Approximation and Complexity of the Capacitated Geometric Median Problem

Abstract

In the Capacitated Geometric Median problem, we are given n points in d-dimensional real space and an integer m, the goal is to locate a new point in space (center) and choose m of the input points to minimize the sum of Euclidean distances from the center to the chosen points. We show that this problem admits an “almost exact” polynomial-time algorithm in the case of fixed d and an approximation scheme PTAS in high dimensions. On the other hand, we prove that, if the dimension of space is not fixed, Capacitated Geometric Median is strongly NP-hard and does not admit a scheme FPTAS unless P \(=\) NP.KeywordsFacility locationGeometric medianOutliersEuclidean distancesApproximation schemeNP-hardness