(Re)packing Equal Disks into Rectangle

The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for...

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Bibliographic Details
Published in:Discrete & computational geometry Vol. 72; no. 4; pp. 1596 - 1629
Main Authors: Fomin, Fedor V., Golovach, Petr A., Inamdar, Tanmay, Saurabh, Saket, Zehavi, Meirav
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2024
Springer Nature B.V
Subjects:
Algorithms
Combinatorics
Computational Mathematics and Numerical Analysis
Disks
Mathematics
Mathematics and Statistics
Packaging
Rectangles
Computational geometry
Circle packing
51E23: Spreads and packing problems
68W40: Analysis of algorithms
Parameterized algorithms
68Q25: Analysis of algorithms and problem complexity
Unit disks
ISSN:0179-5376, 1432-0444
Online Access:Get full text
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Summary:The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for a given packing of equal disks into a rectangle, the question is whether by changing positions of a small number of disks, we can allocate space for packing more disks. More formally, in the repacking problem, for a given set of n equal disks packed into a rectangle and integers k and h , we ask whether it is possible by changing positions of at most h disks to pack n + k disks. Thus the problem of packing equal disks is the special case of our problem with n = h = 0 . While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking problem is NP-hard already for h = 0 . Our main algorithmic contribution is an algorithm that solves the repacking problem in time ( h + k ) O ( h + k ) · | I | O ( 1 ) , where | I | is the input size. That is, the problem is fixed-parameter tractable parameterized by k and h .
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Editor in Charge: Csaba D. Tóth
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-024-00633-1