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

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/epsilon)/epsilon)(d-1)) n log(d+6) n), which is nearly linear for any fixed epsilon and d. Moreover, our method provides the first polynomial- time approximation scheme for k- median 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 problem ( TSP). To obtain the improvement we develop a structure theorem to describe hierarchical decomposition of solutions. The theorem is based on an adaptive decomposition scheme, which guesses at every level of the hierarchy the structure of the optimal solution and accordingly modifies the parameters of the decomposition. We believe that our methodology is of independent interest and may find applications to further geometric problems.