Balas, Kevin ;
Dumitrescu, Adrian ;
Tóth, Csaba
Anchored Rectangle and Square Packings
Abstract
For points p_1,...,p_n in the unit square [0,1]^2, an anchored rectangle packing consists of interiordisjoint axisaligned empty rectangles r_1,...,r_n in [0,1]^2 such that point p_i is a corner of the rectangle r_i (that is, r_i is anchored at p_i) for i=1,...,n. We show that for every set of n points in [0,1]^2, there is an anchored rectangle packing of area at least 7/12O(1/n), and for every n, there are point sets for which the area of every anchored rectangle packing is at most 2/3. The maximum area of an anchored square packing is always at least 5/32 and sometimes at most 7/27.
The above constructive lower bounds immediately yield constantfactor approximations, of 7/12 epsilon for rectangles and 5/32 for squares, for computing anchored packings of maximum area in O(n log n) time. We prove that a simple greedy strategy achieves a 9/47approximation for anchored square packings, and 1/3 for lowerleft anchored square packings. Reductions to maximum weight independent set (MWIS) yield a QPTAS and a PTAS for anchored rectangle and square packings in n^{O(1/epsilon)} and exp(poly(log (n/epsilon))) time, respectively.
BibTeX  Entry
@InProceedings{balas_et_al:LIPIcs:2016:5905,
author = {Kevin Balas and Adrian Dumitrescu and Csaba T{\'o}th},
title = {{Anchored Rectangle and Square Packings}},
booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)},
pages = {13:113:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770095},
ISSN = {18688969},
year = {2016},
volume = {51},
editor = {S{\'a}ndor Fekete and Anna Lubiw},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5905},
URN = {urn:nbn:de:0030drops59054},
doi = {10.4230/LIPIcs.SoCG.2016.13},
annote = {Keywords: Rectangle packing, anchored rectangle, greedy algorithm, charging scheme, approximation algorithm.}
}
2016
Keywords: 

Rectangle packing, anchored rectangle, greedy algorithm, charging scheme, approximation algorithm. 
Seminar: 

32nd International Symposium on Computational Geometry (SoCG 2016)

Issue date: 

2016 
Date of publication: 

2016 