Parameterized Max Min Feedback Vertex Set

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 DP techniques, we first present an algorithm of time textrmtw^O(textrmtw)nO(1), significantly generalizing a recent algorithm of Gaikwad et al. of time textrmvc^O(textrmvc)nO(1), where textrmtw, textrmvc denote the input graph's treewidth and vertex cover respectively. Subsequently, we show that both of these algorithms are essentially optimal, since a textrmvc^o(textrmvc)nO(1) algorithm would refute the ETH. With respect to the natural parameter k, the aforementioned recent work by Gaikwad et al. claimed an FPT branching algorithm with complexity 10k nO(1). We point out that this algorithm is incorrect and present a branching algorithm of complexity 9.34k nO(1).