A QPTAS for stabbing rectangles
Abstract
We consider the following geometric optimization problem: Given $ n $ axisaligned 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 NPhard and that it admits a constantfactor approximation algorithm based on Varadarajan's quasiuniform sampling method. In this work we make progress on rectangle stabbing on two fronts. First, we present a quasipolynomial 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).
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.06571
 Bibcode:
 2021arXiv210706571E
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Data Structures and Algorithms