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Computing Hypervolume Contributions in Low Dimensions: Asymptotically Optimal Algorithm and Complexity Results

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Evolutionary Multi-Criterion Optimization (EMO 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6576))

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Abstract

Given a finite set Y ⊂ ℝd of n mutually non-dominated vectors in d ≥ 2 dimensions, the hypervolume contribution of a point y ∈ Y is the difference between the hypervolume indicator of Y and the hypervolume indicator of Y ∖ {y}. In multi-objective metaheuristics, hypervolume contributions are computed in several selection and bounded-size archiving procedures.

This paper presents new results on the (time) complexity of computing all hypervolume contributions. It is proved that for d = 2,3 the problem has time complexity Θ(n logn), and, for d > 3, the time complexity is bounded below by Ω(n logn). Moreover, complexity bounds are derived for computing a single hypervolume contribution.

A dimension sweep algorithm with time complexity \(\mathcal{O}\)(n logn) and space complexity \(\mathcal{O}(n)\) is proposed for computing all hypervolume contributions in three dimensions. It improves the complexity of the best known algorithm for d = 3 by a factor of \(\sqrt{n}\). Theoretical results are complemented by performance tests on randomly generated test-problems.

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Author information

Authors and Affiliations

  1. DEEI, Faculty of Science and Technology, University of Algarve, Campus de Gambelas, 8005-139, FARO, Portugal

    Michael T. M. Emmerich & Carlos M. Fonseca

  2. CEG-IST, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais 1, 1049-001, Lisboa, Portugal

    Michael T. M. Emmerich & Carlos M. Fonseca

  3. Department of Informatics Engineering, University of Coimbra, Pólo II, 3030-290, Coimbra, Portugal

    Carlos M. Fonseca

Authors
  1. Michael T. M. Emmerich
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  2. Carlos M. Fonseca
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Federal University of Minas Gerais, Av. Antônio Carlos 6627, 31270-901, Belo Horizonte, MG, Brazil

    Ricardo H. C. Takahashi

  2. Department of Mechanical Engineering, Indian Institute of Technology Kanpur, 208 016, Kanpur, India

    Kalyanmoy Deb

  3. Department of Computer Engineering, Centro Federal de Educaçao Tecnológica de Minas Gerais, Av. Amazonas 5253, 30421-169, Belo Horizonte, MG, Brazil

    Elizabeth F. Wanner

  4. Faculty of Economics, University of Catania, Corso Italia 55, 95129, Catania, Italy

    Salvatore Greco

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© 2011 Springer-Verlag Berlin Heidelberg

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Emmerich, M.T.M., Fonseca, C.M. (2011). Computing Hypervolume Contributions in Low Dimensions: Asymptotically Optimal Algorithm and Complexity Results. In: Takahashi, R.H.C., Deb, K., Wanner, E.F., Greco, S. (eds) Evolutionary Multi-Criterion Optimization. EMO 2011. Lecture Notes in Computer Science, vol 6576. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19893-9_9

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  • DOI: https://doi.org/10.1007/978-3-642-19893-9_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-19892-2

  • Online ISBN: 978-3-642-19893-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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