  Recently, there has been substantial interest in clustering research that
takes a beyond worst-case approach to the analysis of algorithms. The typical
idea is to design a clustering algorithm that outputs a near-optimal solution,
provided the data satisfy a natural stability notion. For example, Bilu and
Linial (2010) and Awasthi et al. (2012) presented algorithms that output
near-optimal solutions, assuming the optimal solution is preserved under small
perturbations to the input distances. A drawback to this approach is that the
algorithms are often explicitly built according to the stability assumption and
give no guarantees in the worst case; indeed, several recent algorithms output
arbitrarily bad solutions even when just a small section of the data does not
satisfy the given stability notion.
  In this work, we address this concern in two ways. First, we provide
algorithms that inherit the worst-case guarantees of clustering approximation
algorithms, while simultaneously guaranteeing near-optimal solutions when the
data is stable. Our algorithms are natural modifications to existing
state-of-the-art approximation algorithms. Second, we initiate the study of
local stability, which is a property of a single optimal cluster rather than an
entire optimal solution. We show our algorithms output all optimal clusters
which satisfy stability locally. Specifically, we achieve strong positive
results in our local framework under recent stability notions including metric
perturbation resilience (Angelidakis et al. 2017) and robust perturbation
resilience (Balcan and Liang 2012) for the $k$-median, $k$-means, and
symmetric/asymmetric $k$-center objectives.
