Coloring the nodes of a directed graph

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Abstract

It is an empirical fact that coloring the nodes of a graph can be used to speed up clique search algorithms. In directed graphs transitive subtournaments can play the role of cliques. In order to speed up algorithms to locate large transitive tournaments we propose a scheme for coloring the nodes of a directed graph. The main result of the paper is that in practically interesting situations determining the optimal number of colors in the proposed coloring is an NP-hard problem. A possible conclusion to draw from this result is that for practical transitive tournament search algorithms we have to develop approximate greedy coloring algorithms.

References

  • [1] I. M. Bomze, M. Budinich, P. M. Pardalos, M. Pelillo, The maximum clique problem, in Handbook of Combinatorial Optimization Vol. 4, Eds. D.-Z. Du and P. M. Pardalos, Kluwer Academic Publisher, Boston, MA 1999. ⇒118

  • [2] R. Carraghan, P. M. Pardalos, An exact algorithm for the maximum clique problem, Operation Research Letters 9, 6 (1990), 375-382. ⇒118

  • [3] P. Erdős, L. Moser, On the representation of directed graphs as unions of orderings. Magyar Tud. Akad. Mat. Kutató Int. Közl. 9 (1964), 125-132. ⇒118

  • [4] P. Erdős, J. W. Moon, On sets of consistent arcs in tournament, Canad. Math. Bull. 8 (1965), 269-271. ⇒118

  • [5] M. R. Garey, D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, New York, 2003. ⇒118

  • [6] F. Kárteszi, Introduction to Finite Geometries, North-Holland Pub. Co., Amsterdam, 1976. ⇒120

  • [7] L. Kiviluoto, P. R. J. Östergård, V. P. Vaskelainen, Exact algorithms for finding maximum transitive subtournaments, (manuscript) ⇒118

  • [8] J. Konc, D. Janežič, An improved branch and bound algorithm for the maximum clique problem, MATCH Commun. Math. Comput. Chem. 58, 3 (2007), 569-590. ⇒118

  • [9] D. Kumlander, Some practical algorithms to solve the maximal clique problem, PhD Thesis, Tallin University of Technology, 2005. ⇒118

  • [10] J. W. Moon, Topics on Tournaments, Holt, Reinhart and Winston, New York, 1968. ⇒118

  • [11] P. R. J. Östergård, A fast algorithm for the maximum clique problem, Discrete Appl. Math. 120, 1-3 (2002), 197-207. ⇒118

  • [12] C. H. Papadimitriou, Computational Complexity, Addison-Wesley Publishing Company, Inc., Reading, MA, 1994. ⇒118

  • [13] R. Stearns, The voting problem, Amer. Math. Monthly 66 (1959), 761-763. ⇒ 118

  • [14] E. Tomita, T. Seki, An efficient branch-and-bound algorithm for finding a maximum clique, Discrete Mathematics and Theoretical Computer Science, Lecture Notes in Comp. Sci. 2731 (2003), 278-289. ⇒118

Acta Universitatis Sapientiae, Informatica

The Journal of "Sapientia" Hungarian University of Transylvania

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