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An immune algorithm with stochastic aging and kullback entropy for the chromatic number problem

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Abstract

We present a new Immune Algorithm, IMMALG, that incorporates a Stochastic Aging operator and a simple local search procedure to improve the overall performances in tackling the chromatic number problem (CNP) instances. We characterize the algorithm and set its parameters in terms of Kullback Entropy. Experiments will show that the IA we propose is very competitive with the state-of-art evolutionary algorithms.

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References

  1. Ausiello G. Crescenzi P, Gambosi G, Kann V, Marchetti-Spaccamela A, Protasi M (1999) Complexity and approximation. Springer-Verlag

  2. Barbosa VC, Assis CAG, do Nascimento JO (to appear) Two novel evolutionary formulations of the graph coloring problem. J Combin Optim

  3. Bollobas B (1998) Modern graph theory. Graduate texts in mathematics, vol. 184. Springer-Verlag, Berlin Heidelberg New York

  4. Brooks RL (1941) On colouring the nodes of a network. Cambridge Phil Soc 37:194–197

    MATH  MathSciNet  Google Scholar 

  5. Caramia M, Dell'Olmo P (2001) Iterative coloring extension of a maximum clique. Naval Res Logis 48:518–550

    Article  MATH  MathSciNet  Google Scholar 

  6. Chow FC, Hennessy JL (1990) The priority-based coloring approach to register allocation. ACM Trans Program Languages Syst 12:501–536

    Article  Google Scholar 

  7. Culberson JC, Luo F (1996) Exploring the k-colorable landscape with iterated greedy. Cliques, coloring and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, RI, pp 245–284

  8. Cutello V, Morelli G, Nicosia G, Pavone M (2005) Immune algorithms with aging operators for the string folding problem and the protein folding problem. Lecture Notes Comput Sci 3448:80–90

    Article  Google Scholar 

  9. Cutello V, Nicosia G, Pavone M (2004) Exploring the capability of immune algorithms: a characterization of hypermutation operators. Lecture Notes Comput Sci 3239:263–276

    Article  Google Scholar 

  10. Cutello V, Narzisi G, Nicosia G, Pavone M (2005) Clonal selection algorithms: a comparative case study using effective mutation potentials. Lecture Notes Comput Sci 3627:13–28

    Article  Google Scholar 

  11. Cutello V, Narzisi G, Nicosia G, Pavone M (2005) An immunological algorithm for global numerical optimization. In: Proc. of the seventh international conference on artificial evolution (EA'05). To appear

  12. Cutello V, Nicosia G, Pavone M (2003) A hybrid immune algorithm with information gain for the graph coloring problem. In: Proceedings of genetic and evolutionary computation conference (GECCO) vol. 2723. Springer, pp 171–182

  13. Cutello V, Nicosia G, Pavone M (2004) An immune algorithm with hyper-macromutations for the Dill's 2D hydrophobic-hydrophilic model. Congress on Evolutionary Computation, vol. 1. IEEE Press, pp 1074–1080

  14. Dasgupta D (ed) (1999) Artificial immune systems and their applications. Springer-Verlag, Berlin Heidelberg New York

  15. De Castro LN, Timmis J (2002) Artificial immune systems: a new computational intelligence paradigm. Springer-Verlag, UK

    Google Scholar 

  16. De Castro LN, Von Zuben FJ (2000) The clonal selection algorithm with engineering applications. In: Proceedings of GECCO 2000, workshop on artificial immune systems and their applications, pp 36–37

  17. De Castro LN, Von Zuben FJ (2002) Learning and optimization using the clonal selection principle. IEEE Trans Evolut Comput 6(3):239–251

    Article  Google Scholar 

  18. de Werra D (1985) An introduction to timetabling. European J Oper Res 19:151–162

    Article  MATH  MathSciNet  Google Scholar 

  19. Diestel R (1997) Graph theory. Graduate texts in mathematics, vol. 173. Springer-Verlag, Berlin Heidelberg New York

  20. Eiben AE, Hinterding R, Michalewicz Z (1999) Parameter control in evolutionary algorithms. IEEE Trans Evolut Comput 3(2):124–141

    Article  Google Scholar 

  21. Fleurent C, Ferland JA (1996) Object-oriented implementation of heuristic search methods for graph coloring, maximum clique and satisfability. Cliques, coloring and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, RI, pp 619–652

  22. Forrest S, Hofmeyr SA (2000) Immunology as information processing. Design principles for immune system & other distributed autonomous systems. Oxford Univ. Press, New York

    Google Scholar 

  23. Galinier P, Hao J (1999) Hybrid evolutionary algorithms for graph coloring. J Comb Optim 3(4):379–397

    Article  MATH  MathSciNet  Google Scholar 

  24. Galvin F, Komjáth P (1991) Graph colorings and the axiom of the choice. Period Math Hungar 22:71–75

    Article  MATH  MathSciNet  Google Scholar 

  25. Gamst A (1986) Some lower bounds for a class of frequency assignment problems. IEEE Trans Vehicular Techn 35:8–14

    Google Scholar 

  26. Garey MR, Johnson DS, So HC (1976) An application of graph coloring to printed circuit testing. IEEE Trans Circ Syst CAS-23:591–599

    Article  MATH  MathSciNet  Google Scholar 

  27. Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, New York

    MATH  Google Scholar 

  28. Garrett SM (2005) How do we evaluate artificial immune systems, vol. 13, no. 2 ? Evolutionary Computation, Mit Press, pp 145–178

  29. Glass CA, Prügel-Bennet A (2003) Genetic algorithm for graph coloring: exploration of galinier and hao's algorithm. J Combinat Optim 7(3):229–236

    Article  MATH  Google Scholar 

  30. Glover F, Parker M, Ryan J (1996) Coloring by Tabu branch and bound. Cliques, coloring and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, RI, pp 285–307

  31. Halldórsson MM (1993) A still better performance guarantee for approximate graph coloring. Inf Proc Lett 45:19–23

    Article  MATH  Google Scholar 

  32. Hamiez J, Hao J (2003) An analysis of solution properties of the graph coloring problem. Metaheuristics: Computer Decision-Making. Kluwer, Chapter 15, pp 325–326

  33. Jaynes ET (2003) Probability theory: the logic of science. Cambridge University Press

  34. Jensen TR, Toft B (1995) Graphs coloring problems. Wiley-Interscience Series in Discrete Mathematics and Optimization

  35. Johnson DR, Aragon CR, McGeoch LA, Schevon C (1991) Optimization by simulated annealing: an experimental evaluation; part II, graph coloring and number partitioning. Oper Res 39:378–406

    MATH  Google Scholar 

  36. Johnson DS, Trick MA (eds) (1996) Cliques, coloring and satisfiability: second DIMACS implementation challenge. Am Math Soc, Providence, RI

  37. Kullback S (1959) Statistics and information theory. J. Wiley and Sons, New York

    MATH  Google Scholar 

  38. Leighton FT (1979) A graph coloring algorithm for large scheduling problems. J Res National Bureau Standard 84:489–505

    MATH  MathSciNet  Google Scholar 

  39. Leung K, Duan Q, Xu Z, Wong CW (2001) A new model of simulated evolutionary computation—convergence analysis and specifications. IEEE Trans Evolut Comput 5(1):3–16

    Article  Google Scholar 

  40. Lewandowski G, Condon A (1996) Experiments with parallel graph coloring heuristics and applications of graph coloring. Cliques, coloring and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, RI, pp 309–334

  41. Marino A, Damper RI (2000) Breaking the symmetry of the graph colouring problem with genetic algorithms. Workshop proc. of the genetic and evolutionary computation conference (GECCO'00). Morgan Kaufmann, Las Vegas, NV

  42. Mehrotra A, Trick MA (1996) A column generation approach for graph coloring. INFORMS J Comput 8:344–354

    Article  MATH  Google Scholar 

  43. Morgenstern C (1996) Distributed coloration neighborhood search. Cliques, coloring and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, RI, pp 335–357

  44. Nicosia G, Castiglione F, Motta S (2001) Pattern recognition by primary and secondary response of an artificial immune system. Theory Biosci 120:93–106

    Article  Google Scholar 

  45. Nicosia G, Castiglione F, Motta S (2001) Pattern recognition with a multi-agent model of the immune system. Int. NAISO symposium (ENAIS'2001). ICSC Academic Press, Dubai, UAE, pp 788–794

  46. Nicosia G, Cutello V (2002) Multiple learning using immune algorithms. In: Proceedings of the 4th international conference on recent advances in soft computing. RASC, Nottingham, UK

  47. Seiden PE, Celada F (1992) A model for simulating cognate recognition and response in the immune system. J Theor Biol 158:329–357

    Article  Google Scholar 

  48. Shannon CE (2004) A mathematical theory of communication. Congress on evolutionary computation, vol. 1. IEEE Press, pp 1074–1080 Bell System Technical Journal, vol. 27, pp 379–423 and 623–656 (1948)

  49. Sivia DS (1996) Data analysis. A Bayesian Tutorial. Oxford Science Publications

  50. Tsang EPK (1993) Foundations of constraint satisfaction, vol. 37. Academic Press

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Catania, V.le A. Doria 6, 95125, Catania, Italy

    Vincenzo Cutello, Giuseppe Nicosia & Mario Pavone

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  1. Vincenzo Cutello
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  2. Giuseppe Nicosia
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  3. Mario Pavone
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Correspondence to Giuseppe Nicosia.

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Cutello, V., Nicosia, G. & Pavone, M. An immune algorithm with stochastic aging and kullback entropy for the chromatic number problem. J Comb Optim 14, 9–33 (2007). https://doi.org/10.1007/s10878-006-9036-2

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