Abstract
In this paper, both scatter search (SS) and genetic algorithms (GAs) are studied for the NP-Hard optimization variant of the satisfiability problem, namely MAX-SAT. First, we investigate a new selection strategy based on both fitness and diversity to choose individuals to participate in the reproduction phase of a genetic algorithm. The resulting algorithm is enhanced in two ways leading to two genetic algorithm variants: the first one uses a uniform crossover. The second one uses a specific crossover operator (to MAX-SAT). The crossover operator is combined with an improved stochastic local search (SLS). The crossover operator is used to identify promising regions while the stochastic local search performs an intensified search of solutions around these regions. Secondly, we propose a scatter search variant for MAX-SAT. Both the SS and the GAs implementations share the solution selection strategy, the improved SLS method and the combination operator. Experiments on several instances from MAX-SAT libraries are performed to show and compare the effectiveness of our approaches. The computational experiments show that both (SS) and (GAs) with a stochastic local search (SLS) improvement technique outperform a classical genetic algorithm (without SLS). The two metaheuristics are able of balancing search diversification and intensification which leads to good results. In general, the specific genetic algorithm with a (SLS) improvement technique and a specific combination method provides competitive results and finds solutions of a higher quality than a scatter search.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Alsinet, T., Manya, F., Planes, J.: Improved branch and bound algorithms for MAX-SAT. In: Proceedings of the 6th International Conference on the Theory and Applications of Satisfiability Testing. SAT2003, pp. 408–415 (2003)
Boughaci, D., Drias, H.: Efficient and experimental meta-heuristics for MAX-SAT problems. In: Lecture Notes in Computer Sciences, WEA 2005, vol. 3503/2005, pp. 501–512 (2005)
Boughaci, D., Drias, H., Benhamou, B.: Solving Max-SAT problems using a mimetic evolutionary meta-heuristic. In: Proceedings of 2004 IEEE CIS 2004, pp. 480–484 (2004)
Borchers, B., Furman, J.: A two-phase exact algorithm for Max-SAT and weighted Max-SAT problems. J. Comb. Optim. 2(4), 299–306 (1999)
Cook, S.A.: The complexity of theorem proving procedures. In: Proceedings of the 3rd ACM Symposium on Theory of Computing, pp. 151–158. Ohio (1971)
Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. Commun. CACM 5, 394–397 (1962)
Frank, J.: A study of genetic algorithms to find approximate solutions to hard 3CNF problems In: Proceedings of Golden West International Conference On Artificial Intelligence (1994)
Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman Company, San Francisco (1979)
Givry, S.D., Larrosa, J., Meseguer, P., Schiex, T.: Solving Max-SAT as weighted CSP. In: Proceedings of 9th International Conference on Principles and Practice of Constraint Programming (CP2003), pp. 363–376 (2003)
Glover, F.: Heuristics for integer programming using surrogate constraints. Desci. Sci. 8(1), 156–166 (January 1977)
Glover, F.: Scatter search and path relinking. In: Corne, D., Dorigo, M., Glover, F. (eds.) New Ideas in Optimization, pp: 297–316. McGraw-Hill, New York (1999)
Glover, F., Laguna, M., Mart, R.: Scatter search: advances in evolutionary computation: theory and applications. In: Ghosh, A., Tsutsui, S. (eds.), pp. 519–537. Springer-Verlag, New York (2003)
Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Wokingham, Addison-Wesley (1989)
Gottlieb, J., Marchiori, E., Rossi, C.: Evolutionary algorithms for the satisfiability problem. Evol. Comput. 10(1), 35–50 (2002)
Hao, J.K., Lardeux, F., Saubion, F.: Evolutionary computing for the satisfiability problem. In: Lecture Notes in Computer Science (EvoCOP03), pp. 258–267. UK, Springer (2003)
Hansen, P., Jaumard, B.: Algorithms for the maximum satisfiability problem. J. Comput. 44(4), 279–303 (1990)
Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975)
Hoos, H.H.: An adaptive noise mechanism for WalkSAT. AAAI/IAAI 655–660 (2002)
Hoos, H.H., Stutzle, T.: Stochastic Local Search. Morgan Kaufman, Cambridge, Massachusetts (2005)
Laguna, M., Glover, F.: Scatter Search. Graduate School of Business. University of Colorado, Boulder (1999)
Laguna, M., Marti, R., Campos, V.: Scatter Search for the Linear Ordering Problem. University of Colorado, Boulder (1999)
Lardeux, F., Saubion, F., Hao, J.K.: GASAT: a genetic local search algorithm for the satisfiability problem. In: Journal of Evolutionary Computation, Summer 2006, vol. 14(N2), pp. 223–253. MIT Press (2006)
Marchiori, E., Rossi, C.: A flipping genetic algorithm for hard 3-SAT problems. In: Proceedings of the Genetic and Evolutionary Computation Conference, vol. 1, pp. 393–400 (1999)
Li, C.M., Manya, F., Planes, J.: Detecting disjoint inconsistent subformulas for computing lower bounds for MAX-SAT. In: Proceedings of Twenty-First National Conference on Artificial Intelligence (AAAI 2006) (July 2006)
Li, C.M., Huang, W.Q.: Diversification and determinism in local search for satisfiability. In: Proceedings of SAT 2005, pp. 158–172 (2005)
Mazure, B., Sais, L., Greroire, E.: A tabu search for Sat. In: Proceedings of the AAAI-97/ IAAI-97, pp. 281–285. Providence, Rhode Island (1997)
McAllester, D., Selman, B., Kautz, H.: Evidence for invariants in local search. In: Proceedings of AAAI-97, pp. 321–326 (1997)
Mills, P., Tsang, E.P.K.: Guided local search for solving SAT and weighted MAX-SAT problems. In: Journal of Automated Reasoning, Special Issue on Satisfiability Problems, vol. 24, pp. 205–223. Kluwer (2000)
Rana, S., Whitley, D.: Genetic algorithm behavior in the maxsat domain. In: Eiben, A.E., Back, T., Schoenauer, M., Schwefel, H.-P. (eds.) Proceedings of PPSN-V, Fifth International Conference on Parallel Problem Solving from Nature, vol. 1498 of Lecture Notes in Computer Science, pp. 785–794. Springer Verlag, Berlin, Germany (1998)
Selman, B., Levesque, H., Mitchell, D.: A new method for solving hard satisfiability problems. In: Proceedings of the 10th National Conference on Artificial Intelligence, pp. 440–446. AAAI Press/The MIT Press, Menlo Park, CA, USA (1992)
Selman, B., Kautz, H., Cohen, B.: Noise strategies for local search. In: Proceedings of AAAI-94, pp. 337–343 (1994)
Shen, H., Zhang, H.: Study of lower bound functions for MAX-2-SAT. In: Proceedings of AAAI-02, pp. 185–190 (2002)
Stutzle, T., Hoos, H.H., Roli, A.: A review of the literature on local search algorithms for MAX-SAT. Internet Document (2003)
Talbi, E.G., Muntean, T., Samarandache, I.: Hybridation des algorithmes genetiques avec la recherche tabou. In: Evolution Artificielle EA94. Toulouse, France (September 1994)
Xing, Z., Zhang, W.: MaxSolver: an efficient exact algorithm for (weighted) maximum satisfiability. Artif. Intell. 2, 47–80 (2005)
Zhang, W., Rangan, A., Looks, M.: Backbone guided local search for maximum satisfiability. In: Proceedings 18th International Joint Conference on AI (IJCAI-03), Aug. 9–15, pp. 1179–1184. Acapulco, Mexico (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boughaci, D., Benhamou, B. & Drias, H. Scatter Search and Genetic Algorithms for MAX-SAT Problems. J Math Model Algor 7, 101–124 (2008). https://doi.org/10.1007/s10852-008-9077-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10852-008-9077-x