The Non-Uniform k-Center Problem

In this paper, 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 on the metric space and find the minimum dilation alpha, such that the union of balls of radius alpha*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, or as a wireless router placement problem with routers of different powers/ranges. The NUkC problem generalizes the classic k-center problem when 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 when there are k balls of radius 1 and l balls of radius 0. There are 2-approximation and 3-approximation algorithms known for these problems respectively; the former is best possible unless P=NP and the latter remains unimproved for 15 years. We first observe that no O(1)-approximation is to the optimal dilation is possible unless P=NP, implying that the NUkC problem is more non-trivial 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 Theta(1) centers of each radii. Our techniques also allow us to prove a simple (uni-criteria), optimal 2-approximation to the kCwO problem improving upon the long-standing 3-factor. Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees which have been studied recently in the TCS community. We show NUkC is as hard as the firefighter problem. While we don't know if the converse is true, we are able to adapt ideas from recent works [Chalermsook/Chuzhoy, SODA 2010; Asjiashvili/Baggio/Zenklusen, arXiv 2016] in non-trivial ways to obtain our constant factor bi-criteria approximation.