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 Õ(2(1/ε)O(d2) n), making a significant improvement over the state-of-the-art algorithms that run in time n(d/ε)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-Median and k-Means and efficient bicriteria approximation schemes for k-Median with outliers, k-Means with outliers and k-Center.