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

• Published in: Mathematics Vol. 9; no. 4; p. 453
• Main Authors: Dupin, Nicolas, Nielsen, Frank, Talbi, El-Ghazali
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 23.02.2021; MDPI
• Subjects: [INFO.INFO-CC] Computer Science [cs]/Computational Complexity [cs.CC]; [INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]; [INFO.INFO-RO] Computer Science [cs]/Operations Research [math.OC]; [INFO.INFO-RO]Computer Science [cs]/Operations Research [cs.RO]; [INFO]Computer Science [cs]; [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]; Algorithms; bi-objective optimization; Clustering; Complexity; Computational Complexity; Computational Geometry; Computer Science; Discrete optimization; Dynamic programming; Food science; K-center; Linear programming; Mathematics; Multiple objective analysis; Objectives; Operational research; Operations Research; Optimization; Optimization and Control; P-center; Parallel programming; Pareto Front; Pareto optimization; Polynomials; QA1-939; Sum-diameter clustering; sum-radii clustering; Operational research; Clustering; Pareto Front; Complexity; Sum-diameter clustering; Computational geometry; Bi-objective optimization; Parallel programming; Algorithms; K-center; Discrete optimization; Sum-radii clustering; P-center; Dynamic programming
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math9040453

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.