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 $k$ 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 $D$ 包含一个由 $k$ 个有向偶圈组成的集合 $\mathcal{C}$，且 $D$ 中每个顶点至多属于 $\mathcal{C}$ 中四个圈，要么存在一个大小不超过 $f(k)$ 的子集 $S\subseteq V(D)$，使得 $D-S$ 不含任何有向偶圈。此外，我们给出一个算法，可在时间 $g(k)n^{\mathcal{O}(1)}$ 内找到该论断的两种结果之一，其中 $g\colon \mathbb{N}\to\mathbb{N}$ 为某个可计算函数。这一结果统一了有向图算法理论中两个深入的研究领域：有向图的 Erdős–Pósa 性质研究与偶有向圈问题研究。后者是判定给定有向图是否包含偶有向圈的决策问题，可追溯至 Pólya 1913 年的一个问题。该问题长期悬而未决，直至 Robertson、Seymour 和 Thomas (Ann. of Math. (2) 1999) 与 McCuaig (Electron. J. Combin. 2004; 联合公告于 STOC 1997) 独立发现多项式时间算法后才获解决。偶有向圈问题等价于 Pfaffian 二部图的识别问题，其应用甚至超出离散数学与理论计算机科学范畴。另一方面，Younger 猜想（1973）指出有向圈具有 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。我们的方法建立在解决上述两个问题的技术之上，并将它们整合为一个强大的结构定理，从而为其他重要问题带来进一步的算法应用。