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One-exact approximate Pareto sets

Author

Listed:
  • Arne Herzel

    (University of Kaiserslautern
    Technical University of Munich)

  • Cristina Bazgan

    (Université Paris-Dauphine)

  • Stefan Ruzika

    (University of Kaiserslautern)

  • Clemens Thielen

    (Technical University of Munich)

  • Daniel Vanderpooten

    (Université Paris-Dauphine)

Abstract

Papadimitriou and Yannakakis (Proceedings of the 41st annual IEEE symposium on the Foundations of Computer Science (FOCS), pp 86–92, 2000) show that the polynomial-time solvability of a certain auxiliary problem determines the class of multiobjective optimization problems that admit a polynomial-time computable $$(1+\varepsilon , \dots , 1+\varepsilon )$$ ( 1 + ε , ⋯ , 1 + ε ) -approximate Pareto set (also called an $$\varepsilon $$ ε -Pareto set). Similarly, in this article, we characterize the class of multiobjective optimization problems having a polynomial-time computable approximate $$\varepsilon $$ ε -Pareto set that is exact in one objective by the efficient solvability of an appropriate auxiliary problem. This class includes important problems such as multiobjective shortest path and spanning tree, and the approximation guarantee we provide is, in general, best possible. Furthermore, for biobjective optimization problems from this class, we provide an algorithm that computes a one-exact $$\varepsilon $$ ε -Pareto set of cardinality at most twice the cardinality of a smallest such set and show that this factor of 2 is best possible. For three or more objective functions, however, we prove that no constant-factor approximation on the cardinality of the set can be obtained efficiently.

Suggested Citation

  • Arne Herzel & Cristina Bazgan & Stefan Ruzika & Clemens Thielen & Daniel Vanderpooten, 2021. "One-exact approximate Pareto sets," Journal of Global Optimization, Springer, vol. 80(1), pages 87-115, May.
  • Handle: RePEc:spr:jglopt:v:80:y:2021:i:1:d:10.1007_s10898-020-00951-7
    DOI: 10.1007/s10898-020-00951-7
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    References listed on IDEAS

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    1. Matthias Ehrgott, 2005. "Multicriteria Optimization," Springer Books, Springer, edition 0, number 978-3-540-27659-3, July.
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    Cited by:

    1. Arne Herzel & Stephan Helfrich & Stefan Ruzika & Clemens Thielen, 2023. "Approximating biobjective minimization problems using general ordering cones," Journal of Global Optimization, Springer, vol. 86(2), pages 393-415, June.

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