Approximation schemes for clustering problems

Let k be a fixed integer. We consider the problem of partitioning an input set of points endowed with a distance function into k clusters. We give polynomial time approximation schemes for the following three clustering problems: Metric k-Clustering, ℓ₂² k-Clustering, and ℓ₂² k-Median. In the k-Clustering problem, the objective is to minimize the sum of all intra-cluster distances. In the k-Median problem, the goal is to minimize the sum of distances from points in a cluster to the (best choice of) cluster center. In metric instances, the input distance function is a metric. In ℓ₂² instances, the points are in ℝ^d and the distance between two points x, y is measured by ||x - y||₂² (notice that (ℝ^d, || · ||₂²) is not a metric space). For the first two problems, our results are the first polynomial time approximation schemes. For the third problem, the running time of our algorithms is a vast improvement over previous work.