反馈集的紧局部化
Tight Localizations of Feedback Sets
AI总结：
针对经典NP难的反馈弧集和反馈顶点集问题，提出一种新的启发式算法，通过紧误差界在大规模稀疏图上高效求解，经验验证达到近似比。
摘要：
经典NP难的反馈弧集问题（FASP）和反馈顶点集问题（FVSP）分别要求找到最小的弧集$\varepsilon \subseteq E$或顶点集$ν\subseteq V$，移除它们$G\setminus \varepsilon$、$G\setminus ν$可使给定的多重有向图$G=(V,E)$变为无环图。尽管已知这两个问题都是APX难的，但近似算法或不可近似性证明仍未知。我们针对有向FASP提出了一种新的$\mathcal{O}(|V||E|^4)$启发式算法。虽然已知$r \approx 1.3606$的比率是APX难的下界，但至少通过经验验证，我们实现了$r \leq 2$的近似比。最相关的应用（如电路测试）需要在大型稀疏图上求解FASP，由于我们的方法，这可以在紧误差界内高效完成。
英文摘要：
The classical NP-hard feedback arc set problem (FASP) and feedback vertex set problem (FVSP) ask for a minimum set of arcs $\varepsilon \subseteq E$ or vertices $ν\subseteq V$ whose removal $G\setminus \varepsilon$, $G\setminus ν$ makes a given multi-digraph $G=(V,E)$ acyclic, respectively. Though both problems are known to be APX-hard, approximation algorithms or proofs of inapproximability are unknown. We propose a new $\mathcal{O}(|V||E|^4)$-heuristic for the directed FASP. While a ratio of $r \approx 1.3606$ is known to be a lower bound for the APX-hardness, at least by empirical validation we achieve an approximation of $r \leq 2$. The most relevant applications, such as circuit testing, ask for solving the FASP on large sparse graphs, which can be done efficiently within tight error bounds due to our approach.
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