Negative Slope Coefficient: A Measure to Characterize Genetic Programming Fitness Landscapes

  • Leonardo Vanneschi
  • Marco Tomassini
  • Philippe Collard
  • Sébastien Vérel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3905)


Negative slope coefficient has been recently introduced and empirically proven a suitable hardness indicator for some well known genetic programming benchmarks, such as the even parity problem, the binomial-3 and the artificial ant on the Santa Fe trail. Nevertheless, the original definition of this measure contains several limitations. This paper points out some of those limitations, presents a new and more relevant definition of the negative slope coefficient and empirically shows the suitability of this new definition as a hardness measure for some genetic programming benchmarks, including the multiplexer, the intertwined spirals problem and the royal trees.


Genetic Algorithm Genetic Programming Negative Slope Genetic Operator Fitness Landscape 
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.


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  1. 1.
    Altenberg, L.: The evolution of evolvability in genetic programming. In: Kinnear, K. (ed.) Advances in Genetic Programming, pp. 47–74. MIT Press, Cambridge (1994)Google Scholar
  2. 2.
    Clergue, M., Collard, P., Tomassini, M., Vanneschi, L.: Fitness distance correlation and problem difficulty for genetic programming. In: Langdon, W.B., et al. (eds.) Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 2002, New York City, USA, pp. 724–732. Morgan Kaufmann, San Francisco (2002)Google Scholar
  3. 3.
    Deb, K., Goldberg, D.E.: Analyzing deception in trap functions. In: Whitley, D. (ed.) Foundations of Genetic Algorithms, vol. 2, pp. 93–108. Morgan Kaufmann, San Francisco (1993)Google Scholar
  4. 4.
    Forrest, S., Mitchell, M.: What makes a problem hard for a genetic algorithm? Some anomalous results and their explanation. Machine Learning 13, 285–319 (1993)CrossRefGoogle Scholar
  5. 5.
    Horn, J., Goldberg, D.E.: Genetic algorithm difficulty and the modality of the fitness landscapes. In: Whitley, D., Vose, M. (eds.) Foundations of Genetic Algorithms, vol. 3, pp. 243–269. Morgan Kaufmann, San Francisco (1995)Google Scholar
  6. 6.
    Jones, T.: Evolutionary Algorithms, Fitness Landscapes and Search. PhD thesis, University of New Mexico, Albuquerque (1995)Google Scholar
  7. 7.
    Kinnear Jr., K.E.: Fitness landscapes and difficulty in genetic programming. In: Proceedings of the First IEEEConference on Evolutionary Computing, Piscataway, NY, pp. 142–147. IEEE Press, Los Alamitos (1994)Google Scholar
  8. 8.
    Koza, J.R.: Genetic Programming. The MIT Press, Cambridge (1992)MATHGoogle Scholar
  9. 9.
    Langdon, W.B., Poli, R.: Foundations of Genetic Programming. Springer, Heidelberg (2002)CrossRefMATHGoogle Scholar
  10. 10.
    Madras, N.: Lectures on Monte Carlo Methods. In: American Mathematical Society, Providence, Rhode Island (2002)Google Scholar
  11. 11.
    Manderick, B., de Weger, M., Spiessens, P.: The genetic algorithm and the structure of the fitness landscape. In: Belew, R.K., Booker, L.B. (eds.) Proceedings of the Fourth International Conference on Genetic Algorithms, pp. 143–150. Morgan Kaufmann, San Francisco (1991)Google Scholar
  12. 12.
    Mitchell, M., Forrest, S., Holland, J.: The royal road for genetic algorithms: fitness landscapes and ga performance. In: Varela, F.J., Bourgine, P. (eds.) Toward a Practice of Autonomous Systems, Proceedings of the First European Conference on Artificial Life, pp. 245–254. The MIT Press, Cambridge (1992)Google Scholar
  13. 13.
    Naudts, B., Kallel, L.: A comparison of predictive measures of problem difficulty in evolutionary algorithms. IEEE Transactions on Evolutionary Computation 4(1), 1–15 (2000)CrossRefGoogle Scholar
  14. 14.
    Punch, W.: How effective are multiple populations in genetic programming. In: Koza, J.R., Banzhaf, W., Chellapilla, K., Deb, K., Dorigo, M., Fogel, D.B., Garzon, M., Goldberg, D., Iba, H., Riolo, R.L. (eds.) Genetic Programming 1998: Proceedings of the Third Annual Conference, pp. 308–313. Morgan Kaufmann, San Francisco (1998)Google Scholar
  15. 15.
    Stadler, P.F.: Fitness landscapes. In: Lässig, M., Valleriani (eds.) Biological Evolution and Statistical Physics. Lecture Notes Physics, vol. 585, pp. 187–207. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  16. 16.
    Tomassini, M., Vanneschi, L., Collard, P., Clergue, M.: A study of fitness distance correlation as a difficulty measure in genetic programming. Evolutionary Computation 13(2), 213–239 (2005)CrossRefMATHGoogle Scholar
  17. 17.
    Vanneschi, L.: Theory and Practice for Efficient Genetic Programming. Ph.D. thesis, Faculty of Science, University of Lausanne, Switzerland (2004), Downlodable version at,
  18. 18.
    Vanneschi, L., Clergue, M., Collard, P., Tomassini, M., Vérel, S.: Fitness clouds and problem hardness in genetic programming. In: Deb, K., et al. (eds.) GECCO 2004. LNCS, vol. 3103, pp. 690–701. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Vanneschi, L., Tomassini, M., Clergue, M., Collard, P.: Difficulty of unimodal and multimodal landscapes in genetic programming. In: Cantú-Paz, E., Foster, J.A., Deb, K., Davis, L., Roy, R., O’Reilly, U.-M., Beyer, H.-G., Kendall, G., Wilson, S.W., Harman, M., Wegener, J., Dasgupta, D., Potter, M.A., Schultz, A., Dowsland, K.A., Jonoska, N., Miller, J., Standish, R.K. (eds.) GECCO 2003. LNCS, vol. 2723, pp. 1788–1799. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  20. 20.
    Vanneschi, L., Tomassini, M., Collard, P., Clergue, M.: Fitness distance correlation in genetic programming: a constructive counterexample. In: Congress on Evolutionary Computation (CEC 2003), Canberra, Australia, pp. 289–296. IEEE Press, Piscataway (2003)Google Scholar
  21. 21.
    Vérel, S., Collard, P., Clergue, M.: Where are bottleneck in nk-fitness landscapes? In: CEC 2003: IEEE International Congress on Evolutionary Computation, Canberra, Australia, pp. 273–280. IEEE Press, Piscataway (2003)Google Scholar
  22. 22.
    Weinberger, E.D.: Correlated and uncorrelated fitness landscapes and how to tell the difference. Biol. Cybern. 63, 325–336 (1990)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Leonardo Vanneschi
    • 1
  • Marco Tomassini
    • 2
  • Philippe Collard
    • 3
  • Sébastien Vérel
    • 3
  1. 1.Dipartimento di Informatica, Sistemistica e Comunicazione (D.I.S.Co.)University of Milan-BicoccaMilanItaly
  2. 2.Computer Systems DepartmentUniversity of LausanneLausanneSwitzerland
  3. 3.I3S LaboratoryUniversity of NiceSophia AntipolisFrance

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