The Euclidean k-supplier problem

The k-supplier problem is a fundamental location problem that involves opening k facilities to minimize the maximum distance of any client to an open facility. We consider the k-supplier problem in Euclidean metrics (of arbitrary dimension) and present an algorithm with approximation ratio 1 + √3 < 2.74. This improves upon the previously known 3-approximation algorithm, which also holds for general metrics. Our result is almost best possible as the Euclidean k-supplier problem is NP-hard to approximate better than a factor of ^√7 > 2.64. We also present a nearly linear time algorithm for the Euclidean k-supplier in constant dimensions that achieves an approximation ratio better than three.