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Algorithmica

, Volume 81, Issue 2, pp 703–748 | Cite as

Solving Problems with Unknown Solution Length at Almost No Extra Cost

  • Benjamin Doerr
  • Carola DoerrEmail author
  • Timo Kötzing
Article
  • 23 Downloads
Part of the following topical collections:
  1. Special Issue on Theory of Genetic and Evolutionary Computation

Abstract

Following up on previous work of Cathabard et al. (in: Proceedings of foundations of genetic algorithms (FOGA’11), ACM, 2011) we analyze variants of the (1 + 1) evolutionary algorithm (EA) for problems with unknown solution length. For their setting, in which the solution length is sampled from a geometric distribution, we provide mutation rates that yield for both benchmark functions OneMax and LeadingOnes an expected optimization time that is of the same order as that of the (1 + 1) EA knowing the solution length. More than this, we show that almost the same run times can be achieved even if no a priori information on the solution length is available. We also regard the situation in which neither the number nor the positions of the bits with an influence on the fitness function are known. Solving an open problem from Cathabard et al. we show that, for arbitrary \(s\in {\mathbb {N}}\), such OneMax and LeadingOnes instances can be solved, simultaneously for all \(n\in {\mathbb {N}}\), in expected time \(O(n (\log (n))^2 \log \log (n) \ldots \log ^{(s-1)}(n) (\log ^{(s)}(n))^{1+\varepsilon })\) and \(O(n^2 \log (n) \log \log (n) \ldots \log ^{(s-1)}(n) (\log ^{(s)}(n))^{1+\varepsilon })\), respectively; that is, in almost the same time as if n and the relevant bit positions were known. For the LeadingOnes case, we prove lower bounds of same asymptotic order of magnitude apart from the \((\log ^{(s)}(n))^{\varepsilon }\) factor. Aiming at closing this arbitrarily small remaining gap, we realize that there is no asymptotically best performance for this problem. For any algorithm solving, for all n, all instances of size n in expected time at most T(n), there is an algorithm doing the same in time \(T'(n)\) with \(T'=o(T)\). For OneMax we show results of similar flavor.

Keywords

Black-box optimization Evolutionary computation Runtime analysis Uncertainty Unknown solution length 

Notes

Acknowledgements

Parts of this work have been done while Timo Kötzing was visiting the École Polytechnique. This work was supported in part by the German Research Foundation under Grant FR 2988 (TOSU), by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, and from the Gaspard Monge Program for Optimization and Operations Research (PGMO) of the Jacques Hadamard Mathematical Foundation (FMJH).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.École Polytechnique, CNRS, LIX - UMR 7161PalaiseauFrance
  2. 2.Sorbonne Université, CNRS, LIP6ParisFrance
  3. 3.Hasso-Plattner-InstitutPotsdamGermany

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