TY - GEN

T1 - Packing arc-disjoint cycles in tournaments

AU - Bessy, Stéphane

AU - Bougeret, Marin

AU - Krithika, R.

AU - Sahu, Abhishek

AU - Saurabh, Saket

AU - Thiebaut, Jocelyn

AU - Zehavi, Meirav

N1 - Publisher Copyright:
© Stéphane Bessy, Marin Bougeret, R. Krithika, Abhishek Sahu, Saket Saurabh, Jocelyn Thiebaut, and Meirav Zehavi.

PY - 2019/8/1

Y1 - 2019/8/1

N2 - 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 2O(k log k)nO(1) time and 2O(k)nO(1) time respectively. Moreover, they both admit a kernel with O(k) vertices. We also prove that ACT and ATT cannot be solved in 2o(√k)nO(1) time under the Exponential-Time Hypothesis.

AB - 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 2O(k log k)nO(1) time and 2O(k)nO(1) time respectively. Moreover, they both admit a kernel with O(k) vertices. We also prove that ACT and ATT cannot be solved in 2o(√k)nO(1) time under the Exponential-Time Hypothesis.

KW - Arc-disjoint cycle packing

KW - Kernelization

KW - Parameterized algorithms

KW - Tournaments

UR - http://www.scopus.com/inward/record.url?scp=85071773794&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.MFCS.2019.27

DO - 10.4230/LIPIcs.MFCS.2019.27

M3 - Conference contribution

AN - SCOPUS:85071773794

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019

A2 - Katoen, Joost-Pieter

A2 - Heggernes, Pinar

A2 - Rossmanith, Peter

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019

Y2 - 26 August 2019 through 30 August 2019

ER -