Abstract
We initiate the study of the following natural geometric optimization problem. The input is a set of axisaligned rectangles in the plane. The objective is to find a set of horizontal line segments of minimum total length so that every rectangle is stabbed by some line segment. A line segment stabs a rectangle if it intersects its left and its right boundary. The problem, which we call Stabbing, can be motivated by a resource allocation problem and has applications in geometric network design. To the best of our knowledge, only special cases of this problem have been considered so far.
Stabbing is a weighted geometric set cover problem, which we show to be NPhard. While for general set cover the best possible approximation ratio is Theta(log n), it is an important field in geometric approximation algorithms to obtain better ratios for geometric set cover problems. Chan et al. [SODA'12] generalize earlier results by Varadarajan [STOC'10] to obtain sublogarithmic performances for a broad class of weighted geometric set cover instances that are characterized by having low shallowcell complexity. The shallowcell complexity of Stabbing instances, however, can be high so that a direct application of the framework of Chan et al. gives only logarithmic bounds. We still achieve a constantfactor approximation by decomposing general instances into what we call laminar instances that have low enough complexity.
Our decomposition technique yields constantfactor approximations also for the variant where rectangles can be stabbed by horizontal and vertical segments and for two further geometric set cover problems.
BibTeX  Entry
@InProceedings{chan_et_al:LIPIcs:2018:10009,
author = {Timothy M. Chan and Thomas C. van Dijk and Krzysztof Fleszar and Joachim Spoerhase and Alexander Wolff},
title = {{Stabbing Rectangles by Line Segments  How Decomposition Reduces the ShallowCell Complexity}},
booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)},
pages = {61:161:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770941},
ISSN = {18688969},
year = {2018},
volume = {123},
editor = {WenLian Hsu and DerTsai Lee and ChungShou Liao},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/10009},
URN = {urn:nbn:de:0030drops100094},
doi = {10.4230/LIPIcs.ISAAC.2018.61},
annote = {Keywords: Geometric optimization, NPhard, approximation, shallowcell complexity, line stabbing}
}
Keywords: 

Geometric optimization, NPhard, approximation, shallowcell complexity, line stabbing 
Collection: 

29th International Symposium on Algorithms and Computation (ISAAC 2018) 
Issue Date: 

2018 
Date of publication: 

06.12.2018 