Packing Arc-Disjoint Cycles in Bipartite Tournaments

Abstract

An r-partite tournament is a directed graph obtained by assigning a unique orientation to each edge of a complete undirected r-partite simple graph. Given a bipartite tournament T on n vertices, we explore the parameterized complexity of the problem of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k. Although the maximization version of this problem 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 bipartite tournaments, surprisingly no algorithmic results seem to exist. We show that this problem can be solved in \(2^{\mathcal {O}(k \log k)} n^{\mathcal {O}(1)}\) time and admits a kernel with \(\mathcal {O}(k^2)\) vertices.