Journal of Heuristics

, Volume 4, Issue 1, pp 25–46 | Cite as

Graph Coloring with Adaptive Evolutionary Algorithms

  • A.E. Eiben
  • J.K. van der Hauw
  • J.I. van Hemert


This paper presents the results of an experimental investigation on solving graph coloring problems with Evolutionary Algorithms (EAs). After testing different algorithm variants we conclude that the best option is an asexual EA using order-based representation and an adaptation mechanism that periodically changes the fitness function during the evolution. This adaptive EA is general, using no domain specific knowledge, except, of course, from the decoder (fitness function). We compare this adaptive EA to a powerful traditional graph coloring technique DSatur and the Grouping Genetic Algorithm (GGA) on a wide range of problem instances with different size, topology and edge density. The results show that the adaptive EA is superior to the Grouping (GA) and outperforms DSatur on the hardest problem instances. Furthermore, it scales up better with the problem size than the other two algorithms and indicates a linear computational complexity.

evolutionary algorithms genetic algorithms constraint satisfaction graph coloring grouping problem penalty functions adaptive parameters 


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  1. Angeline, P. (1995). “Adaptive and Self-Adaptive Evolutionary Computation.” In M. Palaniswami, Y. Attikiouzel, R.J. Marks, D. Fogel, and T. Fukuda (eds.), Computational Intelligence: A Dynamic System Perspective. IEEE Press, pp. 152-161.Google Scholar
  2. Arora, S., C. Lund, R. Motwani, M. Sudan, and M. Szegedy. (1992). “Proof Verification and Hardness of Approximation Problems,” Proceedings 33rd IEEE Symposium on the Foundations of Computer Science. IEEE Computer Sociecty, Los Angeles, CA, pp. 14-23.Google Scholar
  3. Bäck, T., D. Fogel, and Z. Michalewicz. (eds.) (1997). Handbook of Evolutionary Computation. Bristol: Institute of Physics Publishing and New York: Oxford University Press.Google Scholar
  4. Belew, R. and L. Booker. (eds.) (1991). Proceedings of the 4th International Conference on Genetic Algorithms. Morgan Kaufmann.Google Scholar
  5. Blum, A. (1989). “An O.n0.4/-Approximation Algorithm for 3-Coloring (and Improved Approximation Algorithms for k-Coloring),” Proceedings of the 21st ACM Symposium on Theory of Computing. New York, ACM, pp. 535-542.Google Scholar
  6. Brélaz, D. (1979). “New Methods to Color Vertices of a Graph,” Communications of the ACM22, 251-256.Google Scholar
  7. Cheeseman, P., B. Kenefsky, and W.M. Taylor. (1991). “Where the Really Hard Problems Are.” In J. Mylopoulos, and R. Reiter (eds.), Proceedings of the 12th IJCAI-91. Morgan Kaufmann, vol. 1, pp. 331-337.Google Scholar
  8. Clearwater, S. and T. Hogg. (1996). “Problem Structure Heuristics and Scaling Behavior for Genetic Algorithms,” Artificial Intelligence81, 327-347.Google Scholar
  9. Coll, P., G. Durán, and P. Moscato. (1995). “A Discussion on Some Design Principles for Efficient Crossover Operators for Graph Coloring Problems,” Anales del XXVII Simposio Brasileiro de Pesquisa Operacional.Google Scholar
  10. Culberson, J. (1996). “On the Futility of Blind Search,” Technical Report TR 96-18, The University of Alberta.Google Scholar
  11. Culberson, J. and F. Luo. (1996). “Exploring the k-Colorable Landscape with Iterated Greedy.” In D. Johnson and M. Trick (eds.), Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge. American Mathematical Society, pp. 245–284. Available by Scholar
  12. Davis, L. (1991). “Order-Based Genetic Algorihms and the Graph Coloring Problem.” Handbook of Genetic Algorithms. New York: Van Nostrand Reinhold, pp. 72-90.Google Scholar
  13. De Jong, K. and W. Spears. (1992). “A Formal Analysis of the Role of Multi-point Crossover in Genetic Algorithms,” Annals of Mathematics and Artificial Intelligence5, 1-26.Google Scholar
  14. Eiben, A., P.-E. Raué, and Z. Ruttkay. (1994). “Genetic Algorithms with Multi-parent Recombination.” In Y. Davidor, H.-P. Schwefel, and R. Männer (eds.), Proceedings of the 3rd Conference on Parallel Problem Solving from Nature, number 866 in Lecture Notes in Computer Science. Springer-Verlag, pp. 78-87.Google Scholar
  15. Eiben, A., P.-E. Raué, and Z. Ruttkay. (1995a). “Constrained Problems.” In L. Chambers (ed.), Practical Handbook of Genetic Algorithms. CRC Press, pp. 307-365.Google Scholar
  16. Eiben, A., P.-E. Raué, and Z. Ruttkay. (1995b). “GA-Easy and GA-Hard Constraint Satisfaction Problems.” In M. Meyer (ed.), Proceedings of the ECAI-94 Workshop on Constraint Processing, number 923 in Lecture Notes in Computer Science. Springer-Verlag, pp. 267-284.Google Scholar
  17. Eiben, A. and Z. Ruttkay. (1996). “Self-adaptivity for Constraint Satisfaction: Learning Penalty Functions,” Proceedings of the 3rd IEEE Conference on Evolutionary Computation. IEEE Press, pp. 258–261.Google Scholar
  18. Eiben, A. and J. van der Hauw. (1996). “Graph Coloring with Adaptive Evolutionary Algorithms,” Technical Report TR-96-11, Leiden University. Also available as http:// Scholar
  19. Eiben, A. and J. van der Hauw. (1997). “Solving 3-SAT by GAS Adapting Constraint Weights,” Proceedings of the 4th IEEE Conference on Evolutionary Computation. IEEE Press, pp. 81-86.Google Scholar
  20. Eiben, A. and Z. Ruttkay. (1997). “Constraint Satisfaction Problems.” In T. Bäck et al. (eds.), Handbook of Evolutionary Computation. Bristol: Institute of Physics Publishing, and New York: Oxford University Press.Google Scholar
  21. Falkenauer, E. (1994). “A New Representation and Operators for Genetic Algorithms Applied to Grouping Problems,” Evolutionary Computation2(2), 123-144.Google Scholar
  22. Falkenauer, E. (1996). “A Hybrid Grouping Genetic Algorithm for Bin Packing,” Journal of Heuristics2, 5-30.Google Scholar
  23. Falkenauer, E. and A. Delchambre. (1992). “A Genetic Algorithm for Bin Packing and Line Balancing,” Proceedings of the IEEE 1992 Int. Conference on Robotics and Automation. IEEE Computer Society Press, pp. 1186- 1192.Google Scholar
  24. Fleurent, C. and J. Ferland. (1996a). “Genetic and Hybrid Algorithms for Graph Coloring.” In I.H.O. G. Laporte and P.L. Hammer (eds.), Annals of Operations Research, number 63 in Metaheuristics in Combinatorial Optimization. J.C. Baltzer AG, Science Publishers, pp. 437-461.Google Scholar
  25. Fleurent, C. and J. Ferland. (1996b). “Object-Oriented Implementation of Heuristic Search Methods for Graph Coloring, Maximum Clique, and Satisfiability.” In M.A. Trick and D.S. Johnson (eds.), Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, volume 26 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, pp. 619-652.Google Scholar
  26. Fogel, D. (1995). Evolutionary Computation. IEEE Press.Google Scholar
  27. Fox, B. and M. McMahon. (1991). “Genetic Operators for Sequencing Problems.” In G. Rawlins (ed.), Foundations of Genetic Algorithms. Morgan Kaufmann, pp. 284–300.Google Scholar
  28. Frank, J. (1996a). “Learning Short-term Weights For GSAT,” Technical Report, University of California at Davis. Available by Scholar
  29. Frank, J. (1996b). “Weighting for Godot: Learning Heuristics For GSAT,” Proceedings of the 13th AAAI-96. AAAI / The MIT Press, pp. 338-343.Google Scholar
  30. Garey, M. and D. Johnson. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freedman and Co.Google Scholar
  31. Glover, F. (1996). “Tabu Search and Adaptive Memory Programming-Advances, Applications, and Challenges,” Interfaces in Computer Science and Operations Research. Norwell, MA: Kluwer Academic Publishers, pp. 1- 75.Google Scholar
  32. Grimmet, G. and C. McDiarmid. (1975). “On Colouring Random Graphs,” Mathematical Proceedings of the Cambridge Philosophical Society77, 313-324.Google Scholar
  33. Hinterding, R., Z. Michalewicz, and A. Eiben. (1997). “Adaptation in Evolutionary Computation: A Survey,” Proceedings of the 4th IEEE Conference on Evolutionary Computation. IEEE Press, pp. 65-69.Google Scholar
  34. Johnson, D., C. Aragon, L. McGeoch, and C. Schevon. (1991). “Optimization by Simulated Annealing: An Experimental Evaluation; Part II, Graph Coloring and Number Partitioning,” Operations Research39(3), 378- 406.Google Scholar
  35. Kronsjo, L. (1987). Algorithms: Their Complexity and Efficiency. Wiley and Sons, second edition.Google Scholar
  36. Kučera, L. (1991). “The Greedy Coloring is a Bad Probabilistic Algorithm,” Journal of Algorithms12, 674-684.Google Scholar
  37. Laszewski, G.V. (1991). “Intelligent Structural Operators For the k-Way Graph Partitioning Problem.” In R. Belew and L. Booker (eds.), Proceeding of the 4th International Conference on Genetic Algorithms. Morgan Kaufmann, pp. 45-52.Google Scholar
  38. Løkketangen, A. and F. Glover. (1996). “Surrogate Constraint Methods with Simple Learning for Satisfiability Problems.” In D. Du, J. Gu, and P. Pardalos (eds.), Proceedings of the DIMACS workshop on Satisfiability Problems: Theory and Applications. American Mathematical Society.Google Scholar
  39. Morris, P. (1993). “The Breakout Method for Escaping From Local Minima,” Proceedings of the 11th National Conference on Artificial Intelligence, AAAI-93. AAAI Press/The MIT Press, pp. 40-45.Google Scholar
  40. Nudel, B. (1983). “Consistent-Labeling Problems and Their Algorithms: Expected Complexities and Theory Based Heuristics,” Artificial Intelligence21, 135-178.Google Scholar
  41. Schwefel, H.-P. (1995). Evolution and Optimum Seeking. Sixth-Generation Computer Technology Series. New York: Wiley.Google Scholar
  42. Selman, B. and H. Kautz. (1993). “Domain-Independent Extensions to GSAT: Solving Large Structured Satisfiability Problems.” In R. Bajcsy (ed.), Proceedings of IJCAI'93. Morgan Kaufmann, pp. 290-295.Google Scholar
  43. Starkweather, T., S. McDaniel, K. Mathias, D. Whitley, and C. Whitley. (1991). “A Comparison of Genetic Sequenceing Operators.” In R. Belew and L. Booker (eds.), Proceedings of the 4th International Conference on Genetic Algorithms. Morgan Kaufmann, pp. 69-76.Google Scholar
  44. Turner, J. (1988). “Almost All k-Colorable Graphs are Easy to Color,” Journal of Algorithms9, 63-82.Google Scholar
  45. Wolpert, D. and W. Macready. (1997). “No Free Lunch Theorems for Optimization,” IEEE Transactions on Evolutionary Computation1(1), 67-82.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • A.E. Eiben
    • 1
  • J.K. van der Hauw
    • 1
  • J.I. van Hemert
    • 1
  1. 1.Leiden UniversityThe Netherlands

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