On Parameterized Independent Feedback Vertex Set

We investigate a generalization of the classical Feedback Vertex Set (FVS) problem from the point of view of parameterized algorithms. Independent Feedback Vertex Set (IFVS) is the “independent” variant of the FVS problem and is defined as follows: given a graph G and an integer k, decide whether there exists F⊆V(G), |F|≤k, such that G[V(G)∖F] is a forest and G[F] is an independent set; the parameter is k. Note that the similarly parameterized version of the FVS problem–where there is no restriction on the graph G[F]–has been extensively studied in the literature. The connected variant CFVS–where G[F] is required to be connected–has received some attention as well. The FVS problem easily reduces to the IFVS problem in a manner that preserves the solution size, and so any algorithmic result for IFVS directly carries over to FVS. We show that IFVS can be solved in O(5knO(1)) time, where n is the number of vertices in the input graph G, and obtain a cubic (O(k3)) kernel for the problem. Note the contrast with the CFVS problem, which does not admit a polynomial kernel unless CoNP⊆NP/Poly.