The Euclidean k-Supplier Problem

Abstract

In the k-supplier problem, we are given a set of clients C and set of facilities F located in a metric (C ∪ F, d), along with a bound k. The goal is to open a subset of k facilities so as to minimize the maximum distance of a client to an open facility, i.e., min _{S ⊆ F: |S| = k} max _v ∈ C d(v,S), where d(v,S) =  min _u ∈ S d(v,u) is the minimum distance of client v to any facility in S. We present a $1+\sqrt{3}<2.74$ approximation algorithm for the k-supplier problem in Euclidean metrics. This improves the previously known 3-approximation algorithm [9] which also holds for general metrics (where it is known to be tight). It is NP-hard to approximate Euclidean k-supplier to better than a factor of $\sqrt{7}\approx 2.65$, even in dimension two [5]. Our algorithm is based on a relation to the edge cover problem. We also present a nearly linear O(n·log² n) time algorithm for Euclidean k-supplier in constant dimensions that achieves an approximation ratio of 2.965, where n = |C ∪ F|.