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.
Linear-time approximation schemes for clustering problems in any dimensions
