Triangle Packing in (Sparse) Tournaments: Approximation and Kernelization
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.
Tournament Triangle packing
Feedback arc set
Approximation algorithms
Parameterized algorithms
14:1-14:13
Regular Paper
Stéphane
Bessy
Stéphane Bessy
Marin
Bougeret
Marin Bougeret
Jocelyn
Thiebaut
Jocelyn Thiebaut
10.4230/LIPIcs.ESA.2017.14
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https://hal-lirmm.ccsd.cnrs.fr/lirmm-01550313
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