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Packing Arc-Disjoint Cycles in Tournaments

Authors Stéphane Bessy, Marin Bougeret, R. Krithika, Abhishek Sahu, Saket Saurabh, Jocelyn Thiebaut, Meirav Zehavi

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Author Details

Stéphane Bessy
  • Université de Montpellier, LIRMM, CNRS, Montpellier, France
Marin Bougeret
  • Université de Montpellier, LIRMM, CNRS, Montpellier, France
R. Krithika
  • Indian Institute of Technology Palakkad, India
Abhishek Sahu
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
  • University of Bergen, Bergen, Norway
Jocelyn Thiebaut
  • Université de Montpellier, LIRMM, CNRS, Montpellier, France
Meirav Zehavi
  • Ben-Gurion University, Beersheba, Israel

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Stéphane Bessy, Marin Bougeret, R. Krithika, Abhishek Sahu, Saket Saurabh, Jocelyn Thiebaut, and Meirav Zehavi. Packing Arc-Disjoint Cycles in Tournaments. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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 in 2^{o(sqrt{k})} n^{O(1)} time under the Exponential-Time Hypothesis.

Subject Classification

ACM Subject Classification
  • Mathematics 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
  • arc-disjoint cycle packing
  • tournaments
  • parameterized algorithms
  • kernelization


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