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 …

This paper provides a randomized approximation scheme for the k-median problem when
the input points lie in the d-dimensional Euclidean space. The worst-case running time is
O(2^O((\log(1/ϵ)/ε)^d-1)n\log^d+6n), which is nearly linear for any fixed ε and d. Moreover,
our method provides the first polynomial-time approximation scheme for and uncapacitated
facility location instances in d-dimensional Euclidean space for any fixed d>2. Our work
extends techniques introduced originally by Arora for the Euclidean traveling salesman …