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Evolutionary Approach to the Maximum Clique Problem: Empirical Evidence on a Larger Scale

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Developments in Global Optimization

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 18))

Abstract

An algorithm for finding a maximum clique in a graph is presented which uses the Comtet regularization of the Motzkin/Straus continuous problem formulation: maximize an indefinite quadratic form over the standard simplex. We shortly review some surprising connections of the problem with dynamic principles of evolutionary game theory, and give a detailed report on our numerical experiences with the method proposed.

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References

  1. T. Bäck and S. Khuri (1994). An evolutionary heuristic for the maximum independent set problem. Proc. 1st IEEE Conf. Evolutionary Comput. Orlando, FL, 531–535.

    Google Scholar 

  2. L. E. Baum and J. A. Eagon (1967). An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology. Bull. Amer. Math. Soc. 73, 360–363.

    Article  MathSciNet  MATH  Google Scholar 

  3. L. E. Baum and G. R. Sell (1968). Growth transformations for functions on manifolds. Pacif. J. Math. 27(2), 211–227.

    Article  MathSciNet  MATH  Google Scholar 

  4. I. M. Bomze (1991). Cross entropy minimization in uninvadable states of complex populations. J. Math. Biol. 30, 73–87.

    Article  MathSciNet  MATH  Google Scholar 

  5. I. M. Bomze (1996). Evolution towards the maximum clique. J. Global Optim., in press.

    Google Scholar 

  6. I. M. Bomze and J. W. Weibull (1995), Does neutral stability imply Lyapunov stability? Games and Econ. Behav. 11., 173–192.

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Comtet (1974). Advanced Combinatorics. Reidel, Dordrecht.

    Book  MATH  Google Scholar 

  8. J. F. Crow and M. Kimura (1970), An Introduction to Population Genetics Theory. Harper & Row, New York.

    MATH  Google Scholar 

  9. W. J. Ewens (1979). Mathematical Population Genetics. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  10. W. J. Ewens and A. Hastings (1995). Aspects of optimality behavior in population genetics theory. In W. Banzhaf and F. H. Eeckman (Eds.), Evolution and Biocomputation: Computational Models of Evolution (pp. 7–17). Springer-Verlag, Berlin.

    Chapter  Google Scholar 

  11. U. Feige, S. Goldwasser, L. Lovász, S. Safra, and M. Szegedy (1991). Approximating clique is almost NP-complete. Proc. 32nd Ann. Symp. Found. Comput. Sci., San Juan, Puerto Rico, 2–12.

    Google Scholar 

  12. M. W. Feldman (1989). Dynamical systems from evolutionary population genetics. In D. L. Stein (Ed.), Lectures in the Sciences of Complexity (pp. 501–526). Addison-Wesley, Redwood City, CA.

    Google Scholar 

  13. R. A. Fisher (1930). The Genetical Theory of Natural Selection. Clarendon Press, Oxford.

    MATH  Google Scholar 

  14. M. R. Garey and D. S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York.

    MATH  Google Scholar 

  15. L. E. Gibbons, D. W. Hearn, and P. M. Pardalos (1995). A continuous based heuristic for the maximum clique problem. In D. S. Johnson and M. Trick (Eds.), Second DIMACS Implementation Challenge. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, in press.

    Google Scholar 

  16. L. E. Gibbons, D. W. Hearn, P. M. Pardalos, and M. V. Ramana (1996). Continuous characterization of the maximum clique problem. Math. Oper. Res., to appear.

    Google Scholar 

  17. T. Grossman (1995). Applying the INN model to the Max Clique problem. In D. S. Johnson and M. Trick (Eds.), Second DIMACS Implementation Challenge. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, in press.

    Google Scholar 

  18. J. Hasselberg, P. M. Pardalos, and G. Vairaktarakis (1993). Test case generators and computational results for the maximum clique problem. J. Global Optim. 3, 463–482.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Hastings and G. A. Fox (1995). Optimization as a technique for studying population genetics equations. In W. Banzhaf and F. H. Eeckman (Eds.), Evolution and Biocomputation: Computational Models of Evolution (pp. 18–26). Springer-Verlag, Berlin.

    Chapter  Google Scholar 

  20. J. Hofbauer and K. Sigmund (1988). The Theory of Evolution and Dynamical Systems. Cambridge University Press.

    MATH  Google Scholar 

  21. R. Horst, P. Pardalos, and V. Thoai (1995). Introduction to Global Optimization. Kluwer, Amsterdam.

    MATH  Google Scholar 

  22. A. Jagota (1995). Approximating maximum clique with a Hopfield network. IEEE Trans. Neural Networks 6(3), 724–735.

    Article  Google Scholar 

  23. A. Jagota, L. Sanchis, and R. Ganesan (1995). Approximately solving maximum clique using neural networks and related heuristics. In D. S. Johnson and M. Trick (Eds.), Second DIMACS Implementation Challenge. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, in press.

    Google Scholar 

  24. S. Karlin (1984). Mathematical models, problems and controversies of evolutionary theory. Bull. Amer. Math. Soc. 10, 221–273.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Kimura (1958). On the change of population fitness by natural selection. Heredity 12., 145–167.

    Article  Google Scholar 

  26. S. E. Levinson, L. R. Rabiner, and M. M. Sondhi (1983). An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition. Bell Syst. Tech. J. 62(4), 1035–1074.

    MathSciNet  MATH  Google Scholar 

  27. V. Losert and E. Akin (1983). Dynamics of games and genes: discrete versus continuous time. J. Math. Biol. 17, 241–251.

    Article  MathSciNet  MATH  Google Scholar 

  28. Yu. Lyubich, G. D. Maistrowskii, and Yu. G. Ol’khovskii (1980), Selection-induced convergence to equilibrium in a single-locus autosomal population. Problems of Information Transmission 16, 66–75.

    MATH  Google Scholar 

  29. T. S. Motzkin and E. G. Straus (1965). Maxima for graphs and a new proof of a theorem of Turán. Canad. J. Math. 17, 533–540.

    Article  MathSciNet  MATH  Google Scholar 

  30. H. Mühlenbein, M. Gorges-Schleuter, and O. Krämer (1988). Evolution algorithms in combinatorial optimization. Parallel Computing 7, 65–85.

    Article  MATH  Google Scholar 

  31. A. S. Murthy, G. Parthasarathy, and V. U. K. Sastry (1994). Clique finding-A genetic approach. Proc. 1st IEEE Conf. Evolutionary Comput. Orlando, FL, 18–21.

    Google Scholar 

  32. P. M. Pardalos and A. T. Phillips (1990). A global optimization approach for solving the maximum clique problem. Int. J. Computer Math. 33, 209–216.

    Article  MATH  Google Scholar 

  33. P. M. Pardalos, M. G. C. Resende, and K. G. Ramakrishnan (1995), editors: Parallel processing of discrete optimization problems. DIMACS series in discrete mathematics and theoretical computer science, Vol. 22. Amer. Math. Society, Providence, RI.

    MATH  Google Scholar 

  34. P. Pardalos and J. Xue (1994). The maximum clique problem. J. Global Optim. 4, 301–328.

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Pelillo (1995). Relaxation labeling networks for the maximum clique problem. J. Artif. Neural Networks, Special issue on “Neural Networks for Optimization,” 2, 313–327.

    Google Scholar 

  36. M. Pelillo (1996). The dynamics of nonlinear relaxation labeling processes. J. Math. Imaging Vision, to appear.

    Google Scholar 

  37. M. Pelillo and A. Jagota (1995). Feasible and infeasible maxima in a quadratic program for maximum clique. J. Artif. Neural Networks, Special issue on “Neural Networks for Optimization,” 2, 411–419.

    Google Scholar 

  38. P. Taylor and L. Jonker (1978). Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145–156.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Statistik, Operations Research und Computerverfahren, Universität Wien, A-1010, Wien, Austria

    Immanuel Bomze

  2. Dipartimento di Matematica Applicata e Informatica, Università “Ca’ Foscari” di Venezia, I-30173, Venezia Mestre, Italy

    Marcello Pelillo & Robert Giacomini

Authors
  1. Immanuel Bomze
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  2. Marcello Pelillo
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  3. Robert Giacomini
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Editor information

Editors and Affiliations

  1. University of Vienna, Vienna, Austria

    Immanuel M. Bomze

  2. József Attila University, Szeged, Hungary

    Tibor Csendes

  3. University of Trier, Trier, Germany

    Reiner Horst

  4. University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

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© 1997 Springer Science+Business Media Dordrecht

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Bomze, I., Pelillo, M., Giacomini, R. (1997). Evolutionary Approach to the Maximum Clique Problem: Empirical Evidence on a Larger Scale. In: Bomze, I.M., Csendes, T., Horst, R., Pardalos, P.M. (eds) Developments in Global Optimization. Nonconvex Optimization and Its Applications, vol 18. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2600-8_6

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  • DOI: https://doi.org/10.1007/978-1-4757-2600-8_6

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4768-0

  • Online ISBN: 978-1-4757-2600-8

  • eBook Packages: Springer Book Archive

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