A PTAS for k-means clustering based on weak coresets

Given a point set P R^d the k-means clustering problem is to find a set C=(c₁,...,c_k) of k points and a partition of P into k clusters C₁,...,C_k such that the sum of squared errors _i=1^k _{p Ci} |p -c_i |₂ ² is minimized. For given centers this cost function is minimized byassigning points to the nearest center.The k-means cost function is probably the most widely used cost function in the area of clustering.In this paper we show that every unweighted point set P has a weak (, k)-coreset of size Poly(k,1/) for the k-means clustering problem, i.e. its size is independent of the cardinality |P| of the point set and the dimension d of the Euclidean space R^d. A weak coreset is a weighted set S P together with a set T such that T contains a (1+)-approximation for the optimal cluster centers from P and for every set of kcenters from T the cost of the centers for S is a (1±)-approximation of the cost for P.We apply our weak coreset to obtain a PTAS for the k-means clustering problem with running time O(nkd + d Poly(k/) + 2^Õ(k/ε)).