  The hitting set problem is a well-known NP-hard optimization problem in
which, given a set of elements and a collection of subsets, the goal is to find
the smallest selection of elements, such that each subset contains at least one
element in the selection. Many geometric set systems enjoy improved
approximation ratios, which have recently been shown to be tight with respect
to the shallow cell complexity of the set system. The algorithms that exploit
the cell complexity, however, tend to be involved and computationally
intensive. This paper shows that a slightly improved asymptotic approximation
ratio for the hitting set problem can be attained using a much simpler
algorithm: solve the linear programming relaxation, take one initial random
sample from the set of elements with probabilities proportional to the
LP-solution, and, while there is an unhit set, take an additional sample from
it proportional to the LP-solution. Our algorithm is a simple generalization of
the elegant net-finder algorithm by Nabil Mustafa. To analyze this algorithm
for the hitting set problem, we generalize the classic Packing Lemma, and the
more recent Shallow Packing Lemma, to the setting of weighted epsilon-nets.
