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

Given a set (A-script) 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-script) (in) (R-openface) n that minimizes the maximum weighted Euclidean distance from c (A-script) to each point in (A-script). In this paper, given (epsilon) > 0, we propose and analyze an algorithm that computes a (1 + (epsilon))-approximate solution to the weighted Euclidean one-center problem. Our algorithm explicitly constructs a small subset (X-script) (subseteq) (A-script), called an (epsilon)- core set of (A-script), for which the optimal solution of the corresponding weighted Euclidean one-center problem is a close approximation to that of (A-script). In addition, we establish that | (X-script)| depends only on (epsilon) 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 + (epsilon))-approximate solution to the weighted Euclidean one-center problem for (A-script) in (O-script)( mn |(X-script)|) arithmetic operations. Our computational results indicate that the size of the (epsilon)-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.