Rigorous packing of unit squares into a circle

Tiago Montanher (), Arnold Neumaier (), Mihály Csaba Markót (), Ferenc Domes () and Hermann Schichl ()
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Tiago Montanher: Wolfgang Pauli Institute
Arnold Neumaier: University of Vienna
Mihály Csaba Markót: Wolfgang Pauli Institute
Ferenc Domes: University of Vienna
Hermann Schichl: University of Vienna

Journal of Global Optimization, 2019, vol. 73, issue 3, No 5, 547-565

Abstract: 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.

Keywords: Square packing into a circle; Interval branch-and-bound; Tiling constraints; Computer-assisted proof; 52C15; 90C26; 65K05; 65G30 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10898-018-0711-5

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