  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
$2^{\mathcal{O}(k \log k)} n^{\mathcal{O}(1)}$ time and has a kernel with
$\mathcal{O}(k)$ 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.
