Threshold Rounding for the Standard LP Relaxation of some Geometric Stabbing Problems

In the rectangle stabbing problem, we are given a set $\text{cR}$ of axis-aligned rectangles in ${\text{RR}}^2$, and the objective is to find a minimum-cardinality set of horizontal and/or vertical lines such that each rectangle is intersected by one of these lines. The standard LP relaxation for this problem is known to have an integrality gap of 2, while a better intergality gap of 1.58… is known for the special case when $\text{cR}$ is a set of horizontal segments. In this paper, we consider two more special cases: when $\text{cR}$ is a set of horizontal and vertical segments, and when $\text{cR}$ is a set of unit squares. We show that the integrality gap of the standard LP relaxation in both cases is stricly less than $2$. Our rounding technique is based on a generalization of the {\it threshold rounding} idea used by Kovaleva and Spieksma (SIAM J. Disc. Math 2006), which may prove useful for rounding the LP relaxations of other geometric covering problems.