锦标赛中有向反馈顶点集的确定性 $(2 + \varepsilon)$-近似算法
A deterministic $(2 + \varepsilon)$-approximation for directed feedback vertex sets in tournaments
- Hamburg University of Technology, Institute for Algorithms and Complexity（汉堡工业大学，算法与复杂性研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出锦标赛中有向反馈顶点集问题的首个确定性多项式时间$(2+\varepsilon)$-近似算法，改进了以往所有结果，并扩展到拟传递有向图。
AI中文摘要：
我们几乎解决了锦标赛中有向反馈顶点集问题的多项式时间可近似性。该问题是顶点覆盖困难的，因此在唯一博弈猜想下，对于任意 $\varepsilon > 0$，在多项式时间内不存在 $(2 - \varepsilon)$-近似算法。在过去28年中，多项工作试图达到这一2的近似性障碍，并设计了近似因子越来越小的算法。其中包括Cai、Deng和Zang（FOCS 1998, SICOMP 2001）的$5/2$-近似算法；Mnich、Vassilevska Williams和V{é}gh（ESA 2016）的$7/3$-近似算法，以及Aprile、Drescher、Fiorini和Huynh（DAM 2023）的另一个$7/3$-近似算法，还有Ghorbani和Mnich（ICALP 2026）的$9/4$-近似算法。我们的主要结果改进了所有这些工作：我们首次给出了锦标赛中有向反馈顶点集问题的确定性多项式时间$(2+\varepsilon)$-近似算法，适用于所有$\varepsilon > 0$。因此，我们几乎回答了Lokshtanov、Misra、Mukherjee、Panolan、Philip和Saurabh（SODA 2020）提出的开放问题，他们要求多项式时间内的确定性2-近似算法。此外，我们将结果扩展到更广泛的拟传递有向图类。
英文摘要：
We nearly settle the polynomial-time approximability of the Directed Feedback Vertex Set problem in tournaments. This problem is Vertex Cover-hard, and thus cannot have a $(2 - \varepsilon)$-approximation for any $\varepsilon > 0$ in polynomial time assuming the Unique Games Conjecture. In the past 28 years, several works have attempted to attain this approximability barrier of 2, and have designed algorithms with smaller and smaller approximation factors. This includes a $5/2$-approximation by Cai, Deng and Zang (FOCS 1998, SICOMP 2001); a $7/3$-approximation by Mnich, Vassilevska Williams and V{é}gh (ESA 2016), another $7/3$-approximation by Aprile, Drescher, Fiorini and Huynh (DAM 2023), and a $9/4$-approximation by Ghorbani and Mnich (ICALP 2026). Our main result improves upon all of those works: we give the first deterministic polynomial-time $(2+\varepsilon)$-approximation for Directed Feedback Vertex Set in tournaments, for all $\varepsilon > 0$. We thereby almost answer an open question by Lokshtanov, Misra, Mukherjee, Panolan, Philip and Saurabh (SODA 2020) who asked for a deterministic 2-approximation in polynomial time. Furthermore, we extend our result to the broader class of quasi-transitive digraphs
