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Approximating the Least Hypervolume Contributor: NP-Hard in General, But Fast in Practice

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Evolutionary Multi-Criterion Optimization (EMO 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5467))

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Abstract

The hypervolume indicator is an increasingly popular set measure to compare the quality of two Pareto sets. The basic ingredient of most hypervolume indicator based optimization algorithms is the calculation of the hypervolume contribution of single solutions regarding a Pareto set. We show that exact calculation of the hypervolume contribution is #P-hard while its approximation is NP-hard. The same holds for the calculation of the minimal contribution. We also prove that it is NP-hard to decide whether a solution has the least hypervolume contribution. Even deciding whether the contribution of a solution is at most (1 + ε) times the minimal contribution is NP-hard. This implies that it is neither possible to efficiently find the least contributing solution (unless P = NP) nor to approximate it (unless NP = BPP).

Nevertheless, in the second part of the paper we present a very fast approximation algorithm for this problem. We prove that for arbitrarily given ε,δ> 0 it calculates a solution with contribution at most (1 + ε) times the minimal contribution with probability at least (1 − δ). Though it cannot run in polynomial time for all instances, it performs extremely fast on various benchmark datasets. The algorithm solves very large problem instances which are intractable for exact algorithms (e.g., 10000 solutions in 100 dimensions) within a few seconds.

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Authors and Affiliations

  1. Universität des Saarlandes, Saarbrücken, Germany

    Karl Bringmann

  2. International Computer Science Institute, Berkeley, CA, USA

    Tobias Friedrich

Authors
  1. Karl Bringmann
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  2. Tobias Friedrich
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Editors and Affiliations

  1. Department of Engineering Science, The University of Auckland, 70 Symonds Street, Room 415, 1001, Auckland, New Zealand

    Matthias Ehrgott

  2. Faculty of Science and Technolgoy, Department of Electronic Engineering and Informatics, Universidade do Algarve, Campus de Gambelas, 8005-139, Faro, Portugal

    Carlos M. Fonseca

  3. Laboratoire d’ Informatique de Nantes -LINA, UMR CNRS 6241, Université de Nantes, 2, Rue de la Houssinière, BP 92208, 44322, Nantes Cedex 03, France

    Xavier Gandibleux

  4. LERIA, Faculty of Sciences, Université d’Angers, 2 Boulevard Lavoisier, 49045, Angers Cedex 01, France

    Jin-Kao Hao

  5. Université de Bretagne-sud - UEB CNRS, UMR 3192 - Lab-STICC, Centre de Recherche,, BP 92116, 56321, Lorient Cedex, France

    Marc Sevaux

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Bringmann, K., Friedrich, T. (2009). Approximating the Least Hypervolume Contributor: NP-Hard in General, But Fast in Practice. In: Ehrgott, M., Fonseca, C.M., Gandibleux, X., Hao, JK., Sevaux, M. (eds) Evolutionary Multi-Criterion Optimization. EMO 2009. Lecture Notes in Computer Science, vol 5467. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01020-0_6

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  • DOI: https://doi.org/10.1007/978-3-642-01020-0_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-01019-4

  • Online ISBN: 978-3-642-01020-0

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