2026-07-22T20:50:14Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:7862 2024-03-06T10:41:16Z ddc:004 open_access

Triangle Packing in (Sparse) Tournaments: Approximation and Kernelization Bessy, Stéphane Bougeret, Marin Thiebaut, Jocelyn Tournament Triangle packing Feedback arc set Approximation algorithms Parameterized algorithms 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Stéphane Bessy and Marin Bougeret and Jocelyn Thiebaut 2017 Is Part Of LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2017.14 urn:nbn:de:0030-drops-78622 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.14 eng https://creativecommons.org/licenses/by/3.0/legalcode