Abstract
In the Maximum Weight Independent Set of Rectangles (MWISR) problem, we are given a collection of weighted axisparallel rectangles in the plane. Our goal is to compute a maximum weight subset of pairwise nonoverlapping rectangles. Due to its various applications, as well as connections to many other problems in computer science, MWISR has received a lot of attention from the computational geometry and the approximation algorithms community. However, despite being extensively studied, MWISR remains not very well understood in terms of polynomial time approximation algorithms, as there is a large gap between the upper and lower bounds, i.e., O(log n\ loglog n) v.s. NPhardness. Another important, poorly understood question is whether one can color rectangles with at most O(omega(R)) colors where omega(R) is the size of a maximum clique in the intersection graph of a set of input rectangles R. Asplund and Grünbaum obtained an upper bound of O(omega(R)^2) about 50 years ago, and the result has remained asymptotically best. This question is strongly related to the integrality gap of the canonical LP for MWISR.
In this paper, we settle above three open problems in a relaxed model where we are allowed to shrink the rectangles by a tiny bit (rescaling them by a factor of 1delta for an arbitrarily small constant delta > 0. Namely, in this model, we show (i) a PTAS for MWISR and (ii) a coloring with O(omega(R)) colors which implies a constant upper bound on the integrality gap of the canonical LP.
For some applications of MWISR the possibility to shrink the rectangles has a natural, wellmotivated meaning. Our results can be seen as an evidence that the shrinking model is a promising way to relax a geometric problem for the purpose of better algorithmic results.
BibTeX  Entry
@InProceedings{adamaszek_et_al:LIPIcs:2015:5293,
author = {Anna Adamaszek and Parinya Chalermsook and Andreas Wiese},
title = {{How to Tame Rectangles: Solving Independent Set and Coloring of Rectangles via Shrinking}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
pages = {4360},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897897},
ISSN = {18688969},
year = {2015},
volume = {40},
editor = {Naveen Garg and Klaus Jansen and Anup Rao and Jos{\'e} D. P. Rolim},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5293},
URN = {urn:nbn:de:0030drops52936},
doi = {10.4230/LIPIcs.APPROXRANDOM.2015.43},
annote = {Keywords: Approximation algorithms, independent set, resource augmentation, rectangle intersection graphs, PTAS}
}
Keywords: 

Approximation algorithms, independent set, resource augmentation, rectangle intersection graphs, PTAS 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015) 
Issue Date: 

2015 
Date of publication: 

13.08.2015 