A constant-factor approximation algorithm for the k-median problem

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 6 2 3 -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 logn)approximation algorithm of Bartal. We also give constant factor approximation algorithms for several natural extensions of the problem. [email protected]. Stanford University, Stanford, CA 94305. Research supported by the Pierre and Christine Lamond Fellowship, NSF Grant IIS-9811904 and NSF Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. [email protected]. Stanford University, Stanford, CA 94305. Research Supported by IBM Cooperative Fellowship, NSF Grant IIS-9811904 and NSF Award CCR-9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. [email protected]. Cornell University, Ithaca, NY 14853. Research partially supported by NSF grants CCR-9700163 ONR grants N00014-98-1-0589 and N00014-96-1-0050. [email protected]. Cornell University, Ithaca, NY 14853. Research partially supported by NSF grants CCR-9700029 and DMS-9805602 and ONR grant N00014-96-1-00500.