Constant Approximation for Individual Preference Stable Clustering

TL;DR

This paper proves the existence of constant-approximation individual preference stable clusterings for general metrics and provides efficient algorithms, advancing understanding of stability and fairness in clustering.

Contribution

It establishes that $O(1)$-IP stable clusterings always exist and offers efficient algorithms to find them, addressing previous gaps in stability guarantees.

Findings

Existence of $O(1)$-IP stable clustering for general metrics

Efficient algorithms for IP stability with average, maximum, and minimum distances

Improved theoretical understanding of stability in clustering

Abstract

Individual preference (IP) stability, introduced by Ahmadi et al. (ICML 2022), is a natural clustering objective inspired by stability and fairness constraints. A clustering is $\alpha$-IP stable if the average distance of every data point to its own cluster is at most $\alpha$ times the average distance to any other cluster. Unfortunately, determining if a dataset admits a $1$-IP stable clustering is NP-Hard. Moreover, before this work, it was unknown if an $o(n)$-IP stable clustering always \emph{exists}, as the prior state of the art only guaranteed an $O(n)$-IP stable clustering. We close this gap in understanding and show that an $O(1)$-IP stable clustering always exists for general metrics, and we give an efficient algorithm which outputs such a clustering. We also introduce generalizations of IP stability beyond average distance and give efficient, near-optimal algorithms in the cases where we consider the maximum and minimum distances within and between clusters.