Clustering under Perturbation Resilience

Abstract

Motivated by the fact that distances between data points in many real-world clustering instances are often based on heuristic measures, Bilu and Linial [6] proposed analyzing objective based clustering problems under the assumption that the optimum clustering to the objective is preserved under small multiplicative perturbations to distances between points. In this paper, we provide several results within this framework. For separable center-based objectives, we present an algorithm that can optimally cluster instances resilient to $(1 + \sqrt{2})$-factor perturbations, solving an open problem of Awasthi et al. [2]. For the k-median objective, we additionally give algorithms for a weaker, relaxed, and more realistic assumption in which we allow the optimal solution to change in a small fraction of the points after perturbation. We also provide positive results for min-sum clustering which is a generally much harder objective than k-median (and also non-center-based). Our algorithms are based on new linkage criteria that may be of independent interest.