eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-09-01
14:1
14:13
10.4230/LIPIcs.ESA.2017.14
article
Triangle Packing in (Sparse) Tournaments: Approximation and Kernelization
Bessy, Stéphane
Bougeret, Marin
Thiebaut, Jocelyn
Given a tournament T and a positive integer k, the C_3-Packing-T asks if there exists a least k (vertex-)disjoint directed 3-cycles in T. This is the dual problem in tournaments of the classical minimal feedback vertex set problem. Surprisingly C_3-Packing-T did not receive a lot of attention in the literature. We show that it does not admit a PTAS unless P=NP, even if we restrict the considered instances to sparse tournaments, that is tournaments with a feedback arc set (FAS) being a matching. Focusing on sparse tournaments we provide a (1+6/(c-1)) approximation algorithm for sparse tournaments having a linear representation where all the backward arcs have "length" at least c. Concerning kernelization, we show that C_3-Packing-T admits a kernel with O(m) vertices, where m is the size of a given feedback arc set. In particular, we derive a O(k) vertices kernel for C_3-Packing-T when restricted to sparse instances. On the negative size, we show that C_3-Packing-T does not admit a kernel of (total bit) size O(k^{2-epsilon}) unless NP is a subset of coNP / Poly. The existence of a kernel in O(k) vertices for C_3-Packing-T remains an open question.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol087-esa2017/LIPIcs.ESA.2017.14/LIPIcs.ESA.2017.14.pdf
Tournament Triangle packing
Feedback arc set
Approximation algorithms
Parameterized algorithms