Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Fekete, Sándor P.; Gurunathan, Vijaykrishna; Juneja, Kushagra; Keldenich, Phillip; Kleist, Linda; Scheffer, Christian https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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Packing Squares into a Disk with Optimal Worst-Case Density

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Abstract

We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is δ = 8/(5π)≈ 0.509. This implies that any set of (not necessarily equal) squares of total area A ≤ 8/5 can always be packed into a disk with radius 1; in contrast, for any ε > 0 there are sets of squares of total area 8/5+ε that cannot be packed, even if squares may be rotated. This settles the last (and arguably, most elusive) case of packing circular or square objects into a circular or square container: The critical densities for squares in a square (1/2), circles in a square (π/(3+2√2) ≈ 0.539) and circles in a circle (1/2) have already been established, making use of recursive subdivisions of a square container into pieces bounded by straight lines, or the ability to use recursive arguments based on similarity of objects and container; neither of these approaches can be applied when packing squares into a circular container. Our proof uses a careful manual analysis, complemented by a computer-assisted part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms. At the same time, our approach showcases the power of a general framework for computer-assisted proofs, based on interval arithmetic.

BibTeX - Entry

@InProceedings{fekete_et_al:LIPIcs.SoCG.2021.36,
  author =	{Fekete, S\'{a}ndor P. and Gurunathan, Vijaykrishna and Juneja, Kushagra and Keldenich, Phillip and Kleist, Linda and Scheffer, Christian},
  title =	{{Packing Squares into a Disk with Optimal Worst-Case Density}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{36:1--36:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13835},
  URN =		{urn:nbn:de:0030-drops-138356},
  doi =		{10.4230/LIPIcs.SoCG.2021.36},
  annote =	{Keywords: Square packing, packing density, tight worst-case bound, interval arithmetic, approximation}
}

Keywords: Square packing, packing density, tight worst-case bound, interval arithmetic, approximation
Seminar: 37th International Symposium on Computational Geometry (SoCG 2021)
Issue date: 2021
Date of publication: 02.06.2021
Supplementary Material: Software (Source Code): https://github.com/phillip-keldenich/squares-in-disk archived at: https://archive.softwareheritage.org/swh:1:dir:7b372039e1b09587f86959e9e1fb013bf2789658


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