A constant-factor approximation algorithm for the k-median problem (extended abstract)

1 Introduction 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$-approximation algorithm for this problem. This improves upon the best previously known result of O(log kloglog k), which was obtained by refining and derandomizing a randomized CJ(log n log log n)-approximation algorithm of Bartal. For the metric k-median problem, we are given n points in a metric space. We must select k of these to be cluster centers, and then assign each input point j to the selected center that is closest to it. If location j is assigned to a center i, we incur a cost proportional to the distance between i and j. The goal is to select the k centers so as to minimize the sum of the assignment costs. We give a 6$-approximation algorithm for this problem, that is, a polynomial-time algorithm that finds a feasible solution of objective function value within a factor of 6; of the optimum. We also give constant factor approximation algorithms for several natural extensions of the problem. Lin & Vitter [18] considered the k-median problem with arbitrary assignment costs, and gave a polynomial-time algorithm that finds, for any c > 0, a solution for which the objective function value is within a factor of 1 + e of the optimum, but is infeasible: it opens (1 + l/c)(ln n + l)k cluster centers. Lin & Vitter also provided evidence that this result is best possible via a reduction from the set cover problem. Consequently, it is quite natural to consider special cases. The problem is solvable in polynomial time on trees [14, 221. However, for general metric spaces, the problem is NP-hard to solve exactly. Arora, Raghavan & R.ao …