  Given a set $P$ of $n$ points and a set $S$ of $m$ disks in the plane, the
disk hitting set problem asks for a smallest subset of $P$ such that every disk
of $S$ contains at least one point in the subset. The problem is NP-hard. In
this paper, we consider a line-constrained version in which all disks have
their centers on a line. We present an $O(m\log^2n+(n+m)\log(n+m))$ time
algorithm for the problem. This improves the previously best result of
$O(m^2\log m+(n+m)\log(n+m))$ time for the weighted case of the problem where
every point of $P$ has a weight and the objective is to minimize the total
weight of the hitting set. Our algorithm actually solves a more general
line-separable problem with a single intersection property: The points of $P$
and the disk centers are separated by a line $\ell$ and the boundary of every
two disks intersect at most once on the side of $\ell$ containing $P$.
