Threshold Rounding for the Standard LP Relaxation of some Geometric Stabbing Problems
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In the rectangle stabbing problem, we are given a set of axis-aligned rectangles in ^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 is a set of horizontal segments. In this paper, we consider two more special cases: when is a set of horizontal and vertical segments, and when 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 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.
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