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 $\nu \subseteq V$ whose removal $G\setminus \varepsilon$, $G\setminus \nu$ 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.

中文翻译：

---

反馈集的严格本地化

经典的 NP 硬反馈弧集问题 (FASP) 和反馈顶点集问题 (FVSP) 要求最小弧集 $\varepsilon \subseteq E$ 或顶点 $\nu \subseteq V$ 其去除 $G\setminus \ varepsilon$, $G\setminus \nu$ 分别使给定的多有向图 $G=(V,E)$ 无环。虽然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是未知的。我们为定向 FASP 提出了一个新的 $\mathcal{O}(|V||E|^4)$-heuristic。虽然已知 $r \approx 1.3606$ 的比率是 APX 硬度的下限，但至少通过经验验证，我们实现了 $r \leq 2$ 的近似值。最相关的应用程序，例如电路测试，要求解决大型稀疏图上的 FASP，由于我们的方法，这可以在严格的误差范围内有效地完成。