  Following the seminal work of Erlebach and van Leeuwen in SODA 2008, we
introduce the minimum ply covering problem. Given a set $P$ of points and a set
$S$ of geometric objects, both in the plane, our goal is to find a subset $S'$
of $S$ that covers all points of $P$ while minimizing the maximum number of
objects covering any point in the plane (not only points of $P$). For objects
that are unit squares and unit disks, this problem is NP-hard and cannot be
approximated by a ratio smaller than 2. We present 2-approximation algorithms
for this problem with respect to unit squares and unit disks. Our algorithms
run in polynomial time when the optimum objective value is bounded by a
constant.
  Motivated by channel-assignment in wireless networks, we consider a variant
of the problem where the selected unit disks must be 3-colorable, i.e., colored
by three colors such that all disks of the same color are pairwise disjoint. We
present a polynomial-time algorithm that achieves a 2-approximate solution,
i.e., a solution that is 6-colorable.
  We also study the weighted version of the problem in dimension one, where $P$
and $S$ are points and weighted intervals on a line, respectively. We present
an algorithm that solves this problem in $O(n + m + M )$-time where $n$ is the
number of points, $m$ is the number of intervals, and $M$ is the number of
pairs of overlapping intervals. This repairs a solution claimed by Nandy,
Pandit, and Roy in CCCG 2017.
