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Updated: Oct 14, 2025

An Efficient and Flexible Cell Aggregation Method for 3D Spheroid Production
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Improved interval methods for solving circle packing problems in the unit square.

Mihály Csaba Markót1

  • 1Faculty of Mathematics, Wolfgang Pauli Institute, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Journal of Global Optimization : an International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management and Engineering
|November 1, 2021
PubMed
Summary
This summary is machine-generated.

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Computer-assisted proofs confirm the densest packings for 31-33 equal circles in a square. Enhanced methods significantly accelerated solutions, achieving results in hours instead of months.

Area of Science:

  • Computational Geometry
  • Optimization Algorithms

Background:

  • Previous work established computer-assisted methods for circle packing problems up to 30 circles.
  • Solving larger instances (31-33 circles) required significantly more computational resources with existing techniques.

Purpose of the Study:

  • To develop and apply improved computational methods for proving optimal circle packings.
  • To solve the densest packing problems for 31, 32, and 33 equal circles in a square.

Main Methods:

  • Developed a new interval-based polygon representation for the local search phase, enhancing efficiency and correctness.
  • Implemented improved strategies for the global search, including symmetry filtering and tile pattern matching.
  • Utilized interval arithmetic and global optimization techniques for rigorous proof generation.
Keywords:
Branch and boundCircle packingGlobal optimizationInterval arithmeticOptimality proof

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Main Results:

  • Successfully determined the densest packings for 31, 32, and 33 non-overlapping equal circles in a square.
  • Achieved optimal solutions in significantly reduced computation times (26, 61, and 13 CPU hours).
  • The new methods are approximately 40-100 times faster than previous approaches.

Conclusions:

  • The enhanced computational approach provides highly precise enclosures for optimal circle packing solutions.
  • The improved methodology is scalable and capable of solving subsequent, larger circle packing instances efficiently.