(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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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
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Abstract	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 .
AbstractList	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 $$n+k$$ n + k disks. Thus the problem of packing equal disks is the special case of our problem with $$n=h=0$$ 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 $$h=0$$ h = 0 . Our main algorithmic contribution is an algorithm that solves the repacking problem in time $$(h+k)^{\mathcal {O}(h+k)}\cdot \|I\|^{\mathcal {O}(1)}$$ ( 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 . 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 equal disks packed into a rectangle and integers and , we ask whether it is possible by changing positions of at most disks to pack disks. Thus the problem of packing equal disks is the special case of our problem with . While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking problem is NP-hard already for . Our main algorithmic contribution is an algorithm that solves the repacking problem in time , where \| \| is the input size. That is, the problem is fixed-parameter tractable parameterized by and . 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.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. 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 $$n+k$$ n+k disks. Thus the problem of packing equal disks is the special case of our problem with $$n=h=0$$ 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 $$h=0$$ h=0. Our main algorithmic contribution is an algorithm that solves the repacking problem in time $$(h+k)^{\mathcal {O}(h+k)}\cdot \|I\|^{\mathcal {O}(1)}$$ (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. 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 . 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.
Author	Inamdar, Tanmay Fomin, Fedor V. Zehavi, Meirav Golovach, Petr A. Saurabh, Saket
Author_xml	– sequence: 1 givenname: Fedor V. surname: Fomin fullname: Fomin, Fedor V. email: Fedor.Fomin@uib.no organization: University of Bergen – sequence: 2 givenname: Petr A. surname: Golovach fullname: Golovach, Petr A. organization: University of Bergen – sequence: 3 givenname: Tanmay surname: Inamdar fullname: Inamdar, Tanmay organization: Indian Institute of Technology, Jodhpur – sequence: 4 givenname: Saket surname: Saurabh fullname: Saurabh, Saket organization: University of Bergen, Institute of Mathematical Sciences – sequence: 5 givenname: Meirav surname: Zehavi fullname: Zehavi, Meirav organization: Ben-Guiron University
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Cites_doi	10.1007/s00454-022-00422-8 10.1016/j.comgeo.2013.08.008 10.1016/S0166-218X(01)00359-6 10.1016/j.cosrev.2016.12.001 10.1145/174644.174650 10.1016/j.tcs.2016.11.034 10.1007/978-3-319-21275-3 10.1007/PL00009472 10.1109/SFCS.1995.492475 10.1007/978-3-662-01206-2 10.1016/j.ejor.2007.01.054 10.1145/210332.210337 10.1080/0025570X.1970.11975991 10.1145/3473713 10.1109/FOCS46700.2020.00098 10.1016/0012-365X(93)E0230-2 10.1007/978-3-319-11421-7_4 10.1006/jctb.2000.2026 10.1007/PL00009306 10.4153/CMB-1965-018-9 10.1145/2455.214106 10.1137/1.9781611973402.2
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Keywords	Computational geometry Circle packing 51E23: Spreads and packing problems 68W40: Analysis of algorithms Parameterized algorithms 68Q25: Analysis of algorithms and problem complexity Unit disks
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Snippet	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...
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SubjectTerms	Algorithms Combinatorics Computational Mathematics and Numerical Analysis Disks Mathematics Mathematics and Statistics Packaging Rectangles
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Title	(Re)packing Equal Disks into Rectangle
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