Abstract
We propose a new heuristic for approximating the maximum clique problem based on a detailed analysis of a class of continuous optimization models which provide a complete characterization of solutions to this NP-hard combinatorial problem. We start from a known continuous formulation of the maximum clique, and tackle the search for local solutions with replicator dynamics Hereby, we add to the objective used in previous works a regularization term that controls the global shape of the energy landscape, that is the function actually maximized by the dynamics. The parameter controlling the regularization is changed during the evolution of the dynamical system to render inefficient local solutions (which formerly were stable) unstable, thus conducting the system to escape from sub-optimal points, and so to improve the final results. The role of this parameter is thus superficially similar to that of temperature in simulated annealing in the sense that its variation allows to find better solutions for the problem at hand. We report on the performances of this approach when applied to selected DIMACS benchmark graphs.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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., vol. 73, pp. 360–363.
L. E. Baum and G. R. Sell (1968), “Growth transformations for functions on manifolds,” Pacif. J. Math., vol. 27, no. 2, pp. 211–227.
I. M. Bomze (1997), “Evolution towards the maximum clique,” J. Global Optim., vol. 10, pp. 143–164.
I. M. Bomze, M. Pelillo, and R. Giacomini (1997), “Evolutionary approach to the maximum clique problem: Empirical evidence on a larger scale,” in: I. M. Bomze, T. Csendes, R. Horst, and P. M. Pardalos (Eds.), Developments in Global Optimization. Kluwer, Dordrecht, pp. 95–108.
I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo (1999), “The maximum clique problem,” to appear in: D. Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization (Vol. 4) Kluwer, Dordrecht.
I. M. Bomze, M. Budinich, M. Pelillo and C. Rossi (1999), “Annealed Replication: a New Heuristic for the Maximum Clique Problem,” to appear in: Discrete Applied Mathematics.
I. M. Bomze and F. Rendi (1998), “Replicator dynamics for evolution towards the maximum clique: variations and experiments,” in: R. De Leone, A. Murli, P.M. Pardalos, and G. Toraldo (Eds.), High Performance Algorithms and Software in Nonlinear Optimization, Kluwer, Dordrecht, pp. 53–68.
M. Budinich (1999), “Bounds on the maximum clique of a graph,” submitted (see http://www.is.infn.it/-mbh/MC_Bounds.ps.Z).
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.
A. H. Gee and R. W. Prager (1994), “Polyhedral combinatorics and neural networks,” Neural Computation, vol. 6, pp. 161–180.
L. E. Gibbons, D. W. Hearn, and P. M. Pardalos (1996), “A continuous based heuristic for the maximum clique problem,” In: D. S. Johnson and M. Trick (Eds.), Cliques,Coloring, and Satisfiability—Second DIMACS Implementation Challenge. American Mathematical Society, Providence, RI, pp. 103–124.
L. E. Gibbons, D. W. Hearn, P. M. Pardalos and M. V. Ramana (1997), “Continuous characterizations of the maximum clique problem,” Math. Oper. Res.,vol. 22, no. 3, pp. 754–768.
J. Hofbauer and K. Sigmund. (1998), The Theory of Evolution and Dynamical Systems,Cambridge University Press, Cambridge, UK.
D. S. Johnson and M. A. Trick (Eds.) (1996), Cliques, Coloring,and Satisfiability: Second DIMACS Implementation Challenge, DI- MACS Series in Discrete Mathematics and Theoretical Computer Science, Vol 26, American Mathematical Society, Providence, RI (sec also http://dimacs.rutgers.edu/Volumes/Vo126.html).
S. Karlin (1984), “Mathematical models, problems and controversies of evolutionary theory,” Bull. Amer. Math. Soc., vol. 10, pp. 221–273.
M. Kimura (1958), “On the change of population fitness by natural selection,” Heredity,vol. 12, pp. 145–167.
S. Kirkpatrick, C.D. Gelatt Jr., and M. P. Vecchi (1983), “Optimization by simulated annealing,” Science vol. 220(4598), pp. 671–679.
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., vol. 62, pp. 1035–1074.
D. W. Matula (1976), “The largest clique size in a random graph,” Technical Report CS 7608, Department of Computer Science, Southern Methodist University.
T. S. Motzkin and E. G. Straus (1965), “Maxima for graphs and a new proof of a theorem of Turin,” Canad. J. Math.,vol. 17, pp. 533–540.
C. H. Papadimitriou and K. Steiglitz (1982), Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs, NJ.
P. M. Pardalos (1996), “Continuous approaches to discrete optimization problems,” In: G. Di Pillo and F. Giannessi (Eds.), Nonlinear Or ‘imization and Applications. Plenum Press, New York, pp. 313–328.
P. M. Pardalos and A. T. Phillips (1990), “A global optimization approach for solving the maximum clique problem,” Int. J. Computer Math., vol. 33, pp. 209–216.
P. M. Pardalos and J. Xue (1994), “The maximum clique problem,” J. Global Optim., vol. 4, pp. 301–328.
M. Pelillo (1995), “Relaxation labeling networks for the maximum clique problem,” J. Artif. Neural Networks, vol. 2, pp. 313–328.
M. Pelillo (1997), “The dynamics of nonlinear relaxation labeling processes,” J. Math. Imaging Vision,vol. 7, no. 4, pp. 309–323.
M. Pelillo and A. Jagota (1995), “Feasible and infeasible maxima in a quadratic program for maximum clique,” J. Artif. Neural Networks, vol. 2, pp. 411–419.
P. Taylor and L. Jonker (1978), “Evolutionarily stable strategies and game dynamics,” Math. Biosci.,vol. 40, pp. 145–156.
A. Torsello and M. Pelillo (1999), “Continuous-time relaxation labeling processes,” Pattern Recognition, accepted for publication.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Bomze, I.M., Budinich, M., Pelillo, M., Rossi, C. (2000). A New “Annealed” Heuristic for the Maximum Clique Problem. In: Pardalos, P.M. (eds) Approximation and Complexity in Numerical Optimization. Nonconvex Optimization and Its Applications, vol 42. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3145-3_6
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3145-3_6
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4829-8
Online ISBN: 978-1-4757-3145-3
eBook Packages: Springer Book Archive