Journal of Combinatorial Optimization

, Volume 13, Issue 2, pp 163–178 | Cite as

On edge orienting methods for graph coloring

Article

Abstract

We consider the problem of orienting the edges of a graph so that the length of a longest path in the resulting digraph is minimum. As shown by Gallai, Roy and Vitaver, this edge orienting problem is equivalent to finding the chromatic number of a graph. We study various properties of edge orienting methods in the context of local search for graph coloring. We then exploit these properties to derive four tabu search algorithms, each based on a different neighborhood. We compare these algorithms numerically to determine which are the most promising and to give potential research directions.

Keywords

Graph coloring Local search Edge orienting 

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References

  1. Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms, and applications. Prentice-Hall, New Jersey, pp 107–108Google Scholar
  2. Barbosa VC, Assis CAG, Do Nascimento JO (2004) Two novel evolutionary formulations of the graph coloring problem. J Comb Optim 8(1):41–63CrossRefMathSciNetMATHGoogle Scholar
  3. Brown JR (1972) Chromatic scheduling and the chromatic number problem. Manag Sci 19(4):456–463MATHGoogle Scholar
  4. Galinier P, Hertz A (2006) A survey of local search methods for graph coloring. Comput & Oper Res (to appear)Google Scholar
  5. Gallai T (1968) On directed paths and circuits. In: Erdös P, Katobna G (eds) Theory of graphs. Academic Press, Tihany, New York, pp 115–118Google Scholar
  6. Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W.H. Freman and Company, NYMATHGoogle Scholar
  7. Glover F, Laguna M (eds) (1997) Tabu search. Kluwer Academic PublishersGoogle Scholar
  8. Herrmann F, Hertz A (2002) Finding the chromatic number by means of critical graphs. ACM J Exp Alg 7(10):1–9MathSciNetGoogle Scholar
  9. Johnson DS, Trick MA (1996) Proceedings of the 2nd DIMACS implementation challenge, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 26, American Mathematical SocietyGoogle Scholar
  10. Kubale M, Jackowski B (1985) A generalized implicit enumeration algorithm for graph coloring. Comm ACM 28(4):412–418CrossRefGoogle Scholar
  11. Mehrotra A, Trick MA (1996) A column generation approach for exact graph coloring. INFORMS J Comput 8(4):344–354MATHGoogle Scholar
  12. Peemöller J (1983) A correction to Brélaz’s modification of Brown’s coloring algorithm. Comm ACM 26(8):593–597CrossRefGoogle Scholar
  13. Roy B (1967) Nombre chromatique et plus longs chemins d’un graphe. Revue AFIRO 1:127–132Google Scholar
  14. Van Laarhoven PJM, Aarts EHL, Lenstra JK (1992) Job-shop scheduling by simulated annealing. Oper Res 40(1):113–125MathSciNetCrossRefMATHGoogle Scholar
  15. Vitaver LM (1962) Determination of minimal coloring of vertices of a graph by means of Boolean powers of the incidence matrix. Dokl Akad Nauk SSSR147 758–759 (in Russian)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • Bernard Gendron
    • 1
  • Alain Hertz
    • 2
  • Patrick St-Louis
    • 1
  1. 1.Département d’informatique et de recherche opérationnelleUniversité de MontréalMontréalCanada
  2. 2.Département de Mathématiques et de Géenie IndustrielÉcole PolytechniqueMontréalCanada

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