Triangle packing in (sparse) tournaments: approximation and kernelization

Stéphane Bessy 1 Marin Bougeret 1 Jocelyn Thiebaut 1
1 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : 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.
Complete list of metadatas

Cited literature [17 references]  Display  Hide  Download

https://hal-lirmm.ccsd.cnrs.fr/lirmm-01550313
Contributor : Marin Bougeret <>
Submitted on : Thursday, June 29, 2017 - 2:45:19 PM
Last modification on : Wednesday, June 19, 2019 - 3:47:45 PM
Long-term archiving on : Thursday, January 18, 2018 - 2:49:16 AM

File

triangle_packing_hal.pdf
Files produced by the author(s)

Licence


Distributed under a Creative Commons Attribution 4.0 International License

Identifiers

Collections

Citation

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

Share

Metrics

Record views

308

Files downloads

234