License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2016.13
URN: urn:nbn:de:0030-drops-59054
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5905/
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Balas, Kevin ; Dumitrescu, Adrian ; Tóth, Csaba

Anchored Rectangle and Square Packings

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LIPIcs-SoCG-2016-13.pdf (0.6 MB)

Abstract

For points p_1,...,p_n in the unit square [0,1]^2, an anchored rectangle packing consists of interior-disjoint axis-aligned 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 7/12-O(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 constant-factor 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 9/47-approximation for anchored square packings, and 1/3 for lower-left 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 =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5905},
  URN =		{urn:nbn:de:0030-drops-59054},
  doi =		{10.4230/LIPIcs.SoCG.2016.13},
  annote =	{Keywords: Rectangle packing, anchored rectangle, greedy algorithm, charging scheme, approximation algorithm.}
}

Keywords: Rectangle packing, anchored rectangle, greedy algorithm, charging scheme, approximation algorithm.
Collection: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue Date: 2016
Date of publication: 10.06.2016

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