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DOI: 10.4230/LIPIcs.SoCG.2021.36
Packing Squares into a Disk with Optimal Worst-Case Density

Authors: Sándor P. Fekete, Vijaykrishna Gurunathan, Kushagra Juneja, Phillip Keldenich, Linda Kleist, and Christian Scheffer

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

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.
Cite as

Sándor P. Fekete, Vijaykrishna Gurunathan, Kushagra Juneja, Phillip Keldenich, Linda Kleist, and Christian Scheffer. Packing Squares into a Disk with Optimal Worst-Case Density. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.36},
  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}
}
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