A tournament is a directed graph in which there is a single arc between every pair of distinct vertices. Given a tournament T on n vertices, we explore the classical and parameterized complexity of the problems of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k and a triangle packing (a set of pairwise arc-disjoint triangles) of size k . We refer to these problems as Arc-disjoint Cycles in Tournaments ( ACT ) and Arc-disjoint Triangles in Tournaments ( ATT ), respectively. Although the maximization version of ACT can be seen as the dual of the well-studied problem of finding a minimum feedback arc set (a set of arcs whose deletion results in an acyclic graph) in tournaments, surprisingly no algorithmic results seem to exist for ACT . We first show that ACT and ATT are both NP -complete. Then, we show that the problem of determining if a tournament has a cycle packing and a feedback arc set of the same size is NP -complete. Next, we prove that ACT is fixed-parameter tractable via a $$2^{\mathcal {O}(k \log k)} n^{\mathcal {O}(1)}$$ 2 O ( k log k ) n O ( 1 ) -time algorithm and admits a kernel with $$\mathcal {O}(k)$$ O ( k ) vertices. Then, we show that ATT too has a kernel with $$\mathcal {O}(k)$$ O ( k ) vertices and can be solved in $$2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}$$ 2 O ( k ) n O ( 1 ) time. Afterwards, we describe polynomial-time algorithms for ACT and ATT when the input tournament has a feedback arc set that is a matching. We also prove that ACT and ATT cannot be solved in $$2^{o(\sqrt{n})} n^{\mathcal {O}(1)}$$ 2 o ( n ) n O ( 1 ) time under the exponential-time hypothesis.

中文翻译：

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在比赛中包装弧不相交循环

锦标赛是一个有向图，其中每对不同的顶点之间都有一条弧线。给定 n 个顶点上的锦标赛 T，我们探讨了确定 T 是否具有大小为 k 的循环包装（一组成对弧不相交循环）和三角形包装（一组成对弧-不相交的三角形）大小为 k 。我们将这些问题分别称为锦标赛中的弧不相交循环 (ACT) 和锦标赛中的弧不相交三角形 (ATT)。尽管 ACT 的最大化版本可以看作是在锦​​标赛中找到最小反馈弧集（一组被删除导致无环图的弧集）这一经过充分研究的问题的对偶，但令人惊讶的是，ACT 似乎不存在算法结果. 我们首先证明 ACT 和 ATT 都是 NP 完全的。然后，我们表明，确定锦标赛是否具有循环包装和相同大小的反馈弧集的问题是 NP 完全的。接下来，我们通过 $$2^{\mathcal {O}(k \log k)} n^{\mathcal {O}(1)}$$ 2 O ( k log k ) 证明 ACT 是固定参数可处理的n O ( 1 ) -time 算法并承认具有 $$\mathcal {O}(k)$$ O ( k ) 顶点的核。然后，我们证明 ATT 也有一个带有 $$\mathcal {O}(k)$$ O ( k ) 个顶点的核，并且可以在 $$2^{\mathcal {O}(k)} n^{\数学 {O}(1)}$$ 2 O ( k ) n O ( 1 ) 次。然后，当输入锦标赛具有匹配的反馈弧集时，我们描述了 ACT 和 ATT 的多项式时间算法。我们还证明了 ACT 和 ATT 不能在 $$2^{o(\sqrt{n})} n^{\mathcal {O}(1)}$$ 2 o ( n ) n O ( 1 ) 时间内解决指数时间假设。