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Rigorous packing of unit squares into a circle

Author

Listed:
  • Tiago Montanher

    (Wolfgang Pauli Institute
    University of Vienna)

  • Arnold Neumaier

    (University of Vienna)

  • Mihály Csaba Markót

    (Wolfgang Pauli Institute
    University of Vienna)

  • Ferenc Domes

    (University of Vienna)

  • Hermann Schichl

    (University of Vienna)

Abstract

This paper considers the task of finding the smallest circle into which one can pack a fixed number of non-overlapping unit squares that are free to rotate. Due to the rotation angles, the packing of unit squares into a container is considerably harder to solve than their circle packing counterparts. Therefore, optimal arrangements were so far proved to be optimal only for one or two unit squares. By a computer-assisted method based on interval arithmetic techniques, we solve the case of three squares and find rigorous enclosures for every optimal arrangement of this problem. We model the relation between the squares and the circle as a constraint satisfaction problem (CSP) and found every box that may contain a solution inside a given upper bound of the radius. Due to symmetries in the search domain, general purpose interval methods are far too slow to solve the CSP directly. To overcome this difficulty, we split the problem into a set of subproblems by systematically adding constraints to the center of each square. Our proof requires the solution of 6, 43 and 12 subproblems with 1, 2 and 3 unit squares respectively. In principle, the method proposed in this paper generalizes to any number of squares.

Suggested Citation

  • Tiago Montanher & Arnold Neumaier & Mihály Csaba Markót & Ferenc Domes & Hermann Schichl, 2019. "Rigorous packing of unit squares into a circle," Journal of Global Optimization, Springer, vol. 73(3), pages 547-565, March.
    • Handle: RePEc:spr:jglopt:v:73:y:2019:i:3:d:10.1007_s10898-018-0711-5
    DOI: 10.1007/s10898-018-0711-5
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    References listed on IDEAS

    as
    1. Josef Kallrath & Steffen Rebennack, 2014. "Cutting ellipses from area-minimizing rectangles," Journal of Global Optimization, Springer, vol. 59(2), pages 405-437, July.
    2. E. G. Birgin & R. D. Lobato & J. M. Martínez, 2017. "A nonlinear programming model with implicit variables for packing ellipsoids," Journal of Global Optimization, Springer, vol. 68(3), pages 467-499, July.
    3. Castillo, Ignacio & Kampas, Frank J. & Pintér, János D., 2008. "Solving circle packing problems by global optimization: Numerical results and industrial applications," European Journal of Operational Research, Elsevier, vol. 191(3), pages 786-802, December.
    4. E. G. Birgin & R. D. Lobato & J. M. Martínez, 2016. "Packing ellipsoids by nonlinear optimization," Journal of Global Optimization, Springer, vol. 65(4), pages 709-743, August.
    5. E G Birgin & J M Martínez & W F Mascarenhas & D P Ronconi, 2006. "Method of sentinels for packing items within arbitrary convex regions," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 57(6), pages 735-746, June.
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    Cited by:

    1. Hu, Zhi-Hua & Zheng, Yu-Xin & Wang, You-Gan, 2022. "Packing computing servers into the vessel of an underwater data center considering cooling efficiency," Applied Energy, Elsevier, vol. 314(C).

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