License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2020.42
URN: urn:nbn:de:0030-drops-122003
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12200/
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### Worst-Case Optimal Covering of Rectangles by Disks

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### 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.

### BibTeX - Entry

```@InProceedings{fekete_et_al:LIPIcs:2020:12200,
author =	{S{\'a}ndor P. Fekete and Utkarsh Gupta and Phillip Keldenich and Christian Scheffer and Sahil Shah},
title =	{{Worst-Case Optimal Covering of Rectangles by Disks}},
booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
pages =	{42:1--42:23},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-143-6},
ISSN =	{1868-8969},
year =	{2020},
volume =	{164},
editor =	{Sergio Cabello and Danny Z. Chen},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address =	{Dagstuhl, Germany},
URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12200},
URN =		{urn:nbn:de:0030-drops-122003},
doi =		{10.4230/LIPIcs.SoCG.2020.42},
annote =	{Keywords: Disk covering, critical density, covering coefficient, tight worst-case bound, interval arithmetic, approximation}
}
```

 Keywords: Disk covering, critical density, covering coefficient, tight worst-case bound, interval arithmetic, approximation Collection: 36th International Symposium on Computational Geometry (SoCG 2020) Issue Date: 2020 Date of publication: 08.06.2020 Supplementary Material: The code of the automatic prover can be found at https://github.com/phillip-keldenich/circlecover. Furthermore, there is a video contribution [Sándor P. Fekete et al., 2020], video available at https://www.ibr.cs.tu-bs.de/users/fekete/Videos/Cover_full.mp4, illustrating the algorithm and proof presented in this paper.

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