# Near-Linear Time Approximation Schemes for Clustering in Doubling Metrics

## TL;DR

This paper introduces nearly linear-time approximation schemes for clustering problems like Facility Location, k-Median, and k-Means in doubling metric spaces, significantly improving computational efficiency over previous methods.

## Contribution

It presents the first nearly linear-time approximation algorithms for these clustering problems in doubling metrics and extends techniques to prize-collecting and outlier variants.

## Key findings

- Achieves nearly linear-time algorithms with complexity depending on dimension and epsilon
- Provides the first efficient schemes for prize-collecting k-Medians and k-Means
- Develops bicriteria approximation schemes for outlier clustering problems

## Abstract

We consider the classic Facility Location, $k$-Median, and $k$-Means problems in metric spaces of doubling dimension $d$. We give nearly linear-time approximation schemes for each problem. The complexity of our algorithms is $2^{(\log(1/\eps)/\eps)^{O(d^2)}} n \log^4 n + 2^{O(d)} n \log^9 n$, making a significant improvement over the state-of-the-art algorithms which run in time $n^{(d/\eps)^{O(d)}}$. Moreover, we show how to extend the techniques used to get the first efficient approximation schemes for the problems of prize-collecting $k$-Medians and $k$-Means, and efficient bicriteria approximation schemes for $k$-Medians with outliers, $k$-Means with outliers and $k$-Center.