Coresets for constrained k-median and k-means clustering in low dimensional Euclidean space

TL;DR

This paper develops streaming algorithms for constrained Euclidean k-median and k-means clustering in low-dimensional space, leveraging coreset techniques to handle various constraints efficiently.

Contribution

It introduces a unified approach using coresets for constrained clustering in streaming models, extending previous methods to a broader class of constraints in low-dimensional Euclidean space.

Findings

Algorithms work for low-dimensional Euclidean space.

Applicable to a wide range of constraints.

Efficient in streaming settings.

Abstract

We study (Euclidean) $k$-median and $k$-means with constraints in the streaming model. There have been recent efforts to design unified algorithms to solve constrained $k$-means problems without using knowledge of the specific constraint at hand aside from mild assumptions like the polynomial computability of feasibility under the constraint (compute if a clustering satisfies the constraint) or the presence of an efficient assignment oracle (given a set of centers, produce an optimal assignment of points to the centers which satisfies the constraint). These algorithms have a running time exponential in $k$, but can be applied to a wide range of constraints. We demonstrate that a technique proposed in 2019 for solving a specific constrained streaming $k$-means problem, namely fair $k$-means clustering, actually implies streaming algorithms for all these constraints. These work for low dimensional Euclidean space. [Note that there are more algorithms for streaming fair $k$-means today, in particular they exist for high dimensional spaces now as well.]