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

TL;DR

This paper improves the approximation bounds for the rectangle stabbing problem by applying a generalized threshold rounding technique to the standard LP relaxation in special geometric cases.

Contribution

It introduces a novel threshold rounding method that reduces the integrality gap below 2 for specific geometric stabbing problems, extending previous results.

Findings

Integrality gap less than 2 for horizontal and vertical segments

Integrality gap less than 2 for unit squares

Generalized threshold rounding technique may apply to other geometric problems

Abstract

In the rectangle stabbing problem, we are given a set $\cR$ of axis-aligned rectangles in $\RR^2$, 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 $\cR$ is a set of horizontal segments. In this paper, we consider two more special cases: when $\cR$ is a set of horizontal and vertical segments, and when $\cR$ 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.