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

  • Michael T. M. Emmerich
  • Carlos M. Fonseca
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6576)

Abstract

Given a finite set Y ⊂ ℝ d 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.

Keywords

multiobjective selection complexity hypervolume indicator 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Michael T. M. Emmerich
    • 1
    • 2
  • Carlos M. Fonseca
    • 1
    • 2
    • 3
  1. 1.DEEI, Faculty of Science and TechnologyUniversity of AlgarveFAROPortugal
  2. 2.CEG-IST, Instituto Superior TécnicoTechnical University of LisbonLisboaPortugal
  3. 3.Department of Informatics EngineeringUniversity of CoimbraCoimbraPortugal

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