complexity on split graphs, no similar characterization is known on other classes of graphs. Towards the understanding of the complexity difference between the two problems, it is natural to study the importance of structural graph parameters. Here we consider graphs of bounded independent set number for which it is known that Weighted Feedback Vertex Set can be solved in polynomial time. We provide a dichotomy result with respect to the size α of a maximum independent set. In particular we show that Weighted Subset Feedback Vertex Set can be solved in polynomial time for graphs with α≤3, whereas we prove that the problem remains NP-hard for graphs with α≥4. Moreover, we show that the (unweighted) Subset Feedback Vertex Set problem can be solved in polynomial time on graphs of bounded independent set number by giving an algorithm with running time nO(α). To complement our results, we demonstrate how our ideas can be extended to other terminal set problems on graphs of bounded independent set size. Node Multiway Cut is a terminal set problem that asks for a vertex set of minimum size that intersects all paths connecting any two terminals. Based on our findings for Subset Feedback Vertex Set, we settle the complexity of Node Multiway Cut as well as its variants where nodes are weighted and/or the terminals are deletable, for every value of the given independent set number.