2 research outputs found
Clustering with Neighborhoods
In the standard planar -center clustering problem, one is given a set
of points in the plane, and the goal is to select center points, so as
to minimize the maximum distance over points in to their nearest center.
Here we initiate the systematic study of the clustering with neighborhoods
problem, which generalizes the -center problem to allow the covered objects
to be a set of general disjoint convex objects rather than just a
point set . For this problem we first show that there is a PTAS for
approximating the number of centers. Specifically, if is the optimal
radius for centers, then in time we can produce a
set of centers with radius . If instead one
considers the standard goal of approximating the optimal clustering radius,
while keeping as a hard constraint, we show that the radius cannot be
approximated within any factor in polynomial time unless , even
when is a set of line segments. When is a set of
unit disks we show the problem is hard to approximate within a factor of
. This hardness result
complements our main result, where we show that when the objects are disks, of
possibly differing radii, there is a approximation
algorithm. Additionally, for unit disks we give an time -approximation to the optimal
radius, that is, an FPTAS for constant whose running time depends only
linearly on . Finally, we show that the one dimensional version of the
problem, even when intersections are allowed, can be solved exactly in time