Title
Rigorous packing of unit squares into a circle
Author
... show all
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 circleInterval branch-and-boundTiling constraintsComputer-assisted proof
Object type
Language
English [eng]
Persistent identifier
https://phaidra.univie.ac.at/o:962173
Appeared in
Title
Journal of Global Optimization
Volume
73
Issue
3
From page
547
To page
565
Publisher
Springer Science and Business Media LLC
Date issued
2018
Access rights
Rights statement
© The Author(s) 2018

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