  We are given a set of weighted unit disks and a set of points in Euclidean
plane. The minimum weight unit disk cover (\UDC) problem asks for a subset of
disks of minimum total weight that covers all given points. \UDC\ is one of the
geometric set cover problems, which have been studied extensively for the past
two decades (for many different geometric range spaces, such as (unit) disks,
halfspaces, rectangles, triangles). It is known that the unweighted \UDC\
problem is NP-hard and admits a polynomial-time approximation scheme (PTAS).
For the weighted \UDC\ problem, several constant approximations have been
developed. However, whether the problem admits a PTAS has been an open
question. In this paper, we answer this question affirmatively by presenting
the first PTAS for \UDC. Our result implies the first PTAS for the minimum
weight dominating set problem in unit disk graphs. Combining with existing
ideas, our result can also be used to obtain the first PTAS for the maxmimum
lifetime coverage problem and an improved constant approximation ratio for the
connected dominating set problem in unit disk graphs.
