We consider the following geometric optimization problem: Given $ n $ axis-aligned rectangles in the plane, the goal is to find a set of horizontal segments of minimum total length such that each rectangle is stabbed. A segment stabs a rectangle if it intersects both its left and right edge. As such, this stabbing problem falls into the category of weighted geometric set cover problems for which techniques that improve upon the general ${\Theta}(\log n)$-approximation guarantee have received a lot of attention in the literature. Chan at al. (2018) have shown that rectangle stabbing is NP-hard and that it admits a constant-factor approximation algorithm based on Varadarajan's quasi-uniform sampling method. In this work we make progress on rectangle stabbing on two fronts. First, we present a quasi-polynomial time approximation scheme (QPTAS) for rectangle stabbing. Furthermore, we provide a simple $8$-approximation algorithm that avoids the framework of Varadarajan. This settles two open problems raised by Chan et al. (2018).

中文翻译：

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用于刺入矩形的 QPTAS

我们考虑以下几何优化问题：给定平面中的 $n$ 个轴对齐的矩形，目标是找到一组总长度最小的水平线段，使得每个矩形都被刺伤。如果线段与其左右边缘都相交，则该线段将刺入矩形。因此，这个刺入问题属于加权几何集覆盖问题的范畴，对于这些问题，改进一般 ${\Theta}(\log n)$-近似保证的技术在文献中受到了很多关注。陈在al。(2018) 已经表明矩形刺入是 NP-hard 并且它允许基于 Varadarajan 的准均匀采样方法的常数因子近似算法。在这项工作中，我们在两个方面取得了矩形刺入的进展。第一的，我们提出了一种用于矩形刺入的准多项式时间近似方案（QPTAS）。此外，我们提供了一个简单的 $8$-近似算法，可以避免 Varadarajan 的框架。这解决了 Chan 等人提出的两个公开问题。(2018)。