In this paper, we introduce and study the Non-Uniform k-Center problem
(NUkC). Given a finite metric space (X,d) and a collection of balls of radii
{r1≥⋯≥rk}, the NUkC problem is to find a placement of their
centers on the metric space and find the minimum dilation α, such that
the union of balls of radius α⋅ri around the ith 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 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) 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 Θ(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.Comment: Adjusted the figur