Abstract
Hybridizing evolutionary algorithms with local search has become a popular trend in recent years. There is empirical evidence for various combinatorial problems where hybrid evolutionary algorithms perform better than plain evolutionary algorithms. Due to the rapid development of a highly active field of research, theory lags far behind and a solid theoretical foundation of hybrid metaheuristics is sorely needed.
We are aiming at a theoretical understanding of why and when hybrid evolutionary algorithms are successful in combinatorial optimization. To this end, we consider a hybrid of a simple evolutionary algorithm, the (1+1) EA, with a powerful local search operator known as variable-depth search (VDS) or Kernighan-Lin. Three combinatorial problems are investigated: Mincut, Knapsack, and Maxsat. More precisely, we focus on simply structured problem instances that contain local optima which are very hard to overcome for many common metaheuristics. The plain (1+1) EA, iterated local search, and simulated annealing need exponential time for optimization, with high probability. In sharp contrast, the hybrid algorithm using VDS finds a global optimum in expected polynomial time. These results demonstrate the usefulness of hybrid evolutionary algorithms with VDS from a rigorous theoretical perspective.
Similar content being viewed by others
References
Aickelin, U., Burke, E.K., Li, J.: An estimation of distribution algorithm with intelligent local search for rule-based nurse rostering. J. Oper. Res. Soc. 58, 1574–1585 (2007)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, New York (2001)
Dawkins, R.: The Selfish Gene. Oxford University Press, New York (1976)
Dorigo, M., Stützle, T.: Ant Colony Optimization. MIT Press, New York (2004)
Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276, 51–81 (2002)
Droste, S., Jansen, T., Wegener, I.: Optimization with randomized search heuristics—the (A)NFL theorem, realistic scenarios, and difficult functions. Theor. Comput. Sci. 287(1), 131–144 (2002)
Feller, W.: An Introduction to Probability Theory and Its Applications, 3rd edn., vol. 1. Wiley, New York (1968)
Fischer, S.: A polynomial upper bound for a mutation-based algorithm on the two-dimensional Ising model. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO’04), pp. 1100–1112. Springer, Berlin (2004)
Friedrich, T., Oliveto, P.S., Sudholt, D., Witt, C.: Theoretical analysis of diversity mechanisms for global exploration. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO’08), pp. 945–952. ACM Press, New York (2008)
Giel, O., Wegener, I.: Evolutionary algorithms and the maximum matching problem. In: Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science (STACS’03), pp. 415–426. Springer, Berlin (2003)
Glover, F.W., Laguna, M.: Tabu Search. Kluwer Academic, Amsterdam (1997)
Hart, W.E.: Locally-adaptive and memetic evolutionary pattern search algorithms. Evol. Comput. 11(1), 29–51 (2003)
Hart, W.E., Krasnogor, N., Smith, J.E. (eds.): Recent Advances in Memetic Algorithms. Studies in Fuzziness and Soft Computing, vol. 166. Springer, Berlin (2004)
Ishibuchi, H., Yoshida, T., Murata, T.: Balance between genetic search and local search in memetic algorithms for multiobjective permutation flowshop scheduling. IEEE Trans. Evol. Comput. 7(2), 204–223 (2003)
Jansen, T., Wegener, I.: Real royal road functions—where crossover provably is essential. Discrete Appl. Math. 149, 111–125 (2005)
Jansen, T., Wegener, I.: A comparison of simulated annealing with a simple evolutionary algorithm on pseudo-Boolean functions of unitation. Theor. Comput. Sci. 386(1–2), 73–93 (2007)
Jerrum, M., Sorkin, G.B.: The Metropolis algorithm for graph bisection. Discrete Appl. Math. 82(1–3), 155–175 (1998)
Kernighan, B., Lin, S.: An efficient heuristic procedure for partitioning graphs. Bell Syst. Tech. J. 49(2), 291–307 (1970)
Krasnogor, N., Gustafson, S.: A study on the use of “self-generation” in memetic algorithms. Natural Comput. 3(1), 53–76 (2004)
Krasnogor, N., Smith, J.: A tutorial for competent memetic algorithms: model, taxonomy, and design issues. IEEE Trans. Evol. Comput. 9(5), 474–488 (2005)
Kumar, R., Banerjee, N.: Analysis of a multiobjective evolutionary algorithm on the 0–1 knapsack problem. Theor. Comput. Sci. 358(1), 104–120 (2006)
Levine, J., Ducatelle, F.: Ant colony optimisation and local search for bin packing and cutting stock problems. J. Oper. Res. Soc. 55(7), 705–716 (2004)
Lin, S., Kernighan, B.W.: An effective heuristic algorithm for the traveling salesman problem. Oper. Res. 21, 498–516 (1973)
Lourenço, H.R., Martin, O., Stützle, T.: Iterated local search. In: Handbook of Metaheuristics. International Series in Operations Research & Management Science, vol. 57, pp. 321–353. Kluwer Academic, Norwell (2002)
Mitzenmacher, M., Upfal, E.: Probability and Computing. Cambridge University Press, Cambridge (2005)
Mladenović, N., Hansen, P.: Variable neighborhood search. Comput. OR 24(11), 1097–1100 (1997)
Neri, F., Toivanen, J., Cascella, G.L., Ong, Y.S.: An adaptive multimeme algorithm for designing HIV multidrug therapies. IEEE/ACM Trans. Comput. Biol. Bioinform. 4(2), 264–278 (2007)
Neumann, F., Reichel, J.: Approximating minimum multicuts by evolutionary multi-objective algorithms. In: Parallel Problem Solving from Nature (PPSN X). LNCS, vol. 5199, pp. 72–81. Springer, Berlin (2008)
Neumann, F., Wegener, I.: Randomized local search, evolutionary algorithms, and the minimum spanning tree problem. Theor. Comput. Sci. 378(1), 32–40 (2007)
Neumann, F., Reichel, J., Skutella, M.: Computing minimum cuts by randomized search heuristics. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2008), pp. 779–786. ACM Press, New York (2008)
Neumann, F., Sudholt, D., Witt, C.: Rigorous analyses for the combination of ant colony optimization and local search. In: Proceedings of the Sixth International Conference on Ant Colony Optimization and Swarm Intelligence (ANTS’08). LNCS, vol. 5217, pp. 132–143. Springer, Berlin (2008)
Oliveto, P.S., Witt, C.: Simplified drift analysis for proving lower bounds in evolutionary computation. In: Parallel Problem Solving from Nature (PPSN X). LNCS, vol. 5199, pp. 82–91. Springer, Berlin (2008)
Oliveto, P.S., He, J., Yao, X.: Computational complexity analysis of evolutionary algorithms for combinatorial optimization: a decade of results. Int. J. Autom. Comput. 4(3), 281–293 (2007)
Ong, Y.S., Lim, M.H., Zhu, N., Wong, K.W.: Classification of adaptive memetic algorithms: a comparative study. IEEE Trans. Syst. Man Cybern., Part B 36(1), 141–152 (2006)
Papadimitriou, C.: Computational Complexity. Addison-Wesley, Reading (1994)
Reichel, J., Skutella, M.: Evolutionary algorithms and matroid optimization problems. In: Proceedings of the Genetic and Evolutionary Compution Conference (GECCO’07), pp. 947–954. ACM Press, New York (2007)
Sindhya, K., Deb, K., Miettinen, K.: A local search based evolutionary multi-objective optimization approach for fast and accurate convergence. In: Parallel Problem Solving from Nature (PPSN X). LNCS, vol. 5199, pp. 815–824. Springer, Berlin (2008)
Smith, J.: Coevolving memetic algorithms: A review and progress report. IEEE Trans. Syst. Man Cybern., Part B 37(1), 6–17 (2007)
Stoer, M., Wagner, F.: A simple min cut algorithm. In: Proceedings of the Second Annual European Symposium on Algorithms (ESA’94), pp. 141–147 (1994)
Sudholt, D.: Crossover is provably essential for the Ising model on trees. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO’05), pp. 1161–1167. ACM Press, New York (2005)
Sudholt, D.: Local search in evolutionary algorithms: the impact of the local search frequency. In: Proceedings of the 17th International Symposium on Algorithms and Computation (ISAAC’06). LNCS, vol. 4288, pp. 359–368. Springer, Berlin (2006)
Sudholt, D.: On the analysis of the (1+1) memetic algorithm. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO’06), pp. 493–500. ACM Press, New York (2006)
Sudholt, D.: Memetic algorithms with variable-depth search to overcome local optima. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO’08), pp. 787–794. ACM Press, New York (2008)
Sudholt, D.: The impact of parametrization in memetic evolutionary algorithms. Theor. Comput. Sci. 410(26), 2511–2528 (2009)
Wegener, I.: Simulated annealing beats Metropolis in combinatorial optimization. In: Proceedings of the 32nd International Colloquium on Automata, Languages and Programming (ICALP’05). LNCS, vol. 3580, pp. 589–601. Springer, Berlin (2005)
Witt, C.: Worst-case and average-case approximations by simple randomized search heuristics. In: Proceedings of the 22nd Symposium on Theoretical Aspects of Computer Science (STACS’05). LNCS, vol. 3404, pp. 44–56. Springer, Berlin (2005)
Zhou, Y., He, J.: A runtime analysis of evolutionary algorithms for constrained optimization problems. IEEE Trans. Evol. Comput. 11(5), 608–619 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
A preliminary version of this article has been presented at GECCO’08 [43].
Rights and permissions
About this article
Cite this article
Sudholt, D. Hybridizing Evolutionary Algorithms with Variable-Depth Search to Overcome Local Optima. Algorithmica 59, 343–368 (2011). https://doi.org/10.1007/s00453-009-9384-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-009-9384-2