Rigorous packing of unit squares into a circle

Tiago de Morais Montanher (Corresponding author), Arnold Neumaier, Mihaly Markot, Ferenc Domes, Hermann Schichl

Publications: Contribution to journalArticlePeer Reviewed

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.
Original languageEnglish
Pages (from-to)547-565
Number of pages19
JournalJournal of Global Optimization
Volume73
Issue number3
DOIs
Publication statusPublished - Mar 2019

Austrian Fields of Science 2012

  • 101016 Optimisation

Keywords

  • Square packing into a circle
  • Interval branch-and-bound
  • Tiling constraints
  • Computer-assisted proof
  • INTERVAL
  • CONVEX
  • SENTINELS
  • OPTIMIZATION

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