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(Re)packing Equal Disks into Rectangle.

Fedor V Fomin1, Petr A Golovach1, Tanmay Inamdar2

  • 1University of Bergen, Bergen, Norway.

Discrete & Computational Geometry
|November 19, 2024
PubMed
Summary
This summary is machine-generated.

This study introduces the disk repacking problem, proving it

Keywords:
Circle packingComputational geometryParameterized algorithmsUnit disks

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Area of Science:

  • Computational Geometry
  • Discrete Mathematics

Background:

  • The equal disk packing problem, arranging non-overlapping disks in a rectangle, is a fundamental geometric challenge with open computational complexity.
  • Algorithmic generalizations are crucial for understanding packing complexities and developing efficient solutions.

Purpose of the Study:

  • To introduce and analyze the disk repacking problem: determining if a small number of disk repositionings can accommodate more disks.
  • To establish the computational complexity of this generalized packing problem.

Main Methods:

  • Formal definition of the repacking problem with parameters n (disks), k (additional disks), and h (repositioned disks).
  • Proof of NP-hardness for the repacking problem, even for small values of k.
  • Development of a fixed-parameter tractable algorithm with runtime dependent on input size |I| and parameters k and h.

Main Results:

  • The disk repacking problem is proven to be NP-hard for k=2.
  • An algorithm is presented that solves the repacking problem in time O(f(k, h) * |I|^c), demonstrating fixed-parameter tractability.

Conclusions:

  • The repacking problem offers a tractable algorithmic approach to disk packing challenges.
  • The fixed-parameter tractable algorithm provides an efficient method for solving instances of the repacking problem.