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 1/2 , then one can achieve a (1 + f())-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+ for some constant > 0, we can obtain a PTAS. In particular, under this assumption, for any > 0 we achieve a (1 + )-approximation to the k-means optimal in time polynomial in n and k, and exponential in 1/ and 1/. We thus decouple the strength of the assumption from the quality of the approximation ratio. We...

• Type: Journal
• Year: 2010
• Where: FOCS
• Authors: Pranjal Awasthi, Avrim Blum, Or Sheffet
• Keywords: Balcan Et Al, Et Al, FOCS 2010, K-means, Theoretical Computer Science