Packing Arc-Disjoint Cycles in Tournaments

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 {\sc Arc-disjoint Cycles in Tournaments} (\act) and {\sc Arc-disjoint Triangles in Tournaments} (\att), respectively. Although the maximization version of \act\ can be seen as the linear programming 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\ and \att\ are fixed-parameter tractable, they can be solved in $2^{\oo(k \log k)} n^{\oo(1)}$ time and $2^{\oo(k)} n^{\oo(1)}$ time respectively. Moreover, they both admit a kernel with $\oo(k)$ vertices. We also prove that \act\ and \att\ cannot be solved in $2^{o(\sqrt{k})} n^{\oo(1)}$ time under the Exponential-Time Hypothesis