(Re)packing Equal Disks into Rectangle
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| Publicat a: | Discrete & Computational Geometry vol. 72, no. 4 (Dec 2024), p. 1596 |
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| Autor principal: | |
| Altres autors: | , , , |
| Publicat: |
Springer Nature B.V.
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| Matèries: | |
| Accés en línia: | Citation/Abstract Full Text - PDF |
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| Resum: | 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<inline-graphic xlink:href="454_2024_633_Article_IEq1.gif" /> disks. Thus the problem of packing equal disks is the special case of our problem with n=h=0<inline-graphic xlink:href="454_2024_633_Article_IEq2.gif" />. 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<inline-graphic xlink:href="454_2024_633_Article_IEq3.gif" />. Our main algorithmic contribution is an algorithm that solves the repacking problem in time (h+k)O(h+k)·|I|O(1)<inline-graphic xlink:href="454_2024_633_Article_IEq4.gif" />, where |I| is the input size. That is, the problem is fixed-parameter tractable parameterized by k and h. |
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| ISSN: | 0179-5376 1432-0444 |
| DOI: | 10.1007/s00454-024-00633-1 |
| Font: | Science Database |