  In COCOA 2015, Korman et al. studied the following geometric covering
problem: given a set $S$ of $n$ line segments in the plane, find a minimum
number of line segments such that every cell in the arrangement of the line
segments is covered. Here, a line segment $s$ covers a cell $f$ if $s$ is
incident to $f$. The problem was shown to be NP-hard, even if the line segments
in $S$ are axis-parallel, and it remains NP-hard when the goal is cover the
"rectangular" cells (i.e., cells that are defined by exactly four axis-parallel
line segments).
  In this paper, we consider the approximability of the problem. We first give
a PTAS for the problem when the line segments in $S$ are in any orientation,
but we can only select the covering line segments from one orientation. Then,
we show that when the goal is to cover the rectangular cells using line
segments from both horizontal and vertical line segments, then the problem is
APX-hard. We also consider the parameterized complexity of the problem and
prove that the problem is FPT when parameterized by the size of an optimal
solution. Our FPT algorithm works when the line segments in $S$ have two
orientations and the goal is to cover all cells, complementing that of Korman
et al. in which the goal is to cover the "rectangular" cells.
