Triangle packing in (sparse) tournaments: approximation and kernelization - LIRMM - Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier Access content directly
Conference Papers Year : 2017

Triangle packing in (sparse) tournaments: approximation and kernelization

Stéphane Bessy
Marin Bougeret
Jocelyn Thiebaut


Given a tournament T and a positive integer k, the C_3-Packing-T problem 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 ⊆ coNP/Poly. The existence of a kernel in O(k) vertices for C_3-Packing-T remains an open question.
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lirmm-01550313 , version 1 (29-06-2017)




Stéphane Bessy, Marin Bougeret, Jocelyn Thiebaut. Triangle packing in (sparse) tournaments: approximation and kernelization. ESA 2017 - 25th Annual European Symposium on Algorithms, Sep 2017, Vienne, Austria. pp.14:1--14:13, ⟨10.4230/LIPIcs.ESA.2017.14⟩. ⟨lirmm-01550313⟩
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