May 13, 2002

We give polynomial time approximation schemes for the problem of partitioning an input set of n points into a fixed number k of clusters so as to minimize the sum over all clusters of the total pairwise distances in a cluster. Our algorithms work for arbitrary metric spaces as well as for points in R^d where the distance between two points x, y is measured by ||x - y||²₂ (notice that R^d, ||.||²₂) is not a metric space). Our algorithms can be modified to handle other objective functions, such as minimizing the sum over all clusters of the total distance to the best choice for cluster center.