Quick-Sort Style Approximation Algorithms for Generalizations of Feedback Vertex Set in Tournaments

A feedback vertex set (FVS) in a digraph is a subset of vertices whose removal makes the digraph acyclic. In other words, it hits all cycles in the digraph. Lokshtanov et al. [TALG '21] gave a factor 2 randomized approximation algorithm for finding a minimum weight FVS in tournaments. We generalize the result by presenting a factor \(2\alpha\) randomized approximation algorithm for finding a minimum weight FVS in digraphs of independence number \(\alpha\); a generalization of tournaments which are digraphs with independence number \(1\). Using the same framework, we present a factor \(2\) randomized approximation algorithm for finding a minimum weight Subset FVS in tournaments: given a vertex subset \(S\) in addition to the graph, find a subset of vertices that hits all cycles containing at least one vertex in \(S\). Note that FVS in tournaments is a special case of Subset FVS in tournaments in which \(S = V(T)\).