Unified Polynomial Dynamic Programming Algorithms for P-Center Variants in a 2D Pareto Front †
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Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91400 Orsay, France
2
Sony Computer Science Laboratories Inc., Tokyo 141-0022, Japan
3
CNRS UMR 9189-CRIStAL-Centre de Recherche en Informatique Signal et Automatique de Lille, Université Lille, F-59000 Lille, France
*
Author to whom correspondence should be addressed.
†
This paper is an extended version of our paper published in OLA 2020, International Conference in Optimization and Learning, Cadiz, Spain, 17–19 February 2020.
Academic Editor: Frank Werner
Mathematics 2021, 9(4), 453; https://doi.org/10.3390/math9040453
Received: 21 December 2020
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Revised: 12 February 2021
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Accepted: 16 February 2021
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Published: 23 February 2021
(This article belongs to the Special Issue Mathematical Methods for Operations Research Problems)
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
Keywords:
discrete optimization; operational research; computational geometry; complexity; algorithms; dynamic programming; clustering; k-center; p-center; sum-radii clustering; sum-diameter clustering; bi-objective optimization; Pareto Front; parallel programming
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MDPI and ACS Style
Dupin, N.; Nielsen, F.; Talbi, E.-G. Unified Polynomial Dynamic Programming Algorithms for P-Center Variants in a 2D Pareto Front. Mathematics 2021, 9, 453. https://doi.org/10.3390/math9040453
AMA Style
Dupin N, Nielsen F, Talbi E-G. Unified Polynomial Dynamic Programming Algorithms for P-Center Variants in a 2D Pareto Front. Mathematics. 2021; 9(4):453. https://doi.org/10.3390/math9040453
Chicago/Turabian StyleDupin, Nicolas, Frank Nielsen, and El-Ghazali Talbi. 2021. "Unified Polynomial Dynamic Programming Algorithms for P-Center Variants in a 2D Pareto Front" Mathematics 9, no. 4: 453. https://doi.org/10.3390/math9040453
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