  We initiate the study of the following natural geometric optimization
problem. The input is a set of axis-aligned rectangles in the plane. The
objective is to find a set of horizontal line segments of minimum total length
so that every rectangle is stabbed by some line segment. A line segment stabs a
rectangle if it intersects its left and its right boundary. The problem, which
we call Stabbing, can be motivated by a resource allocation problem and has
applications in geometric network design. To the best of our knowledge, only
special cases of this problem have been considered so far.
  Stabbing is a weighted geometric set cover problem, which we show to be
NP-hard. A constrained variant of Stabbing turns out to be even APX-hard. While
for general set cover the best possible approximation ratio is $\Theta(\log
n)$, it is an important field in geometric approximation algorithms to obtain
better ratios for geometric set cover problems. Chan et al. [SODA'12]
generalize earlier results by Varadarajan [STOC'10] to obtain sub-logarithmic
performances for a broad class of weighted geometric set cover instances that
are characterized by having low shallow-cell complexity. The shallow-cell
complexity of Stabbing instances, however, can be high so that a direct
application of the framework of Chan et al. gives only logarithmic bounds. We
still achieve a constant-factor approximation by decomposing general instances
into what we call laminar instances that have low enough complexity.
  Our decomposition technique yields constant-factor approximations also for
the variant where rectangles can be stabbed by horizontal and vertical segments
and for two further geometric set cover problems.
