Journal of Heuristics

, Volume 24, Issue 1, pp 1–24 | Cite as

Variations on memetic algorithms for graph coloring problems



Graph vertex coloring with a given number of colors is a well-known and much-studied NP-complete problem. The most effective methods to solve this problem are proved to be hybrid algorithms such as memetic algorithms or quantum annealing. Those hybrid algorithms use a powerful local search inside a population-based algorithm. This paper presents a new memetic algorithm based on one of the most effective algorithms: the hybrid evolutionary algorithm (HEA) from Galinier and Hao (J Comb Optim 3(4): 379–397, 1999). The proposed algorithm, denoted HEAD—for HEA in Duet—works with a population of only two individuals. Moreover, a new way of managing diversity is brought by HEAD. These two main differences greatly improve the results, both in terms of solution quality and computational time. HEAD has produced several good results for the popular DIMACS benchmark graphs, such as 222-colorings for \({<}{} \texttt {dsjc1000.9}{>}\), 81-colorings for \({<}{} \texttt {flat1000\_76\_0}{>}\) and even 47-colorings for \({<}{} \texttt {dsjc500.5}{>}\) and 82-colorings for \({<}{} \texttt {dsjc1000.5}{>}\).


Combinatorial optimization Metaheuristics Coloring Graph Evolutionary 


  1. Aardal, K., Hoesel, S., Koster, A., Mannino, C., Sassano, A.: Models and solution techniques for frequency assignment problems. Q. J. Belg. Fr. Ital. Oper. Res. Soc. 1(4), 261–317 (2003)MathSciNetMATHGoogle Scholar
  2. Allignol, C., Barnier, N., Gondran, A.: Optimized flight level allocation at the continental scale. In: International Conference on Research in Air Transportation (ICRAT 2012), Berkeley, California, USA, 22–25 May 2012 (2012)Google Scholar
  3. Barnier, N., Brisset, P.: Graph coloring for air traffic flow management. Ann. Oper. Res. 130(1–4), 163–178 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. Dib, M., Caminada, A., Mabed, H.: Frequency management in radio military networks. In: INFORMS Telecom 2010, 10th INFORMS Telecommunications Conference Montreal, Canada (2010)Google Scholar
  5. Dubois, N., de Werra, D.: Epcot: an efficient procedure for coloring optimally with Tabu search. Comput. Math. Appl. 25(10–11), 35–45 (1993)MathSciNetCrossRefMATHGoogle Scholar
  6. Fleurent, C., Ferland, J.: Genetic and hybrid algorithms for graph coloring. Ann. Oper. Res. 63, 437–464 (1996)CrossRefMATHGoogle Scholar
  7. Galinier, P., Hao, J.-K.: Hybrid evolutionary algorithms for graph coloring. J. Comb. Optim. 3(4), 379–397 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. Galinier, P., Hertz, A.: A survey of local search methods for graph coloring. Comput. Oper. Res. 33, 2547–2562 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. Galinier, P., Hertz, A., Zufferey, N.: An adaptive memory algorithm for the \(k\)-coloring problem. Discret. Appl. Math. 156(2), 267–279 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. Galinier, P., Hamiez, J.-P., Hao, J.-K., Porumbel, D.C.: Recent advances in graph vertex coloring. In: Zelinka, I.,  Snásel, V.,  Abraham, A. (eds) Handbook of Optimization, Vol. 38 of Intelligent Systems Reference Library, pp. 505–528. Springer, Berlin (2013)Google Scholar
  11. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of \({\cal{NP}}\)-Completeness. Freeman, San Francisco (1979)MATHGoogle Scholar
  12. Gusfield, D.: Partition-distance: a problem and class of perfect graphs arising in clustering. Inf. Process. Lett. 82(3), 159–164 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. Hao, J.-K.: Memetic algorithms in discrete optimization. In: Neri, F.,  Cotta, C.,  Moscato, P. (eds.) Handbook of Memetic Algorithms, Vol. 379 of Studies in Computational Intelligence, pp. 73–94. Springer, Berlin (2012)Google Scholar
  14. Hao, J.-K., Wu, Q.: Improving the extraction and expansion method for large graph coloring. Discret. Appl. Math. 160(16–17), 2397–2407 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. Held, S., Cook, W., Sewell, E.C.: Safe lower bounds for graph coloring. In: Günlük, O., Woeginger, G.J. (eds.) Integer Programming and Combinatoral Optimization. IPCO 2011. Lecture Notes in Computer Science, Vol. 6655. Springer, Berlin, Heidelberg (2011)Google Scholar
  16. Hertz, A., de Werra, D.: Using Tabu search techniques for graph coloring. Computing 39(4), 345–351 (1987)MathSciNetCrossRefMATHGoogle Scholar
  17. Hertz, A., Plumettaz, M., Zufferey, N.: Variable space search for graph coloring. Discret. Appl. Math. 156(13), 2551–2560 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. Johnson, D.S., Trick, M.: Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, 1993, Vol. 26 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, Providence (1996)Google Scholar
  19. Johnson, D.S., Aragon, C.R., McGeoch, L.A., Schevon, C.: Optimization by Simulated annealing: an experimental evaluation. Part II, graph coloring and number partitioning. Oper. Res. 39(3), 378–406 (1991)CrossRefMATHGoogle Scholar
  20. Karp, R.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)CrossRefGoogle Scholar
  21. Leighton, F.T.: A graph coloring algorithm for large scheduling problems. J. Res. Natl. Bur. Stand. 84(6), 489–506 (1979)MathSciNetCrossRefMATHGoogle Scholar
  22. Lewis, R.: Graph coloring and recombination. In: Kacprzyk, J., Pedrycz, W. (eds.) Handbook of Computational Intelligence, pp. 1239–1254. Springer, Berlin (2015). (Ch. Graph Coloring and Recombination)Google Scholar
  23. Lü, Z., Hao, J.-K.: A memetic algorithm for graph coloring. Eur. J. Oper. Res. 203(1), 241–250 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. Malaguti, E., Toth, P.: A survey on vertex coloring problems. Int. Trans. Oper. Res. 17(1), 1–34 (2010)MathSciNetCrossRefMATHGoogle Scholar
  25. Malaguti, E., Monaci, M., Toth, P.: An exact approach for the vertex coloring problem. Discret. Optim. 8(2), 174–190 (2011)MathSciNetCrossRefMATHGoogle Scholar
  26. Porumbel, D.C., Hao, J.-K., Kuntz, P.: An evolutionary approach with diversity guarantee and well-informed grouping recombination for graph coloring. Comput. Oper. Res. 37, 1822–1832 (2010)CrossRefMATHGoogle Scholar
  27. Titiloye, O., Crispin, A.: Graph coloring with a distributed hybrid quantum annealing algorithm. In: O’Shea, J., Nguyen, N., Crockett, K., Howlett, R., Jain, L. (eds.) Agent and Multi-Agent Systems: Technologies and Applications. Lecture Notes in Computer Science, vol. 6682, pp. 553–562. Springer, Berlin (2011a)CrossRefGoogle Scholar
  28. Titiloye, O., Crispin, A.: Quantum annealing of the graph coloring problem. Discret. Optim. 8(2), 376–384 (2011b)MathSciNetCrossRefMATHGoogle Scholar
  29. Titiloye, O., Crispin, A.: Parameter tuning patterns for random graph coloring with quantum annealing. PLoS ONE 7(11), e50060 (2012)CrossRefGoogle Scholar
  30. Wood, D.C.: A technique for coloring a graph applicable to large-scale timetabling problems. Comput. J. 12, 317–322 (1969)MathSciNetCrossRefMATHGoogle Scholar
  31. Wu, Q., Hao, J.-K.: Coloring large graphs based on independent set extraction. Comput. Oper. Res. 39(2), 283–290 (2012)MathSciNetCrossRefMATHGoogle Scholar
  32. Zufferey, N., Amstutz, P., Giaccari, P.: Graph colouring approaches for a satellite range scheduling problem. J. Sched. 11(4), 263–277 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.UTBM, OPERAUniv. Bourgogne Franche-ComtéBelfortFrance
  2. 2.MAIAA, ENACFrench Civil Aviation UniversityToulouseFrance

Personalised recommendations