Packing Arc-Disjoint Cycles in Tournaments

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