Abstract
An algorithm for approximately solving the maximum clique is presented which uses relaxation labeling neural network techniques, focusing on the continuous problem formulation: maximize a quadratic form over the standard simplex. We employ somewhat surprising connections of the latter 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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L. E. Baum and G. R. Sell (1968). Growth transformations for functions on manifolds. Pacif. J. Math. 27(2), 211–227.
I. M. Bomze (1991). Cross entropy minimization in uninvadable states of complex populations. J. Math. Biol. 30, 73–87.
I. M. Bomze (1995). Evolution towards the maximum clique. J. Global. Optim., in press.
I. M. Bomze, M. Pelillo, and R. Giacomini (1996). Evolutionary approach to the maximum clique problem: Empirical evidence on a larger scale. In I. M. Bomze, T. Csendes, R. Horst, and P. Pardalos (Eds.), Developments in Global Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands.
C. Bron and J. Kerbosch (1973). Finding all cliques of an undirected graph. Comm. ACM 16(9), 575–577.
R. Carraghan and P. M. Pardalos (1990). An exact algorithm for the maximum clique problem. Oper. Res. Lett. 9, 375–382.
B. Carter and K. Park (1993). How good are genetic algorithms at finding large cliques: An experimental study. Technical Report BU-CS-93-015, Computer Science Department., Boston University.
L. Comtet (1974). Advanced Combinatorics. Reidel, Dordrecht.
J. F. Crow and M. Kimura (1970), An Introduction to Population Genetics Theory. Harper & Row, New York.
R. A. Fisher (1930). The Genetical Theory of Natural Selection. Clarendon Press, Oxford.
M. R. Garey and D. S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York.
L. E. Gibbons et al. (1996). Continuous characterization of the maximum clique problem. Math. Oper. Res., to appear.
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 (pp. 18–26). Springer-Verlag, Berlin.
J. Hofbauer and K. Sigmund (1988). The Theory of Evolution and Dynamical Systems. Cambridge University Press.
R. A. Hummel and S. W. Zucker (1983). On the foundations of relaxation labeling processes. IEEE Trans. Pattern Anal. Machine Intell. 5(3), 267–287.
A. Jagota (1995). Approximating maximum clique with a Hopfield network. IEEE Trans. Neural Networks 6(3), 724–735.
M. Kimura (1958). On the change of population fitness by natural selection. Heredity 12, 145–167.
S. Y. Lin and Z. Chen (1992). A flexible parallel architecture for relaxation labeling algorithms. IEEE Trans. Signal Process. 40(5), 1231–1240.
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.
J. Maynard-Smith (1982). Evolution and the Theory of Games. Cambridge University Press.
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.
H. Mühlenbein, M. Gorges-Schleuter, and O. Krämer (1988). Evolution algorithms in combinatorial optimization. Parallel Computing 7, 65–85.
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.
P. Pardalos and J. Xue (1994). The maximum clique problem. J. Global Optim. 4, 301–328.
M. Pelillo (1994). On the dynamics of relaxation labeling processes. Proc. IEEE Int. Conf. Neural Networks, Orlando, FL, 1006–1011.
M. Pelillo and A. Jagota (1995). Feasible and infeasible maxima in a quadratic program for maximum clique. J. Artif. Neural Networks 2(4), 411–419.
A. Rosenfeld, R. A. Hummel, and S. W. Zucker (1976). Scene labeling by relaxation operations. IEEE Trans. Syst. Man Cybern. 6(6), 420–433.
P. Taylor and L. Jonker (1978). Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145–156.
J. W. Weibull (1995). Evolutionary Game Theory. MIT Press, Cambridge, MA.
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Pelillo, M., Bomze, I.M. (1996). Parallelizable evolutionary dynamics principles for solving the maximum clique problem. In: Voigt, HM., Ebeling, W., Rechenberg, I., Schwefel, HP. (eds) Parallel Problem Solving from Nature — PPSN IV. PPSN 1996. Lecture Notes in Computer Science, vol 1141. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61723-X_1031
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DOI: https://doi.org/10.1007/3-540-61723-X_1031
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