Abstract
Randomized search heuristics such as evolutionary algorithms, simulated annealing, and ant colony optimization are a broadly used class of general-purpose algorithms. Analyzing them via classical methods of theoretical computer science is a growing field. While several strong runtime analysis results have appeared in the last 20 years, a powerful complexity theory for such algorithms is yet to be developed. We enrich the existing notions of black-box complexity by the additional restriction that not the actual objective values, but only the relative quality of the previously evaluated solutions may be taken into account by the black-box algorithm. Many randomized search heuristics belong to this class of algorithms.
We show that the new ranking-based model can give more realistic complexity estimates. The class of all binary-value functions has a black-box complexity of O(logn) in the previous black-box models, but has a ranking-based complexity of Θ(n).
On the other hand, for the class of all OneMax functions, we present a ranking-based black-box algorithm that has a runtime of Θ(n/logn), which shows that the OneMax problem does not become harder with the additional ranking-basedness restriction.
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Notes
That is, {g(Om z (x 1)),…,g(Om z (x s ))}∩([g(Om z (x i )),g(Om z (y))]∪[g(Om z (y)),g(Om z (x i ))])={g(Om z (x i )),g(Om z (y))}.
References
Aaronson, S.: Lower bounds for local search by quantum arguments. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC’04), pp. 465–474. ACM Press, New York (2004)
Aldous, D.: Minimization algorithms and random walk on the d-cube. Ann. Probab. 11, 403–413 (1983)
Anil, G., Wiegand, R.P.: Black-box search by elimination of fitness functions. In: Proceedings of Foundations of Genetic Algorithms (FOGA’09), pp. 67–78. ACM Press, New York (2009)
Chvátal, V.: Mastermind. Combinatorica 3, 325–329 (1983)
Doerr, B., Johannsen, D., Kötzing, T., Lehre, P.K., Wagner, M., Winzen, C.: Faster black-box algorithms through higher arity operators. In: Proceedings of Foundations of Genetic Algorithms (FOGA’11), pp. 163–172. ACM Press, New York (2011)
Doerr, B., Kötzing, T., Winzen, C.: Too fast unbiased black-box algorithms. In: Proceedings of the 13th Annual Genetic and Evolutionary Computation Conference (GECCO’11), pp. 2043–2050. ACM Press, New York (2011)
Doerr, B., Winzen, C.: Towards a complexity theory of randomized search heuristics: ranking-based black-box complexity. In: Proceedings of Computer Science Symposium in Russia (CSR’11), pp. 15–28. Springer, Berlin (2011)
Droste, S., Jansen, T., Tinnefeld, K., Wegener, I.: A new framework for the valuation of algorithms for black-box optimization. In: Proceedings of Foundations of Genetic Algorithms (FOGA’03), pp. 253–270. Morgan Kaufmann, San Mateo (2003)
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.: Upper and lower bounds for randomized search heuristics in black-box optimization. Theory Comput. Syst. 39, 525–544 (2006)
Dubhashi, D., Panconesi, A.: Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, Cambridge (2009)
Erdős, P., Rényi, A.: On two problems of information theory. Magy. Tud. Akad. Mat. Kut. Intéz. Közl. 8, 229–243 (1963)
Fournier, H., Teytaud, O.: Lower bounds for comparison based evolution strategies using vc-dimension and sign patterns. Algorithmica 59, 387–408 (2011)
Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1990)
Goodrich, M.T.: On the algorithmic complexity of the mastermind game with black-peg results. Inf. Process. Lett. 109, 675–678 (2009)
Hromkovič, J.: Algorithmics for Hard Problems: Introduction to Combinatorial Optimization, Randomization, Approximation, and Heuristics. Springer, Berlin (2003)
Lehre, P.K., Witt, C.: Black-box search by unbiased variation. In: Proceedings of Genetic and Evolutionary Computation Conference (GECCO’10), pp. 1441–1448. ACM Press, New York (2010). A journal version of this paper is to appear in Algorithmica. doi:10.1007/s00453-012-9616-8
Llewellyn, D.C., Tovey, C., Trick, M.: Local optimization on graphs. Discrete Appl. Math. 23, 157–178 (1989). Erratum 46, 93–94, 1993
Michalewicz, Z., Fogel, D.B.: How to Solve It—Modern Heuristics, 2nd edn. Springer, Berlin (2004)
Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)
Neumann, F., Wegener, I.: Randomized local search, evolutionary algorithms, and the minimum spanning tree problem. Theor. Comput. Sci. 378, 32–40 (2007)
Robbins, H.: A remark on Stirling’s formula. Am. Math. Mon. 62, 26–29 (1955)
Teytaud, O., Gelly, S.: General lower bounds for evolutionary algorithms. In: Proceedings of the 9th International Conference on Parallel Problem Solving from Nature—PPSN IX (PPSN’06), pp. 21–31. Springer, Berlin (2006)
Yao, A.C.-C.: Probabilistic computations: toward a unified measure of complexity. In: Proceedings of 18th Annual Symposium on Foundations of Computer Science (FOCS’77), pp. 222–227 (1977)
Zhang, S.: New upper and lower bounds for randomized and quantum local search. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC’06), pp. 634–643. ACM Press, New York (2006)
Acknowledgements
Carola Winzen is a recipient of the Google Europe Fellowship in Randomized Algorithms. This work is supported in part by this Google Fellowship.
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A preliminary version of the results appeared in [7].
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Doerr, B., Winzen, C. Ranking-Based Black-Box Complexity. Algorithmica 68, 571–609 (2014). https://doi.org/10.1007/s00453-012-9684-9
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DOI: https://doi.org/10.1007/s00453-012-9684-9