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Abstract

It is widely acknowledged that some of the most powerful algorithms for graph coloring involve the combination of evolutionary-based methods with exploitative local search-based techniques. This chapter conducts a review and discussion of such methods, principally focussing on the role that recombination plays in this process. In particular we observe that, while in some cases recombination seems to be usefully combining substructures inherited from parents, in other cases it is merely acting as a macro perturbation operator, helping to reinvigorate the search from time to time.

Keywords

Local Search Tabu Search Graph Coloring Greedy Randomize Adaptive Search Procedure Color Classis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
EA

evolutionary algorithm

GGA

grouping genetic algorithm

GPX

greedy partition crossover

GRASP

greedy randomized adaptive search procedure

RAM

random access memory

References

  1. [63.1]
    N. Zufferey, P. Amstutz, O. Giaccari: Graph colouring approaches for a satellite range scheduling problem, J. Sched. 11(4), 263–277 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [63.2]
    M. Carter: A survey of practical applications of examination timetabling algorithms, Oper. Res. 34(2), 193–202 (1986)MathSciNetCrossRefGoogle Scholar
  3. [63.3]
    R. Lewis: A survey of metaheuristic-based techniques for university timetabling problems, OR Spectrum 30(1), 167–190 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [63.4]
    R. Lewis, J. Thompson: On the application of graph colouring techniques in round-robin sports scheduling, Comput. Oper. Res. 38(1), 190–204 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [63.5]
    K. Aardel, S. van Hoesel, A. Koster, C. Mannino, A. Sassano: Models and solution techniques for the frequency assignment problems, 4OR: Q. J. Belg. Fr. Ital. Oper. Res. Soc. 1(4), 1–40 (2002)MathSciNetGoogle Scholar
  6. [63.6]
    C.M. Valenzuela: A study of permutation operators for minimum span frequency assignment using an order based representation, J. Heuristics 7, 5–21 (2001)CrossRefzbMATHGoogle Scholar
  7. [63.7]
    K. Appel, W. Haken: Solution of the four color map problem, Sci. Am. 4, 108121 (1977)MathSciNetGoogle Scholar
  8. [63.8]
    M. Gamache, A. Hertz, J. Ouellet: A graph coloring model for a feasibility problem in monthly crew scheduling with preferential bidding, Comput. Oper. Res. 34, 2384–2395 (2007)CrossRefzbMATHGoogle Scholar
  9. [63.9]
    G. Chaitin: Register allocation and spilling via graph coloring, ACM SIGPLAN Notices 39(4), 66–74 (2004)MathSciNetCrossRefGoogle Scholar
  10. [63.10]
    M.R. Garey, D.D. Johnson: Computers and Intractability – A guide to NP-completeness, 1st edn. (W. H. Freeman, San Francisco 1979)zbMATHGoogle Scholar
  11. [63.11]
    M. Karp: Reducibility among combinatorial problems. In: Complexity of Computer Computations, The IBM Research Symposia Series, Vol. 1972, ed. by R.E. Miller, J.W. Thatcher, J.D. Bohlinger (Plenum Press, New York 1972) pp. 85–103CrossRefGoogle Scholar
  12. [63.12]
    D. Welsh, M. Powell: An upper bound for the chromatic number of a graph and its application to timetabling problems, Comput. J. 12, 317–322 (1967)zbMATHGoogle Scholar
  13. [63.13]
    D. Brélaz: New methods to color the vertices of a graph, Commun. ACM 22(4), 251–256 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [63.14]
    P. Spinrad, G. Vijayan: Worse case analysis of a graph colouring algorithm, Discrete Appl. Math. 12, 89–92 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [63.15]
    R. Brown: Chromatic scheduling and the chromatic number problem, Manag. Sci. 19(4), 451–463 (1972)zbMATHGoogle Scholar
  16. [63.16]
    S. Korman: The graph-colouring problem. In: Combinatorial Optimization, ed. by N. Christofides, A. Mingozzi, P. Toth, C. Sandi (Wiley, New York 1979) pp. 211–235Google Scholar
  17. [63.17]
    M. Kubale, B. Jackowski: A generalized implicit enumeration algorithm for graph colouring, Communications ACM 28(28), 412–418 (1985)CrossRefGoogle Scholar
  18. [63.18]
    J. Culberson, F. Luo: Exploring the k-colorable landscape with iterated greedy, Proc. 2nd DIMACS Implement. Chall. (1996), pp. 245–284Google Scholar
  19. [63.19]
    C. Mumford: New order-based crossovers for the graph coloring problem, Lect. Notes Comput. Sci. 4193, 880–889 (2006)CrossRefGoogle Scholar
  20. [63.20]
    E. Erben: A grouping genetic algorithm for graph colouring and exam timetabling, Lect. Notes Comput. Sci. 2079, 132–158 (2001)CrossRefzbMATHGoogle Scholar
  21. [63.21]
    R. Lewis: A general-purpose hill-climbing method for order independent minimum grouping problems: A case study in graph colouring and bin packing, Comput. Oper. Res. 36(7), 2295–2310 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [63.22]
    M. Chams, A. Hertz, O. Dubuis: Some experiments with simulated annealing for coloring graphs, Eur. J. Oper. Res. 32, 260–266 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [63.23]
    D. Johnson, C. Aragon, L. McGeoch, C. Schevon: Optimization by simulated annealing: An experimental evaluation; part II, graph coloring and number partitioning, Oper. Res. 39, 378–406 (1991)CrossRefzbMATHGoogle Scholar
  24. [63.24]
    A. Hertz, D. de Werra: Using tabu search techniques for graph coloring, Computing 39(4), 345–351 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [63.25]
    M. Laguna, R. Marti: A GRASP for coloring sparse graphs, Comput. Optim. Appl. 19, 165–178 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [63.26]
    M. Chiarandini, T. Stützle: An application of iterated local search to graph coloring, Proc. Comput. Symp. Graph Color. Gen. (2002) pp. 112–125Google Scholar
  27. [63.27]
    L. Paquete, T. Stützle: An experimental investigation of iterated local search for coloring graphs, applications of evolutionary computing, Lect. Notes Comput. Sci. 2279, 121–130 (2002)zbMATHGoogle Scholar
  28. [63.28]
    C. Avanthay, A. Hertz, N. Zufferey: A variable neighborhood search for graph coloring, Eur. J. Oper. Res. 151, 379–388 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. [63.29]
    J. Thompson, K. Dowsland: An improved ant colony optimisation heuristic for graph colouring, Discrete Appl. Math. 156, 313–324 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [63.30]
    R. Dorne, J.-K. Hao: A new genetic local search algorithm for graph coloring, Lect. Notes Comput. Sci. 1498, 745–754 (1998)CrossRefGoogle Scholar
  31. [63.31]
    A.E. Eiben, J.K. van der Hauw, J.I. van Hemert: Graph coloring with adaptive evolutionary algorithms, J. Heuristics 4(1), 25–46 (1998)CrossRefzbMATHGoogle Scholar
  32. [63.32]
    C. Fleurent, J. Ferland: Genetic and hybrid algorithms for graph colouring, Ann. Oper. Res. 63, 437–461 (1996)CrossRefzbMATHGoogle Scholar
  33. [63.33]
    P. Galinier, J.-K. Hao: Hybrid evolutionary algorithms for graph coloring, J. Comb. Optim. 3, 379–397 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [63.34]
    Z. Lü, J.-K. Hao: A memetic algorithm for graph coloring, Eur. J. Oper. Res. 203(1), 241–250 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. [63.35]
    D. Porumbel, J.-K. Hao, P. Kuntz: An evolutionary approach with diversity guarantee and well-informed grouping recombination for graph coloring, Comput. Oper. Res. 37, 1822–1832 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  36. [63.36]
    C. Morgenstern: Distributed coloration neighborhood search, Discrete Math. Theor. Comput. Sci. 26, 335–358 (1996)zbMATHGoogle Scholar
  37. [63.37]
    I. Blochliger, N. Zufferey: A graph coloring heuristic using partial solutions and a reactive tabu scheme, Comput. Oper. Res. 35, 960–975 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  38. [63.38]
    E. Malaguti, M. Monaci, P. Toth: A metaheuristic approach for the vertex coloring problem, INFORMS J. Comput. 20(2), 302–316 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  39. [63.39]
    A. Hertz, M. Plumettaz, N. Zufferey: Variable space search for graph coloring, Discrete Appl. Math. 156(13), 2551–2560 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  40. [63.40]
    N.J. Radcliffe: Forma analysis and random respectful recombination, Proc. 4th Int. Conf. Genet. Algorithms (1991) pp. 222–229Google Scholar
  41. [63.41]
    E. Falkenauer: Genetic Algorithms and Grouping Problems, 1st edn. (Wiley, New York 1998)zbMATHGoogle Scholar
  42. [63.42]
    E. Coll, G. Duran, P. Moscato: A discussion on some design principles for efficient crossover operators for graph coloring problems, An. XXVII Simp. Bras. Pesqui. Oper. (1995)Google Scholar
  43. [63.43]
    R. Abbasian, M. Mouhoub, A. Jula: Solving graph coloring problems using cultural algorithms, Proc. 24th Florida Artif. Intell. Res. Soc. Conf. (2011)Google Scholar
  44. [63.44]
    J. Munkres: Algorithms for the assignment and transportation problems, J. Soc. Ind. Appl. Math. 5(1), 32–38 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  45. [63.45]
    D. Bertsekas: Auction algorithms for network flow problems: A tutorial introduction, Comput. Optim. Appl. 1, 7–66 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  46. [63.46]
    A. Tucker, J. Crampton, S. Swift: RGFGA: An efficient representation and crossover for grouping genetic algorithms, Evol. Comput. 13(4), 477–499 (2005)CrossRefGoogle Scholar
  47. [63.47]
    R. Lewis, E. Pullin: Revisiting the restricted growth function genetic algorithm for grouping problems, Evol. Comput. 19(4), 693–704 (2011)CrossRefGoogle Scholar
  48. [63.48]
    E. Falkenauer: A new representation and operators for genetic algorithms applied to grouping problems, Evol. Comput. 2(2), 123–144 (1994)CrossRefGoogle Scholar
  49. [63.49]
    C. Glass, A. Prugel-Bennett: Genetic algorithms for graph coloring: Exploration of Galnier and Hao's algorithm, J. Comb. Optim. 7, 229–236 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  50. [63.50]
    R. Lewis, J. Thompson, C. Mumford, J. Gillard: A wide-ranging computational comparison of high-performance graph colouring algorithms, Comput. Oper. Res. 39(9), 1933–1950 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  51. [63.51]
    P. Galinier, A. Hertz: A survey of local search algorithms for graph coloring, Comput. Oper. Res. 33, 2547–2562 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  52. [63.52]
    J. Culberson: Graph coloring page, http://web.cs.ualberta.ca/~joe/Coloring/ (2010)
  53. [63.53]
    M. Carter, G. Laporte, S.Y. Lee: Examination timetabling: Algorithmic strategies and applications, J. Oper. Res. Soc. 47, 373–383 (1996)CrossRefGoogle Scholar
  54. [63.54]
    T. Hogg, B. Huberman, C. Williams: Refining the phase transition in combinatorial search, Artif. Intell. 81(1/2), 127–154 (1996)MathSciNetCrossRefGoogle Scholar
  55. [63.55]
    K. Smith-Miles, L. Lopes: Measuring instance difficulty for combinatorial optimization problems, Comput. Oper. Res. 39(5), 875–889 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of MathematicsCardiff UniversityCardiffUK

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