A nearly linear-time approximation scheme for the Euclidean k-median problem

In the k-median problem we are given a set S of n points in a metric space and a positive integer k. The objective is to locate k medians among the points so that the sum of the distances from each point in S to its closest median is minimized. The k-median problem is a well-studied, NP-hard, basic clustering problem which is closely related to facility location. We examine the version of the problem in Euclidean space. Obtaining approximations of good quality had long been an elusive goal and only recently Arora, Raghavan and Rao gave a randomized polynomial-time approximation scheme for the Euclidean plane by extending techniques introduced originally by Arora for Euclidean TSP. For any fixed epsilon > 0, their algorithm outputs a (1 + epsilon)-approximation in O(nkn(O(1/epsilon))logn) time. In this paper we provide a randomized approximation scheme for points in d-dimensional Euclidean space, with running time O(2(1/epsilon d)nlognlogk), which is nearly linear for any fixed epsilon and d. Our algorithm provides the first polynomial time approximation scheme for k-median instances in d-dimensional Euclidean space for any fixed d > 2. To obtain the drastic running time improvement we develop a structure theorem to describe hierarchical decomposition of solutions. The theorem is based on a novel adaptive decomposition scheme, which guesses at every level of the hierarchy the structure of the optimal solution and modifies accordingly the parameters of the decomposition. We believe that our methodology is of independent interest and can find applications to further geometric problems.