Center-based clustering under perturbation stability

Center-based clustering under perturbation stability

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by Pranjal Awasthi , Avrim Blum

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@TECHREPORT{Awasthi_center-basedclustering,
    author = {Pranjal Awasthi and Avrim Blum},
    title = {Center-based clustering under perturbation stability},
    institution = {},
    year = {}
}

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Abstract

Optimal clustering under most popular objective functions is NP-hard, and therefore unlikely to be efficiently solvable in the worst case. Recently, Bilu and Linial [11] suggested an approach aimed instead at understanding the complexity of clustering instances which arise in practice. They argue that such instances should be 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 constantmagnitude perturbations of the underlying metric. Specifically, we show that for center-based clustering instances which are stable to O(1) perturbations, the popular Single-Linkage algorithm combined with dynamic programming will find the optimal clustering. Keywords: Clustering, k-median, k-means, Stability Conditions

Citations

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