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)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.

Allocation of Starting Points in Global Optimization Problems

“…Nevertheless, uniform sampling continues to draw attention, and investigations on this topic are of serious interest [8]. Approaches based on the p-location problem [9] and p-center methodology [10] can also be used for solving the problems considered in our paper. However, we aimed to check the efficiency of a global optimization approach.…”

Kit of Uniformly Deployed Sets for p-Location Problems

“…Definitions of weighted versions of these problems can be found in [2]. Related problems concerning dispersion of p facilities are referred to in [3][4][5]. Work [6] is a survey on location models in the public sector, which comprises the deployment of fire brigades or police stations and the design of the medical emergency service system.…”

“…The original p-median and weighted p-median problems are easy to model by means of linear integer programming with classical allocation constraints, and solve via a standard integer programming method [4,15]. Nevertheless, this approach proved to be ineffective Mathematics 2023, 11, 2418 2 of 14 when bigger instances of the problems have to be solved.…”

Special Issue “Mathematical Methods for Operations Research Problems”

“…In the third paper [3], the authors cluster the Pareto Front for a multi-objective optimization problem in a given number of clusters and identify isolated points. In particular, K-center problems and some variants are investigated and a unified formulation is given, where both discrete and continuous variants, partial K-center problems and their min-sum K-radii on a line are considered.…”