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 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.