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Worst-Case Optimal Covering of Rectangles by Disks

Authors Sándor P. Fekete , Utkarsh Gupta , Phillip Keldenich , Christian Scheffer , Sahil Shah

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Sándor P. Fekete
  • Department of Computer Science, TU Braunschweig, Germany
Utkarsh Gupta
  • Department of Computer Science & Engineering, IIT Bombay, India
Phillip Keldenich
  • Department of Computer Science, TU Braunschweig, Germany
Christian Scheffer
  • Department of Computer Science, TU Braunschweig, Germany
Sahil Shah
  • Department of Computer Science & Engineering, IIT Bombay, India

Acknowledgements

We thank Sebastian Morr and Arne Schmidt for helpful discussions. Major parts of the research by Utkarsh Gupta and Sahil Shah were carried out during a stay at TU Braunschweig.

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Sándor P. Fekete, Utkarsh Gupta, Phillip Keldenich, Christian Scheffer, and Sahil Shah. Worst-Case Optimal Covering of Rectangles by Disks. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 42:1-42:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.42

Abstract

We provide the solution for a fundamental problem of geometric optimization by giving a complete characterization of worst-case optimal disk coverings of rectangles: For any λ ≥ 1, the critical covering area A^*(λ) is the minimum value for which any set of disks with total area at least A^*(λ) can cover a rectangle of dimensions λ× 1. We show that there is a threshold value λ₂ = √{√7/2 - 1/4} ≈ 1.035797…, such that for λ < λ₂ the critical covering area A^*(λ) is A^*(λ) = 3π(λ²/16 + 5/32 + 9/(256λ²)), and for λ ≥ λ₂, the critical area is A^*(λ)=π(λ²+2)/4; these values are tight. For the special case λ=1, i.e., for covering a unit square, the critical covering area is 195π/256 ≈ 2.39301…. The proof uses a careful combination of manual and automatic analysis, demonstrating the power of the employed interval arithmetic technique.

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
  • Theory of computation → Computational geometry
Keywords
  • Disk covering
  • critical density
  • covering coefficient
  • tight worst-case bound
  • interval arithmetic
  • approximation

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