Skip to main content
Log in

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

  • Published:
Journal of Mathematical Modelling and Algorithms

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  MATH  Google Scholar 

  2. Bloechliger, I., Zufferey, N.: A graph coloring heuristic using partial solutions and a reactive tabu scheme. Comput. Oper. Res. 35, 960–975 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chow, F.C., Hennessy, J.L.: The priority-based coloring approach to register allocation. ACM Trans. Program. Lang. Syst. 12, 501–536 (1990)

    Article  Google Scholar 

  6. Costa, D., Hertz, A.: Ants can color graphs. J. Oper. Res. Soc. 48, 295–305 (1997)

    MATH  Google Scholar 

  7. de Werra, D.: An introduction to timetabling. Eur. J. Oper. Res. 19(2), 151–162(1985)

    Article  MATH  Google Scholar 

  8. Dorigo, M.: Optimization, learning, and natural algorithms. Ph.D. dissertation (in Italian), Dipartimento di Elettronica, Politecnico di Milano, Italy (1992)

  9. Dorigo, M., Blum, C.: Ant colony optimization theory: a survey. Theor. Comp. Sci. 344(2–3), 243–278 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  11. Dorigo, M., Stutzle, T.: Ant Colony Optimization. MIT Press, Massachusetts Institute of Technology, Cambridge (2004)

    Book  MATH  Google Scholar 

  12. Douiri, S.M., Elbernoussi, S.: New algorithm for the sum coloring problem. Int. J. Contemp. Math. Sci. 6(10), 453–463 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Fleurent, C., Ferland, J.: Genetic and hybrid algorithms for graph coloring. Ann. Oper. Res. 63, 437–464 (1996)

    Article  MATH  Google Scholar 

  14. Galinier, P., Hao, J.K.: Hybrid evolutionary algorithms for graph coloring. J. Comb. Optim. 3, 379–397 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gamst, A.: Some lower bounds for a class of frequency assignment problem. IEEE Trans. Veh. Technol. 35, 8–14 (1999)

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

  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)

  19. Kubicka, E., Schwenk, A.J.: An introduction to chromatic sums. In: Proceedings of the ACM Computer Science Conference, pp. 39–45 (1989)

  20. Li, Y., Lucet, C., Moukrim, A., Sghiouer, K.: Greedy Algorithms for the Minimum Sum Coloring Problem. In: International Workshop: Logistics and Transport (2009)

  21. Lucet, C., Mendes, F., Moukrim, A.: An exact method for graph coloring. Comput. Oper. Res. 33(8), 2189–2207 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Malafiejski, M.: Sum coloring of graphs. Graph Colorings, Contemp. Math. 352, 55–65 (2004)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  24. Poorzahedy, H., Abulghasemi, F.: Application of ant system to network design problem. Transportation 32, 251–273 (2005)

    Article  Google Scholar 

  25. Salari, E., Eshghi, K.: An ACO algorithm for the graph coloring problem. Int. J. Contemp. Math. Sci. 3(6), 293–304 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Stecke, K.: Design planning, scheduling and control problems of flexible manufacturing. Ann. Oper. Res. 3, 3–12 (1985)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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 2001

  29. Zufferey, N., Amstutz, P., Giaccari, P.: Graph colouring approaches for a satellite range scheduling problem. J. Sched. 11, 263–277 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sidi Mohamed Douiri.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Douiri, S.M., Elbernoussi, S. A New Ant Colony Optimization Algorithm for the Lower Bound of Sum Coloring Problem. J Math Model Algor 11, 181–192 (2012). https://doi.org/10.1007/s10852-012-9172-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10852-012-9172-x

Keywords

Navigation