Center-based Clustering under Perturbation Stability

TL;DR

This paper proves that for center-based clustering objectives like k-median, k-means, and k-center, instances stable to small perturbations can be efficiently clustered to find the optimal solution, extending previous conjectures.

Contribution

It confirms the conjecture that stable instances under small perturbations are efficiently solvable for center-based clustering objectives, providing algorithms and hardness results.

Findings

Efficiently finds optimal clustering for stable instances with factor-3 perturbations in metric spaces without Steiner points.

Uses Single-Linkage plus dynamic programming to achieve optimal clustering.

Shows NP-hardness under weaker stability conditions.

Abstract

Clustering under most popular objective functions is NP-hard, even to approximate well, and so unlikely to be efficiently solvable in the worst case. Recently, Bilu and Linial \cite{Bilu09} suggested an approach aimed at bypassing this computational barrier by using properties of instances one might hope to hold in practice. In particular, they argue that instances in practice should be stable to small perturbations in the metric space and give an efficient algorithm for clustering instances of the Max-Cut problem that are stable to perturbations of size $O(n^{1/2})$. 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). Specifically, we show we can efficiently find the optimal clustering assuming only stability to factor-3 perturbations of the underlying metric in spaces without Steiner points, and stability to factor $2+\sqrt{3}$ perturbations for general metrics. In particular, we show for such instances that the popular Single-Linkage algorithm combined with dynamic programming will find the optimal clustering. We also present NP-hardness results under a weaker but related condition.