## 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.

Original language | American English |
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Journal | Discrete and Computational Geometry |

DOIs | |

State | Accepted/In press - 1 Jan 2024 |

## Keywords

- 51E23: Spreads and packing problems
- 68Q25: Analysis of algorithms and problem complexity
- 68W40: Analysis of algorithms
- Circle packing
- Computational geometry
- Parameterized algorithms
- Unit disks

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics