The Non-Uniform k -Center Problem

Abstract

In this article, we introduce and study the Non-Uniform k -Center (NUkC) problem. Given a finite metric space ( X , d ) and a collection of balls of radii { r 1 ≥ … ≥ r k }, the NUkC problem is to find a placement of their centers in the metric space and find the minimum dilation α, such that the union of balls of radius α ⋅ r i around the i th center covers all the points in X . This problem naturally arises as a min-max vehicle routing problem with fleets of different speeds. The NUkC problem generalizes the classic k -center problem, wherein all the k radii are the same (which can be assumed to be 1 after scaling). It also generalizes the k -center with outliers (kCwO for short) problem, in which there are k balls of radius 1 and ℓ (number of outliers) balls of radius 0. Before this work, there was a 2-approximation and 3-approximation algorithm known for these problems, respectively; the former is best possible unless P=NP. We first observe that no O (1)-approximation to the optimal dilation is possible unless P=NP, implying that the NUkC problem is harder than the above two problems. Our main algorithmic result is an ( O (1), O (1))- bi-criteria approximation result: We give an O (1)-approximation to the optimal dilation; however, we may open Θ(1) centers of each radii. Our techniques also allow us to prove a simple (uni-criterion), optimal 2-approximation to the kCwO problem improving upon the long-standing 3-factor approximation for this problem. Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees that have been studied recently in the TCS community. We show NUkC is at least as hard as the firefighter problem. While we do nt know whether the converse is true, we are able to adapt ideas from recent works [1, 3] in non-trivial ways to obtain our constant factor bi-criteria approximation.