We prove the existence of a computable function $f\colon\mathbb{N}\to\mathbb{N}$ such that for every integer $k$ and every digraph $D$ either contains a collection $\mathcal{C}$ of directed cycles of even length such that no vertex of $D$ belongs to more than four cycles in $\mathcal{C}$, or there exists a set $S\subseteq V(D)$ of size at most $f(k)$ such that $D-S$ has no directed cycle of even length. Moreover, we provide an algorithm that finds one of the two outcomes of this statement in time $g(k)n^{\mathcal{O}(1)}$ for some computable function $g\colon \mathbb{N}\to\mathbb{N}$. Our result unites two deep fields of research from the algorithmic theory for digraphs: The study of the Erd\H{o}s-P\'osa property of digraphs and the study of the Even Dicycle Problem. The latter is the decision problem which asks if a given digraph contains an even dicycle and can be traced back to a question of P\'olya from 1913. It remained open until a polynomial time algorithm was finally found by Robertson, Seymour, and Thomas (Ann. of Math. (2) 1999) and, independently, McCuaig (Electron. J. Combin. 2004; announced jointly at STOC 1997). The Even Dicycle Problem is equivalent to the recognition problem of Pfaffian bipartite graphs and has applications even beyond discrete mathematics and theoretical computer science. On the other hand, Younger's Conjecture (1973), states that dicycles have the Erd\H{o}s-P\'osa property. The conjecture was proven more than two decades later by Reed, Robertson, Seymour, and Thomas (Combinatorica 1996) and opened the path for structural digraph theory as well as the algorithmic study of the directed feedback vertex set problem. Our approach builds upon the techniques used to resolve both problems and combines them into a powerful structural theorem that yields further algorithmic applications for other prominent problems.

翻译：我们证明存在一个可计算函数$f\colon\mathbb{N}\to\mathbb{N}$，使得对任意整数$k$和任意有向图$D$，要么$D$包含一个偶长有向回路集合$\mathcal{C}$，其中$D$的每个顶点至多属于$\mathcal{C}$中的四条回路，要么存在一个大小至多为$f(k)$的顶点子集$S\subseteq V(D)$，使得$D-S$不含偶长有向回路。进一步，我们给出一个算法，能在时间$g(k)n^{\mathcal{O}(1)}$内找到该结论的两种情形之一，其中$g\colon \mathbb{N}\to\mathbb{N}$为某个可计算函数。我们的结果将关于有向图的算法理论中两个深度研究领域统一起来：有向图的Erdős–Pósa性质研究与偶有向回路问题研究。后者是判定给定有向图是否包含偶有向回路的决策问题，可追溯至Pólya于1913年提出的一个疑问。该问题直到Robertson、Seymour和Thomas（Ann. of Math. (2) 1999）以及McCuaig（Electron. J. Combin. 2004；联合公布于STOC 1997）独立发现多项式时间算法后才得以解决。偶有向回路问题等价于Pfaffian二部图的识别问题，其应用甚至超越了离散数学与理论计算机科学。另一方面，Younger猜想（1973）指出有向回路具有Erdős–Pósa性质。该猜想在二十多年后由Reed、Robertson、Seymour和Thomas（Combinatorica 1996）证明，为有向图结构理论及有向反馈顶点集问题的算法研究开辟了道路。我们的方法建立在这两个问题的解决技术之上，将其整合为一个强大的结构定理，并为其他重要问题带来进一步的算法应用。