Advertisement

An Analysis of Solution Properties of the Graph Coloring Problem

  • Jean-Philippe Hamiez
  • Jin-Kao Hao
Part of the Applied Optimization book series (APOP, volume 86)

Abstract

This paper concerns the analysis of solution properties of the Graph Coloring Problem. For this purpose, we introduce a property based on the notion of representative sets which are sets of vertices that are always colored the same in a set of solutions. Experimental results on well-studied DIMACS graphs show that many of them contain such sets and give interesting information about the diversity of the solutions. We also show how such an analysis may be used to improve a tabu search algorithm.

Keywords

Graph coloring Solution analysis Representative sets Tabu search. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. A.L. Barabasi and R. Albert. Emergence of scaling in random networks. Science, 286: 509–512, 1999.MathSciNetCrossRefGoogle Scholar
  2. D. Brélaz. New methods to color the vertices of a graph. Communications of the ACM, 22 (4): 251–256, 1979.zbMATHCrossRefGoogle Scholar
  3. P. Cheeseman, B. Kanefsky, and W.M. Taylor. Where the really hard problems are. In J. Mylopoulos and R. Reiter, editors, Proceedings of the Twelfth International Joint Conference on Artificial Intelligence, volume 1, pages 331–337. Morgan Kaufmann Publishers, 1991.Google Scholar
  4. J.C. Culberson and I.P. Gent. Well out of reach: Why hard problems are hard. Technical Report APES-13–1999, APES Research Group, 1999. http://www.dcs.st-and.ac.ukl —apes/reports/apes-13–1999.ps.gz.
  5. L. Davis, editor. Handbook of Genetic Algorithms. Van Nostrand Reinhold, 1991.Google Scholar
  6. R. Dorne and J.K. Hao. A new genetic local search algorithm for graph coloring. In A.E. Eiben, T. Back, M. Schoenauer, and H.P. Schwefel, editors, Parallel Problem Solving from Nature - PPSN V, volume 1498 of Lecture Notes in Computer Science, pages 745–754. Springer-Verlag, 1998a.Google Scholar
  7. R. Dorne and J.K. Hao. Tabu search for graph coloring, T-colorings and set T-colorings. In S. Voss, S. Martello, I.H. Osman, and C. Roucairol, editors, Meta-Heuristics: Advances and Trends in Local Search Paradigms for Optimization, pages 77–92. Kluwer Academic Publishers, 1998b.Google Scholar
  8. N. Dubois and D. De Werra. EPCOT: an efficient procedure for coloring optimally with tabu search. Computers and Mathematics with Applications, 25 (10/11): 35–45, 1993.zbMATHCrossRefGoogle Scholar
  9. C. Fleurent and J.A. Ferland. Genetic and hybrid algorithms for graph coloring. Annals of Operations Research, 63: 437–461, 1996a.zbMATHCrossRefGoogle Scholar
  10. C. Fleurent and J.A. Ferland. Object-oriented implementation of heuristic search methods for graph coloring, maximum clique, and satisfiability. In D.S. Johnson and M.A. Trick, editors, Cliques, Coloring, and Satisfiability, volume 26 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 619–652. American Mathematical Society, 1996b.Google Scholar
  11. N. Funabiki and T. Higashino. A minimal-state processing search algorithm for graph coloring problems. IEICE Transactions on Fundamentals, E83-A(7): 1420–1430, 2000.Google Scholar
  12. P. Galinier. Etude des métaheuristiques pour la résolution du problème de satisfaction de contraintes et de la coloration de graphes. PhD thesis, University of Montpellier II, France, 1999.Google Scholar
  13. P. Galinier and J.K. Hao. Hybrid evolutionary algorithms for graph coloring. Journal of Combinatorial Optimization, 3 (4): 379–397, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  14. A. Gamst. Some lower bounds for a class of frequency assignment problems. IEEE Transactions on Vehicular Technology, 35 (1): 8–14, 1986.CrossRefGoogle Scholar
  15. M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completness. W.H. Freeman and Company, 1979.Google Scholar
  16. F. Glover and M. Laguna. Tabu Search. Kluwer Academic Publishers, 1997.Google Scholar
  17. C.P. Gomes, B. Selman, and H.A. Kautz. Boosting combinatorial search through randomization. In C. Rich, J. Mostow, B.G. Buchanan, and R. Uthurusamy, editors, Proceedings of the Fifteenth National Conference on Artificial Intelligence and Tenth Conference on Innovative Applications of Artificial Intelligence, pages 431–437. AAAI Press/MIT Press, 1998.Google Scholar
  18. M.M. Halldarsson. A still better performance guarantee for approximate graph coloring. Information Processing Letters, 45 (1): 19–23, 1993.MathSciNetCrossRefGoogle Scholar
  19. J.P. Hamiez and J.K. Hao. Scatter search for graph coloring. In P. Collet, E. Lut-ton, M. Schoenauer, C. Fonlupt, and J.K. Hao, editors, Artificial Evolution, volume 2310 of Lecture Notes in Computer Science, pages 168–179. Springer-Verlag, 2002.Google Scholar
  20. A. Hertz and D. De Werra. Using tabu search techniques for graph coloring. Computing, 39: 345–351, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  21. A. Hertz, B. Jaumard, and M. Poggi De Aragâo. Local optima topology for the k-coloring problem. Discrete Applied Mathematics, 49 (1–3): 257–280, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  22. T. Hogg. Refining the phase transition in combinatorial search. Artificial Intelligence, 81 (1–2): 127–154, 1996.MathSciNetCrossRefGoogle Scholar
  23. A. Jagota. An adaptive, multiple restarts neural network algorithm for graph coloring. European Journal of Operational Research, 93 (2): 257–270, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  24. D.S. Johnson, C.R. Aragon, L.A. McGeoch, and C. Schevon. Optimization by simulated annealing: an experimental evaluation; Part II, Graph coloring and number partitioning. Operations Research, 39 (3): 378–406, 1991.zbMATHCrossRefGoogle Scholar
  25. D.S. Johnson and M.A. Trick, editors. Cliques Coloring and Satisfiabilityvolume 26 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society,1996.Google Scholar
  26. P.T. Leighton. A graph coloring algorithm for large scheduling problems. Journal of Research of the National Bureau of Standards, 84: 489–506, 1979.MathSciNetzbMATHCrossRefGoogle Scholar
  27. C. Lund and M. Yannakakis. On the hardness of approximating minimization problems. Journal of the ACM, 41 (5): 960–981, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  28. R. Monasson, R. Zecchina, S. Kirkpatrick, B. Selman, and L. Troyansky. Determining computational complexity from characteristic ‘phase transitions’. Nature, 400 (8): 133–137, 1999.MathSciNetGoogle Scholar
  29. C.A. Morgenstern. Distributed coloration neighborhood search. In D.S. Johnson and M.A. Trick, editors, Cliques, Coloring, and Satisfiability, volume 26 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 335–357. American Mathematical Society, 1996.Google Scholar
  30. E.P.K. Tsang. Foundations of Constraint Satisfaction. Academic Press, 1993.Google Scholar
  31. T. Walsh. Search in a small world. In T. Dean, editor, Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence, volume 2, pages 1172–1177. Morgan Kaufmann Publishers, 1999.Google Scholar
  32. T. Walsh. Search on high degree graphs. In B. Nebel, editor, Proceedings of the Seventeenth International Joint Conference on Artificial Intelligence, volume 1, pages 266–271. Morgan Kaufmann Publishers, 2001.Google Scholar
  33. D.J. Watts and S.H. Strogatz. Collective dynamics of ‘small-world’ networks. Nature, 393: 440–442, 1998.CrossRefGoogle Scholar
  34. D. Welsh. Codes and Cryptography. Oxford University Press, 1988.Google Scholar
  35. M. Yokoo. Why adding more constraints makes a problem easier for hill-climbing algorithms: analyzing landscapes of CSPs. In G. Smolka, editor, Principles and Practice of Constraint Programming - CP97, volume 1330 of Lecture Notes in Computer Science, pages 356–370. Springer-Verlag, 1997.Google Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Jean-Philippe Hamiez
    • 1
  • Jin-Kao Hao
    • 2
  1. 1.LGI2P, École des Mines d’Alès (site EERIE)Parc Scientifique Georges BesseNîmes CEDEX 01France
  2. 2.LERIA, Université d’Angers (U.F.R. des Sciences)Angers CEDEX 01France

Personalised recommendations