Polynomial Time Algorithm for $2$-Stable Clustering Instances

TL;DR

This paper presents a polynomial time algorithm for 2-stable clustering instances, advancing the understanding of stable instances where clustering remains consistent under perturbations, and addresses an open problem in the field.

Contribution

It introduces a polynomial time algorithm specifically for 2-stable clustering instances, filling a gap left by previous work on higher stability levels.

Findings

Polynomial time algorithm for 2-stable instances

Improves upon previous algorithms for higher stability levels

Answers an open question in clustering stability research

Abstract

Clustering with most objective functions is NP-Hard, even to approximate well in the worst case. Recently, there has been work on exploring different notions of stability which lend structure to the problem. The notion of stability, $\alpha$-perturbation resilience, that we study in this paper was originally introduced by Bilu et al.~\cite{Bilu10}. The works of Awasthi et al~\cite{Awasthi12} and Balcan et al.~\cite{Balcan12} provide a polynomial time algorithm for $3$-stable and $(1+\sqrt{2})$-stable instances respectively. This paper provides a polynomial time algorithm for $2$-stable instances, improving on and answering an open question in ~\cite{Balcan12}.