发表机构
The Institute of Mathematical Sciences, HBNI（数学科学研究所，HBNI）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对局部半完全和有向拟传递有向图，提出随机化多项式时间因子-2近似算法求解加权有向反馈顶点集，改进现有结果并达到最优近似比。
AI 中文摘要
有向图的\emph{有向反馈顶点集}是一组顶点，删除这些顶点会破坏所有有向环。\textsc{有向反馈顶点集}（\textsc{DFVS}）问题要求找到基数最小或总权重最小的这样的集合。尽管在{唯一博弈猜想}下，一般的\textsc{DFVS}不允许常数因子近似，但锦标赛图由于Lokshtanov等人[SODA'20]的工作，允许随机化因子-$2$近似。我们将这一保证扩展到两类更广泛的结构化有向图，这两类图也都包含稀疏有向图。我们的第一个也是主要的结果是，对于\emph{局部半完全有向图}（\textsf{LSD}s）上的加权\textsc{DFVS}，存在一个随机化多项式时间因子-$2$近似，该类严格推广了半完全有向图和锦标赛图。据我们所知，这是\textsf{LSD}s上\textsc{DFVS}的第一个非平凡常数因子近似，即使在无权设置下也是如此。我们的第二个结果是，对于\emph{有向拟传递图}上的加权\textsc{DFVS}，存在一个随机化多项式时间因子-$2$近似，改进了Ghorbani和Mnich~[ICALP'26]最近的确定性$9/4$-近似。该算法源于我们组合框架的简单递归应用。因子$2$在{唯一博弈猜想}下是最优的，因为锦标赛图是\textsf{LSD}s以及有向拟传递图的子类。
英文摘要
A \emph{directed feedback vertex set} of a digraph is a set of vertices whose removal destroys all directed cycles. The \textsc{Directed Feedback Vertex Set} (\textsc{DFVS}) problem asks for such a set of minimum cardinality or minimum total weight. Although general \textsc{DFVS} admits no constant-factor approximation under the {Unique Games Conjecture}, tournaments admit a randomized factor-$2$ approximation due to Lokshtanov et al. [SODA'20]. We extend this guarantee to two broader classes of structured digraphs, both of which also contain sparse digraphs. Our first and main result is a randomized polynomial-time factor-$2$ approximation for weighted \textsc{DFVS} on \emph{locally semicomplete digraphs} (\textsf{LSD}s), a class that strictly generalizes semicomplete digraphs and tournaments. To the best of our knowledge, this is the first non-trivial constant-factor approximation for \textsc{DFVS} on \textsf{LSD}s, even in the unweighted setting. Our second result is a randomized polynomial-time factor-$2$ approximation for weighted \textsc{DFVS} on \emph{quasi-transitive digraphs}, improving the recent deterministic $9/4$-approximation of Ghorbani and Mnich~[ICALP'26]. The algorithm follows from a simple recursive application of our composition framework. The factor $2$ is optimal under the {Unique Games Conjecture}, since tournaments are subclass of \textsf{LSD}s as well as quasi-transitive digraphs.
Comments17 pages
