We prove that there exists a function f:N→R such that every directed graph G contains either k directed odd cycles where every vertex of G is contained in at most two of them, or a set of at most f(k) vertices meeting all directed odd cycles. We give a polynomial-time algorithm for fixed k which outputs one of the two outcomes. This extends the half-integral Erdős-Pósa theorem for undirected odd cycles by Reed [Combinatorica 1999] to directed graphs.

中文翻译：

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有向奇数循环的半积分 Erdős-Pósa 定理


我们证明存在一个函数 f：N→R，使得每个有向图 G 都包含 k 个有向奇数循环，其中 G 的每个顶点最多包含其中两个，或者一组最多 f（k） 个顶点满足所有有向奇数循环。我们给出了一个固定 k 的多项式时间算法，它输出两个结果之一。这将 Reed [Combinatorica 1999] 的无向奇数循环的半积分 Erdős-Pósa 定理扩展到有向图。