is a classical -complete problem and we study it from two perspectives: (1) by restricting the cycles in the packing to be of a fixed length, and (2) by restricting the inputs to bipartite tournaments. Focusing first on (where the cycles in the packing are required to be of length ), we show -completeness in oriented graphs with girth for each and study the parameterized complexity of the problem with respect to two parameterizations (solution size and vertex cover size) for in oriented graphs. Moving on to in bipartite tournaments, we show that every bipartite tournament either contains arc-disjoint cycles or has a feedback arc set of size at most . This result adds to the set of Erdös-Pósa-type results known in the combinatorics literature for packing and covering problems.

中文翻译：

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在有向图中包装弧不相交循环

是一个经典的完全问题，我们从两个角度研究它：（1）通过将打包中的循环限制为固定长度，以及（2）通过限制双向锦标赛的输入。首先关注（其中包装中的循环需要具有长度 ），我们展示了每个周长的有向图的完整性，并研究了问题相对于两个参数化（解决方案大小和顶点覆盖大小）的参数化复杂性对于有向图。继续讨论二分锦标赛，我们证明每个二分锦标赛要么包含弧不相交循环，要么最多具有大小为 的反馈弧集。该结果添加到了组合数学文献中已知的用于包装和覆盖问题的 Erdös-Pósa 型结果集。