PARAMETERIZED MAX MIN FEEDBACK VERTEX SET

Abstract

Given a graph G and an integer k, Max Min FVS asks whether there exists a minimal set of vertices of size at least k whose deletion destroys all cycles. We present several results that improve upon the state of the art of the parameterized complexity of this problem with respect to both structural and natural parameters. Using standard dynamic programming techniques, we first present an algorithm of time twO(tw)nO(1)significantly generalizing a recent algorithm of Gaikwad et al. of time vcO(vo)nO(1)where tw, vc denote the input graph's treewidth and vertex cover, respectively. Subsequently, we show that both of these algorithms are essentially optimal, since a vcO(vo)nO(1)algorithm would refute the Exponential Time Hypothesis. With respect to the natural parameter k, the aforementioned recent work by Gaikwad et al. claimed a fixed-parameter tractable branching algorithm with complexity 10knO(1)We point out that this algorithm is incorrect and present a branching algorithm of complexity 9.34knO(1)