Approximation Schemes for Clustering Problems in Finite Metrics and High Dimensinal Spaces

Let k be a fixed integer. We consider the problem ofpartitioning an input set of points endowed with a distancefunction into k clusters. We give polynomial timeapproximation schemes for the following three clustering problems:Metric k-Clustering, l 22k-Clustering, and l22 k-Median.In the k-Clustering problem, the objective is to minimizethe sum of all intra-cluster distances. In the k-Medianproblem, the goal is to minimize the sum of distances from pointsin a cluster to the (best choice of) cluster center. In metricinstances, the input distance function is a metric. In l22 instances, the points are in Rd and the distance between two points x,yis measured by x−y 22 (noticethat (R d, ṡ 22 is nota metric space). For the first two problems, our results are thefirst polynomial time approximation schemes. For the third problem,the running time of our algorithms is a vast improvement overprevious work.