Enclosing weighted points with an almost-unit ball

Given a set of n points with positive real weights in d-dimensional space, we consider an approximation to the problem of placing a unit ball, in order to maximize the sum of the weights of the points enclosed by the ball. Given an approximation parameter ε < 1, we present an O(n/εd−1) expected time algorithm that determines a ball of radius 1+ ε enclosing a weight at least as large as the weight of the optimal unit ball. This is the first approximate algorithm for the weighted version of the problem in d-dimensional space. We also present a matching lower bound for a certain class of algorithms for the problem.