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### Documents authored by Gupta, Utkarsh

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##### Worst-Case Optimal Covering of Rectangles by Disks

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

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

##### 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.

##### Cite as

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)

```@InProceedings{fekete_et_al:LIPIcs.SoCG.2020.42,
author =	{Fekete, S\'{a}ndor P. and Gupta, Utkarsh and Keldenich, Phillip and Scheffer, Christian and Shah, Sahil},
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 =	{Cabello, Sergio and Chen, Danny Z.},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.42},
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}
}```
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