# A New “Annealed” Heuristic for the Maximum Clique Problem

## 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.

## Keywords

Maximum Clique Heuristic.## Preview

Unable to display preview. Download preview PDF.

## References

- [1]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.MathSciNetMATHCrossRefGoogle Scholar - [2]L. E. Baum and G. R. Sell (1968), “Growth transformations for functions on manifolds,”
*Pacif. J. Math*., vol. 27, no. 2, pp. 211–227.MathSciNetMATHCrossRefGoogle Scholar - [3]I. M. Bomze (1997), “Evolution towards the maximum clique,”
*J. Global Optim*., vol. 10, pp. 143–164.MathSciNetMATHCrossRefGoogle Scholar - [4]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.CrossRefGoogle Scholar - [5]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.Google Scholar - [6]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*. Google Scholar - [7]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.CrossRefGoogle Scholar - [8]M. Budinich (1999), “Bounds on the maximum clique of a graph,” submitted (see http://www.is.infn.it/-mbh/MC_Bounds.ps.Z).
- [9]J. F. Crow and M. Kimura. (1970),
*An Introduction to Population Genetics Theory*. Harper & Row, New York.Google Scholar - [10]R. A. Fisher. (1930),
*The Genetical Theory of Natural Selection*. Clarendon Press, Oxford.MATHGoogle Scholar - [11]A. H. Gee and R. W. Prager (1994), “Polyhedral combinatorics and neural networks,”
*Neural Computation*, vol. 6, pp. 161–180.CrossRefGoogle Scholar - [12]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.Google Scholar - [13]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.MathSciNetMATHCrossRefGoogle Scholar - [14]J. Hofbauer and K. Sigmund. (1998),
*The Theory of Evolution and Dynamical Systems*,Cambridge University Press, Cambridge, UK.Google Scholar - [15]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). - [16]S. Karlin (1984), “Mathematical models, problems and controversies of evolutionary theory,”
*Bull. Amer. Math. Soc*., vol. 10, pp. 221–273.MathSciNetMATHCrossRefGoogle Scholar - [17]M. Kimura (1958), “On the change of population fitness by natural selection,”
*Heredity*,vol. 12, pp. 145–167.CrossRefGoogle Scholar - [18]S. Kirkpatrick, C.D. Gelatt Jr., and M. P. Vecchi (1983), “Optimization by simulated annealing,”
*Science*vol. 220(4598), pp. 671–679.MathSciNetMATHCrossRefGoogle Scholar - [19]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.MathSciNetMATHGoogle Scholar - [20]D. W. Matula (1976), “The largest clique size in a random graph,”
*Technical Report CS 7608*, Department of Computer Science, Southern Methodist University.Google Scholar - [21]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.MathSciNetMATHCrossRefGoogle Scholar - [22]C. H. Papadimitriou and K. Steiglitz (1982),
*Combinatorial Optimization: Algorithms and Complexity*. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar - [23]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.Google Scholar - [24]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.MATHCrossRefGoogle Scholar - [25]P. M. Pardalos and J. Xue (1994), “The maximum clique problem,”
*J. Global Optim*., vol. 4, pp. 301–328.MathSciNetMATHCrossRefGoogle Scholar - [26]M. Pelillo (1995), “Relaxation labeling networks for the maximum clique problem,”
*J. Artif. Neural Networks*, vol. 2, pp. 313–328.Google Scholar - [27]M. Pelillo (1997), “The dynamics of nonlinear relaxation labeling processes,”
*J*.*Math. Imaging Vision*,*vol*. 7, no. 4, pp. 309–323.MathSciNetCrossRefGoogle Scholar - [28]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.Google Scholar - [29]P. Taylor and L. Jonker (1978), “Evolutionarily stable strategies and game dynamics,”
*Math. Biosci*.,vol. 40, pp. 145–156.MathSciNetMATHCrossRefGoogle Scholar - [30]A. Torsello and M. Pelillo (1999), “Continuous-time relaxation labeling processes,”
*Pattern Recognition*, accepted for publication.Google Scholar