Packing Arc-Disjoint Cycles in Tournaments

Abstract

A tournament is a directed graph in which there is a single arc between every pair of distinct vertices. Given a tournament T on n vertices, we explore the classical and parameterized complexity of the problems of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k and a triangle packing (a set of pairwise arc-disjoint triangles) of size k. We refer to these problems as Arc-disjoint Cycles in Tournaments (ACT) and Arc-disjoint Triangles in Tournaments (ATT), respectively. Although the maximization version of ACT 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 tournaments, surprisingly no algorithmic results seem to exist for ACT. We first show that ACT and ATT are both NP-complete. Then, we show that the problem of determining if a tournament has a cycle packing and a feedback arc set of the same size is NP-complete. Next, we prove that ACT and ATT are fixed-parameter tractable, they can be solved in 2 O(k log k) n O(1) time and 2 O(k) n O(1) time respectively. Moreover, they both admit a kernel with O(k) vertices. We also prove that ACT and ATT cannot be solved intime under the Exponential-Time Hypothesis. ACM Subject ClassificationMathematics of computing → Graph theory; Theory of computation → Complexity classes; Theory of computation → Parameterized complexity and exact algorithms; Theory of computation → Design and analysis of algorithms; Mathematics of computing → Graph algorithms Keywords and phrases arc-disjoint cycle packing, tournaments, parameterized algorithms, kernelization

Packing arc-disjoint cycles in oriented graphs

“…Our remaining arguments for the case when r = 4 also extend easily so that the statements of Observation 3.1, Observation 3.2, Observation 3.3, Observation 3.4 and Lemma 3.1 can be generalized by replacing G with G r and 4 with r. It may be noted that the graph G r obtained has girth r and when r is even, G r is bipartite. Further, it is known from the literature that Arc-Disjoint 3-Cycle Packing is NP-complete for tournaments [5]. Hence, we have the following theorem, obtained as a generalization of Lemma 3.2.…”

“…Tournaments and bipartite tournaments are well-studied special classes of digraphs with interesting structural and algorithmic properties. While Arc Disjoint Cycle Packing is known to be NP-complete in tournaments [5], the complexity of this problem in bipartite tournaments is still open. Further, Arc Disjoint 4-Cycle Packing is also open in bipartite tournaments.…”

“…Bessy et. al., have shown that Arc Disjoint 3-Cycle Packing is NP-Complete on tournaments [5]. From this, it follows easily (by reduction via subdivision of arcs) that, for each q ≥ 1, Arc Disjoint 3q-Cycle Packing is NP-complete…”

“…Therefore, studying this problem on a subclass of directed graphs and studying Disjoint r-Cycle Packing (where the length of each cycle in the solution set is required to be r) are natural directions of research. Both Vertex Disjoint Cycle Packing and Arc Disjoint Cycle Packing are NP-complete but FPT on tournaments [5,6]. However, these problems are W [1]-hard on bipartite digraphs [3,31].…”

A Note on Arc-Disjoint Cycles in Bipartite Tournaments

“…Given a directed graph and a non-negative integer k, Feedback Arc Set is the problem of determining if the graph has a feedback arc set of size at most k. Finding a minimum feedback arc set in tournaments and bipartite tournaments is NP-hard [1,5,8,10]. However, it is known that for each non-negative integer k, every tournament either contains k arc-disjoint cycles or has a feedback arc set of size at most 5k [4] and results from [6,12] improve the bound of 5k to 3.7k. In this note, we prove an analogous result for bipartite tournaments 1 .…”