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arXiv · 1802.07090

The Parameterized Complexity of Packing Arc-Disjoint Cycles in Tournaments

R. Krithika·Abhishek Sahu·Saket Saurabh·Meirav Zehavi
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Abstract

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.

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R. Krithika, Abhishek Sahu, Saket Saurabh, Meirav Zehavi. 2018-02-20. The Parameterized Complexity of Packing Arc-Disjoint Cycles in Tournaments. https://arxiv.org/abs/1802.07090

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