A New Ant Colony Optimization Algorithm for the Lower Bound of Sum Coloring Problem

  • Sidi Mohamed Douiri
  • Souad Elbernoussi


We consider an undirected graph G = (V, E), the minimum sum coloring problem (MSCP) asks to find a valid vertex coloring of G, using natural numbers (1,2,...), the aim is to minimize the total sum of colors. In this paper we are interested in the elaboration of an approximate solution for the minimum sum coloring problem (MSCP), more exactly we try to give a lower bound for MSCP by looking for a decomposition of the graph based on the metaheuristic of ant colony optimization (ACO). We test different instances to validate our approach.


Minimum sum coloring problem Lower bound for MSCP Ant colony optimization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bar-Noy, A., Bellareb, M., Halldorsson, M.M., Shachnai, H., Tamir, T.: On chromatic sums and distributed resource allocation. Inf. Comput. 140(2), 183–202 (1998)MATHCrossRefGoogle Scholar
  2. 2.
    Bloechliger, I., Zufferey, N.: A graph coloring heuristic using partial solutions and a reactive tabu scheme. Comput. Oper. Res. 35, 960–975 (2008)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Cheng, C.B., Mao, C.P.: A modified ant colony system for solving the travelling salesman problem with time windows. Math. Comput. Model. 46, 1225–1235 (2007)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chow, F.C., Hennessy, J.L.: The priority-based coloring approach to register allocation. ACM Trans. Program. Lang. Syst. 12, 501–536 (1990)CrossRefGoogle Scholar
  6. 6.
    Costa, D., Hertz, A.: Ants can color graphs. J. Oper. Res. Soc. 48, 295–305 (1997)MATHGoogle Scholar
  7. 7.
    de Werra, D.: An introduction to timetabling. Eur. J. Oper. Res. 19(2), 151–162(1985)MATHCrossRefGoogle Scholar
  8. 8.
    Dorigo, M.: Optimization, learning, and natural algorithms. Ph.D. dissertation (in Italian), Dipartimento di Elettronica, Politecnico di Milano, Italy (1992)Google Scholar
  9. 9.
    Dorigo, M., Blum, C.: Ant colony optimization theory: a survey. Theor. Comp. Sci. 344(2–3), 243–278 (2005)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Dorigo, M., Di Caro, G.: Ant colony optimisation: a new meta-heuristic. In: Proceedings of the 1999 Congress on Evolutionary Computation, vol. 2, pp. 1470–1477 (1999)Google Scholar
  11. 11.
    Dorigo, M., Stutzle, T.: Ant Colony Optimization. MIT Press, Massachusetts Institute of Technology, Cambridge (2004)MATHCrossRefGoogle Scholar
  12. 12.
    Douiri, S.M., Elbernoussi, S.: New algorithm for the sum coloring problem. Int. J. Contemp. Math. Sci. 6(10), 453–463 (2011)MathSciNetMATHGoogle Scholar
  13. 13.
    Fleurent, C., Ferland, J.: Genetic and hybrid algorithms for graph coloring. Ann. Oper. Res. 63, 437–464 (1996)MATHCrossRefGoogle Scholar
  14. 14.
    Galinier, P., Hao, J.K.: Hybrid evolutionary algorithms for graph coloring. J. Comb. Optim. 3, 379–397 (1999)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Gamst, A.: Some lower bounds for a class of frequency assignment problem. IEEE Trans. Veh. Technol. 35, 8–14 (1999)CrossRefGoogle Scholar
  16. 16.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)MATHGoogle Scholar
  17. 17.
    Kokosinski, Z., Kawarciany, K.: On sum coloring of graphs with parallel genetic algorithms. In: ICANNGA’07, 2007, Part I, LNCS 4431, pp. 211–219 (2007)Google Scholar
  18. 18.
    Kroon, L.G., Sen, A., Deng, H., Roy, A.: The optimal cost chromatic partition problem for trees and interval graphs. In: Graph-Theoretical Concepts in Computer Science, LNCS, pp. 279–292 (1996)Google Scholar
  19. 19.
    Kubicka, E., Schwenk, A.J.: An introduction to chromatic sums. In: Proceedings of the ACM Computer Science Conference, pp. 39–45 (1989)Google Scholar
  20. 20.
    Li, Y., Lucet, C., Moukrim, A., Sghiouer, K.: Greedy Algorithms for the Minimum Sum Coloring Problem. In: International Workshop: Logistics and Transport (2009)Google Scholar
  21. 21.
    Lucet, C., Mendes, F., Moukrim, A.: An exact method for graph coloring. Comput. Oper. Res. 33(8), 2189–2207 (2006)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Malafiejski, M.: Sum coloring of graphs. Graph Colorings, Contemp. Math. 352, 55–65 (2004)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Moukrim, A., Sghiouer, K., Lucet, C., Li, Y.: Lower bounds for the minimal sum coloring problem. Electron. Notes Discrete Math. 36, 663–670 (2010)CrossRefGoogle Scholar
  24. 24.
    Poorzahedy, H., Abulghasemi, F.: Application of ant system to network design problem. Transportation 32, 251–273 (2005)CrossRefGoogle Scholar
  25. 25.
    Salari, E., Eshghi, K.: An ACO algorithm for the graph coloring problem. Int. J. Contemp. Math. Sci. 3(6), 293–304 (2008)MathSciNetMATHGoogle Scholar
  26. 26.
    Stecke, K.: Design planning, scheduling and control problems of flexible manufacturing. Ann. Oper. Res. 3, 3–12 (1985)CrossRefGoogle Scholar
  27. 27.
    Thomassen, C., Erdos, P., Alavi, Y., Malde, P.J., Schwenk, A.J.: Tight bounds on the chromatic sum of a connected graph. J. Graph Theory 13(3), 353–357 (1989)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Walkowiak, K.: Graph coloring using ant algorithms. In: Proceedings of the Conference on Computer Recognition Systems KOSYR, pp. 199–204, Milkow, 28–31 May 2001Google Scholar
  29. 29.
    Zufferey, N., Amstutz, P., Giaccari, P.: Graph colouring approaches for a satellite range scheduling problem. J. Sched. 11, 263–277 (2008)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Research laboratory Mathematics Computing and Applications, Faculty of Sciences-AgdalUniversity Mohammed VRP, RabatMorocco

Personalised recommendations