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Publications
Center-based Clustering under Perturbation Stability

# Center-based Clustering under Perturbation Stability,Computing Research Repository,Pranjal Awasthi,Avrim Blum,Or Sheffet

Center-based Clustering under Perturbation Stability
Optimal clustering is a notoriously hard task. Recently, Bilu and Linial suggested an approach towards understanding the complexity of clustering instances which arise in practice. They argue that such instances are often stable to perturbations in the metric space and give an efficient algorithm for clustering instances which are stable to perturbations of size $O(n^{1/2})$ for max-cut based clustering. In addition, they conjecture that instances stable to as little as O(1) perturbations should be solvable in polynomial time. In this paper we prove that this conjecture is true for any center-based clustering objective (such as $k$-median, $k$-means, and $k$-center), i.e., we can efficiently find the optimal clustering assuming only stability to {\em constant}-magnitude perturbations of the underlying metric. Specifically, we show that for center-based clustering instances which are stable to 3 perturbations, a combination of the popular single-linkage heuristic together with dynamic programming will find the optimal clustering. In fact, our algorithm works for a weaker notion of stability we call {\em $\alpha$-center stability}, and we show that for this weaker notion, beating the factor of 3 is NP-hard. We also study how the notion of stability to perturbations relates to other definitions of clustering stability that have been proposed recently.