Metric Perturbation Resilience

TL;DR

This paper investigates the concept of perturbation resilience in clustering, providing an exact algorithm for 2-perturbation resilient instances and establishing tight complexity bounds, thus advancing understanding of clustering stability under perturbations.

Contribution

It introduces an exact algorithm for 2-perturbation resilient clustering problems with center-based objectives and proves the tightness of this result under complexity assumptions.

Findings

Exact algorithm for 2-perturbation resilient clustering

Improved bounds over previous algorithms for perturbation resilience

Proved tightness of the algorithm's applicability under complexity assumptions

Abstract

We study the notion of perturbation resilience introduced by Bilu and Linial (2010) and Awasthi, Blum, and Sheffet (2012). A clustering problem is $\alpha$-perturbation resilient if the optimal clustering does not change when we perturb all distances by a factor of at most $\alpha$. We consider a class of clustering problems with center-based objectives, which includes such problems as k-means, k-median, and k-center, and give an exact algorithm for clustering 2-perturbation resilient instances. Our result improves upon the result of Balcan and Liang (2016), who gave an algorithm for clustering $1+\sqrt{2}\approx 2.41$ perturbation resilient instances. Our result is tight in the sense that no polynomial-time algorithm can solve $(2-\varepsilon)$-perturbation resilient instances unless NP = RP, as was shown by Balcan, Haghtalab, and White (2016). We show that the algorithm works on instances satisfying a slightly weaker and more natural condition than perturbation resilience, which we call metric perturbation resilience.