A $2$-Approximation for Directed Feedback Vertex Set in Locally Semicomplete and Quasi-Transitive Digraphs

2026-09-16

AI Summary

本文针对局部半完全有向图和拟传递有向图中的定向反馈顶点集问题，提出了一种随机化多项式时间2-近似算法。

Abstract

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.

Problem

• Directed Feedback Vertex Set
• Locally Semicomplete Digraphs
• Quasi-Transitive Digraphs

Innovation

• directed feedback vertex set
• locally semicomplete digraphs
• quasi-transitive digraphs
• factor-2 approximation