# Approximation Schemes for Capacitated Clustering in Doubling Metrics

## TL;DR

This paper introduces the first quasi-polynomial and polynomial time approximation schemes for capacitated clustering problems in doubling metrics, significantly advancing solutions for redistricting applications.

## Contribution

It presents the first QPTAS for both capacitated k-median and k-means in doubling metrics, and the first PTAS for capacitated k-median in R^2, improving prior bicriteria algorithms.

## Key findings

- First QPTAS for capacitated k-median and k-means in doubling metrics.
- First PTAS for capacitated k-median in R^2.
- Significant improvement over previous bicriteria algorithms.

## Abstract

Motivated by applications in redistricting, we consider the uniform capacitated k-median and uniform capacitated k-means problems in bounded doubling metrics. We provide the first QPTAS for both problems and the first PTAS for the uniform capacitated k-median problem for points in R^2 . This is the first improvement over the bicriteria QPTAS for capacitated k-median in low-dimensional Euclidean space of Arora, Raghavan, Rao [STOC 1998] (1 + {\epsilon}-approximation, 1 + {\epsilon}-capacity violation) and arguably the first polynomial-time approximation algorithm for a non-trivial metric.