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 $S = V(T)$.

翻译：反馈顶点集（FVS）是指有向图中一个顶点子集，移除该子集后图变为无环图，即该子集覆盖了图中的所有环。Lokshtanov等人[TALG '21]提出了一个因子为2的随机化近似算法，用于在锦标赛中寻找最小权重的FVS。我们通过提出一个因子为$2\alpha$的随机化近似算法，将这一结果推广至独立数为$\alpha$的有向图（独立数为1的有向图即锦标赛）中寻找最小权重FVS。基于相同框架，我们进一步提出一个因子为2的随机化近似算法，用于在锦标赛中寻找最小权重的子集FVS：给定图及一个顶点子集$S$，寻找一个覆盖所有包含至少一个$S$中顶点的环的顶点子集。注意，锦标赛中的FVS是该问题中$S = V(T)$时的特例。