License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2017.14
URN: urn:nbn:de:0030-drops-78622
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7862/
Bessy, Stéphane ;
Bougeret, Marin ;
Thiebaut, Jocelyn
Triangle Packing in (Sparse) Tournaments: Approximation and Kernelization
Abstract
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.
BibTeX - Entry
@InProceedings{bessy_et_al:LIPIcs:2017:7862,
author = {St{\'e}phane Bessy and Marin Bougeret and Jocelyn Thiebaut},
title = {{Triangle Packing in (Sparse) Tournaments: Approximation and Kernelization}},
booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)},
pages = {14:1--14:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-049-1},
ISSN = {1868-8969},
year = {2017},
volume = {87},
editor = {Kirk Pruhs and Christian Sohler},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7862},
URN = {urn:nbn:de:0030-drops-78622},
doi = {10.4230/LIPIcs.ESA.2017.14},
annote = {Keywords: Tournament Triangle packing, Feedback arc set, Approximation algorithms, Parameterized algorithms}
}
Keywords: |
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Tournament Triangle packing, Feedback arc set, Approximation algorithms, Parameterized algorithms |
Collection: |
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25th Annual European Symposium on Algorithms (ESA 2017) |
Issue Date: |
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2017 |
Date of publication: |
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01.09.2017 |