Triangle Packing in (Sparse) Tournaments: Approximation and Kernelization

Authors Stéphane Bessy, Marin Bougeret, Jocelyn Thiebaut

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Stéphane Bessy
Marin Bougeret
Jocelyn Thiebaut

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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)


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


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  1. Faisal N. Abu-Khzam. A quadratic kernel for 3-set packing. In International Conference on Theory and Applications of Models of Computation, pages 81-87. Springer, 2009. Google Scholar
  2. Giorgio Ausiello, Pierluigi Crescenzi, Giorgio Gambosi, Viggo Kann, Alberto Marchetti-Spaccamela, and Marco Protasi. Complexity and approximation: Combinatorial optimization problems and their approximability properties. Springer Science &Business Media, 2012. Google Scholar
  3. Piotr Berman and Marek Karpinski. On some tighter inapproximability results. In International Colloquium on Automata, Languages, and Programming, pages 200-209. Springer, 1999. Google Scholar
  4. Stephane Bessy, Marin Bougeret, and Jocelyn Thiebaut. Triangle packing in (sparse) tournaments: approximation and kernelization. Technical report, HAL LIRMM, lirmm-01550313, v1, 2017. URL:
  5. Mao-Cheng Cai, Xiaotie Deng, and Wenan Zang. A min-max theorem on feedback vertex sets. Mathematics of Operations Research, 27(2):361-371, 2002. Google Scholar
  6. Pierre Charbit, Stéphan Thomassé, and Anders Yeo. The minimum feedback arc set problem is NP-hard for tournaments. Combinatorics, Probability and Computing, 16(01):1-4, 2007. Google Scholar
  7. Marek Cygan. Improved approximation for 3-dimensional matching via bounded pathwidth local search. In Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium on, pages 509-518. IEEE, 2013. Google Scholar
  8. Holger Dell and Dániel Marx. Kernelization of packing problems. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete algorithms, SODA'12, 2012. Google Scholar
  9. Venkatesan Guruswami, C. Pandu Rangan, Maw-Shang Chang, Gerard J. Chang, and C. K. Wong. The vertex-disjoint triangles problem. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 26-37. Springer, 1998. Google Scholar
  10. Danny Hermelin and Xi Wu. Weak compositions and their applications to polynomial lower bounds for kernelization. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 104-113. Society for Industrial and Applied Mathematics, 2012. Google Scholar
  11. Bart M. P. Jansen and Astrid Pieterse. Sparsification upper and lower bounds for graph problems and Not-All-Equal SAT. Algorithmica, pages 1-26, 2015. Google Scholar
  12. Claire Kenyon-Mathieu and Warren Schudy. How to rank with few errors. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 95-103. ACM, 2007. Google Scholar
  13. Matthias Mnich, Virginia Vassilevska Williams, and László A. Végh. A 7/3-Approximation for Feedback Vertex Sets in Tournaments. In 24th Annual European Symposium on Algorithms, ESA 2016, pages 67:1-67:14, 2016. Google Scholar
  14. Hannes Moser. A problem kernelization for graph packing. In International Conference on Current Trends in Theory and Practice of Computer Science, pages 401-412. Springer, 2009. Google Scholar
  15. George L. Nemhauser and Leslie E. Trotter Jr. Properties of vertex packing and independence system polyhedra. Mathematical Programming, 6(1):48-61, 1974. Google Scholar
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