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

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.
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Contributor : Nicolas Dupin Connect in order to contact the contributor
Submitted on : Wednesday, February 24, 2021 - 10:23:52 AM
Last modification on : Tuesday, January 4, 2022 - 6:49:18 AM

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Nicolas Dupin, Frank Nielsen, El-Ghazali Talbi. Unified Polynomial Dynamic Programming Algorithms for P-Center Variants in a 2D Pareto Front. Mathematics , MDPI, 2021, 9 (4), pp.453. ⟨10.3390/math9040453⟩. ⟨hal-03150807⟩



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