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Triangle Packing in (Sparse) Tournaments: Approximation and Kernelization

Authors Stéphane Bessy, Marin Bougeret, Jocelyn Thiebaut

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  • Tournament Triangle packing
  • Feedback arc set
  • Approximation algorithms
  • Parameterized algorithms

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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.

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Stéphane Bessy, Marin Bougeret, and Jocelyn Thiebaut. Triangle Packing in (Sparse) Tournaments: Approximation and Kernelization. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ESA.2017.14

Author Details

Stéphane Bessy
Marin Bougeret
Jocelyn Thiebaut

References

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