Stability yields a PTAS for k-Median and k-Means Clustering

We consider k-median clustering in finite metric spaces and k-means clustering in Euclidean spaces, in the setting where k is part of the input (not a constant). For the k-means problem, Ostrovsky et al. [18] show that if the optimal (k - 1)-means clustering of the input is more expensive than the optimal k-means clustering by a factor of 1/epsilon(2), then one can achieve a (1 + f(epsilon))-approximation to the k-means optimal in time polynomial in n and k by using a variant of Lloyd's algorithm. In this work we substantially improve this approximation guarantee. We show that given only the condition that the (k - 1)-means optimal is more expensive than the k-means optimal by a factor 1 + alpha for some constant alpha > 0, we can obtain a PTAS. In particular, under this assumption, for any epsilon > 0 we achieve a (1 + epsilon)-approximation to the k-means optimal in time polynomial in n and k, and exponential in 1/epsilon and 1/alpha. We thus decouple the strength of the assumption from the quality of the approximation ratio. We also give a PTAS for the k-median problem in finite metrics under the analogous assumption as well. For k-means, we in addition give a randomized algorithm with improved running time of n(O(1))(k log n)(poly(1/epsilon,) (1/alpha)). Our technique also obtains a PTAS under the assumption of Balcan et al. [4] that all (1 + alpha) approximations are delta-close to a desired target clustering, in the case that all target clusters have size greater than delta n and alpha > 0 is constant. Note that the motivation of Balcan et al. [4] is that for many clustering problems, the objective function is only a proxy for the true goal of getting close to the target. From this perspective, our improvement is that for k-means in Euclidean spaces we reduce the distance of the clustering found to the target from O(delta) to delta when all target clusters are large, and for k-median we improve the "largeness" condition needed in [4] to get exactly delta-close from O(delta n) to delta n. Our results are based on a new notion of clustering stability.