An Algorithm and a Core Set Result for the Weighted Euclidean One-Center Problem

Given a set 𝒜 of m points in n-dimensional space with corresponding positive weights, the weighted Euclidean one-center problem, which is a generalization of the minimum enclosing ball problem, involves the computation of a point c 𝒜 ∈ ℝ n that minimizes the maximum weighted Euclidean distance from c 𝒜 to each point in 𝒜. In this paper, given ϵ > 0, we propose and analyze an algorithm that computes a (1 + ϵ)-approximate solution to the weighted Euclidean one-center problem. Our algorithm explicitly constructs a small subset 𝒳 ⫅ 𝒜, called an ϵ-core set of 𝒜, for which the optimal solution of the corresponding weighted Euclidean one-center problem is a close approximation to that of 𝒜. In addition, we establish that ∣ 𝒳∣ depends only on ϵ and on the ratio of the smallest and largest weights, but is independent of the number of points m and the dimension n. This result subsumes and generalizes the previously known core set results for the minimum enclosing ball problem. Our algorithm computes a (1 + ϵ)-approximate solution to the weighted Euclidean one-center problem for 𝒜 in 𝒪(mn∣𝒳∣) arithmetic operations. Our computational results indicate that the size of the ϵ-core set computed by the algorithm is, in general, significantly smaller than the theoretical worst-case estimate, which contributes to the efficiency of the algorithm, especially for large-scale instances. We shed some light on the possible reasons for this discrepancy between the theoretical estimate and the practical performance.