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Article

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

1
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 / Revised: 12 February 2021 / Accepted: 16 February 2021 / 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 M<N points, K-center and min-sum-K-radii variants are, respectively, solvable in O(K(M+1)NlogN) and O(K(M+1)N2) 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. View Full-Text
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 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 Style

Dupin, 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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