Quick-Sort Style Approximation Algorithms for Generalizations of Feedback Vertex Set in Tournaments
<|reference_start|>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 $S = V(T)$.<|reference_end|>
@article{gupta2024quick-sort,
    title={Quick-Sort Style Approximation Algorithms for Generalizations of
  Feedback Vertex Set in Tournaments},
    author={Sushmita Gupta and Sounak Modak and Saket Saurabh and Sanjay
  Seetharaman},
    journal={arXiv preprint arXiv:2402.06407},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.06407},
    primaryClass={cs.DS}
}