A Constant-Factor Approximation Algorithm for the k-Median Problem

摘要

We present the first constant-factor approximation algorithm for the metric k-median problem. The k-median problem is one of the most well-studied clustering problems, i.e., those problems in which the aim is to partition a given set of points into clusters so that the points within a cluster are relatively close with respect to some measure. For the metric k-median problem, we are given n points in a metric space. We select k of these to be cluster centers and then assign each point to its closest selected center. If point j is assigned to a center i, the cost incurred is proportional to the distance between i and j. The goal is to select the k centers that minimize the sum of the assignment costs. We give a 623-approximation algorithm for this problem. This improves upon the best previously known result of O(log k  log log  k), which was obtained by refining and derandomizing a randomized O(log  n  log log  n)-approximation algorithm of Bartal.