# On the Local Structure of Stable Clustering Instances

## TL;DR

This paper investigates the structure of stable clustering instances under various data assumptions, showing local optima are close to global solutions, which explains the effectiveness of local search algorithms in practice.

## Contribution

It establishes that structured data satisfying stability or separability conditions have local optima near global solutions, providing theoretical guarantees for local search methods.

## Key findings

- Local optima are close to global optima in structured instances.
- Local Search algorithm performs well under stability and separability conditions.
- Structural results apply to k-median and k-means clustering.

## Abstract

We study the classic $k$-median and $k$-means clustering objectives in the beyond-worst-case scenario. We consider three well-studied notions of structured data that aim at characterizing real-world inputs: Distribution Stability (introduced by Awasthi, Blum, and Sheffet, FOCS 2010), Spectral Separability (introduced by Kumar and Kannan, FOCS 2010), Perturbation Resilience (introduced by Bilu and Linial, ICS 2010). We prove structural results showing that inputs satisfying at least one of the conditions are inherently "local". Namely, for any such input, any local optimum is close both in term of structure and in term of objective value to the global optima. As a corollary we obtain that the widely-used Local Search algorithm has strong performance guarantees for both the tasks of recovering the underlying optimal clustering and obtaining a clustering of small cost. This is a significant step toward understanding the success of local search heuristics in clustering applications.