The Parameterized Complexity of Packing Arc-Disjoint Cycles in Tournaments
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Given a directed graph D on n vertices and a positive integer k, the Arc-Disjoint Cycle Packing problem is to determine whether D has k arc-disjoint cycles. This problem is known to be W[1]-hard in general directed graphs. In this paper, we initiate a systematic study on the parameterized complexity of the problem restricted to tournaments. We show that the problem is fixed-parameter tractable and admits a polynomial kernel when parameterized by the solution size k. In particular, we show that it can be solved in 2^O(k k) n^O(1) time and has a kernel with O(k) vertices. The primary ingredient in both these results is a min-max theorem that states that every tournament either contains k arc-disjoint triangles or has a feedback arc set of size at most 6k. Our belief is that this combinatorial result is of independent interest and could be useful in other problems related to cycles in tournaments.
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