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Unified Polynomial Dynamic Programming Algorithms for P-Center Variants in a 2D Pareto Front

Author

Listed:
  • Nicolas Dupin

    (Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91400 Orsay, France)

  • Frank Nielsen

    (Sony Computer Science Laboratories Inc., Tokyo 141-0022, Japan)

  • El-Ghazali Talbi

    (CNRS UMR 9189-CRIStAL-Centre de Recherche en Informatique Signal et Automatique de Lille, Université Lille, F-59000 Lille, France)

Abstract

With many efficient solutions for a multi-objective optimization problem, this paper aims to cluster the Pareto Front in a given number of clusters K and to detect isolated points. K -center problems and variants are investigated with a unified formulation considering the discrete and continuous versions, partial K -center problems, and their min-sum- K -radii variants. In dimension three (or upper), this induces NP-hard complexities. In the planar case, common optimality property is proven: non-nested optimal solutions exist. This induces a common dynamic programming algorithm running in polynomial time. Specific improvements hold for some variants, such as K -center problems and min-sum K -radii on a line. When applied to N points and allowing to uncover M < N points, K-center and min-sum- K -radii variants are, respectively, solvable in O ( K ( M + 1 ) N log N ) and O ( K ( M + 1 ) N 2 ) time. Such complexity of results allows an efficient straightforward implementation. Parallel implementations can also be designed for a practical speed-up. Their application inside multi-objective heuristics is discussed to archive partial Pareto fronts, with a special interest in partial clustering variants.

Suggested Citation

  • Nicolas Dupin & Frank Nielsen & El-Ghazali Talbi, 2021. "Unified Polynomial Dynamic Programming Algorithms for P-Center Variants in a 2D Pareto Front," Mathematics, MDPI, vol. 9(4), pages 1-30, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:453-:d:504614
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    References listed on IDEAS

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