eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-08-20
27:1
27:14
10.4230/LIPIcs.MFCS.2019.27
article
Packing Arc-Disjoint Cycles in Tournaments
Bessy, Stéphane
1
Bougeret, Marin
1
Krithika, R.
2
Sahu, Abhishek
3
Saurabh, Saket
3
4
Thiebaut, Jocelyn
1
Zehavi, Meirav
5
Université de Montpellier, LIRMM, CNRS, Montpellier, France
Indian Institute of Technology Palakkad, India
The Institute of Mathematical Sciences, HBNI, Chennai, India
University of Bergen, Bergen, Norway
Ben-Gurion University, Beersheba, Israel
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol138-mfcs2019/LIPIcs.MFCS.2019.27/LIPIcs.MFCS.2019.27.pdf
arc-disjoint cycle packing
tournaments
parameterized algorithms
kernelization