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Packing Squares into a Disk with Optimal Worst-Case Density

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

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ACM Subject Classification
  • Theory of computation → Packing and covering problems
  • Theory of computation → Computational geometry
Keywords
  • Square packing
  • packing density
  • tight worst-case bound
  • interval arithmetic
  • approximation

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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.

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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) https://doi.org/10.4230/LIPIcs.SoCG.2021.36

Author Details

Sándor P. Fekete
  • Department of Computer Science, TU Braunschweig, Germany
Vijaykrishna Gurunathan
  • Department of Computer Science & Engineering, IIT Bombay, India
Kushagra Juneja
  • Department of Computer Science & Engineering, IIT Bombay, India
Phillip Keldenich
  • Department of Computer Science, TU Braunschweig, Germany
Linda Kleist
  • Department of Computer Science, TU Braunschweig, Germany
Christian Scheffer
  • Department of Computer Science, TU Braunschweig, Germany

Supplementary Materials

References

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