  Given a set $P$ of $n$ points and a set $S$ of $n$ weighted disks in the
plane, the disk coverage problem is to compute a subset of disks of smallest
total weight such that the union of the disks in the subset covers all points
of $P$. The problem is NP-hard. In this paper, we consider a line-separable
unit-disk version of the problem where all disks have the same radius and their
centers are separated from the points of $P$ by a line $\ell$. We present an
$O(n^{3/2}\log^2 n)$ time algorithm for the problem. This improves the
previously best work of $O(n^2\log n)$ time. Our result leads to an algorithm
of $O(n^{{7}/{2}}\log^2 n)$ time for the halfplane coverage problem (i.e.,
using $n$ weighted halfplanes to cover $n$ points), an improvement over the
previous $O(n^4\log n)$ time solution. If all halfplanes are lower ones, our
algorithm runs in $O(n^{{3}/{2}}\log^2 n)$ time, while the previous best
algorithm takes $O(n^2\log n)$ time. Using duality, the hitting set problems
under the same settings can be solved with similar time complexities.
