  Given a set $P$ of $n$ weighted points and a set $S$ of $m$ disks in the
plane, the hitting set problem is to compute a subset $P'$ of points of $P$
such that each disk contains at least one point of $P'$ and the total weight of
all points of $P'$ is minimized. The problem is known to be NP-hard. In this
paper, we consider a line-constrained version of the problem in which all disks
are centered on a line $\ell$. We present an $O((m+n)\log(m+n)+\kappa \log m)$
time algorithm for the problem, where $\kappa$ is the number of pairs of disks
that intersect. For the unit-disk case where all disks have the same radius,
the running time can be reduced to $O((n + m)\log(m + n))$. In addition, we
solve the problem in $O((m + n)\log(m + n))$ time in the $L_{\infty}$ and $L_1$
metrics, in which a disk is a square and a diamond, respectively. Our
techniques can also be used to solve other geometric hitting set problems. For
example, given in the plane a set $P$ of $n$ weighted points and a set $S$ of
$n$ half-planes, we solve in $O(n^4\log n)$ time the problem of finding a
minimum weight hitting set of $P$ for $S$. This improves the previous best
algorithm of $O(n^6)$ time by nearly a quadratic factor.
