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The Parameterized Complexity of Packing Arc-Disjoint Cycles in Tournaments
Given a directed graph on vertices and a positive integer , the
Arc-Disjoint Cycle Packing problem is to determine whether has
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 . In particular, we show that it can be solved in
time and has a kernel with
vertices. The primary ingredient in both these results is a
min-max theorem that states that every tournament either contains
arc-disjoint triangles or has a feedback arc set of size at most . Our
belief is that this combinatorial result is of independent interest and could
be useful in other problems related to cycles in tournaments