Near-Linear Time Approximation Schemes for Clustering in Doubling Metrics

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.