{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/worst-case-optimal-covering-of-rectangles-by","title":"Worst-Case Optimal Covering of Rectangles by Disks","arxiv_id":"2003.08236","date":"2020-03-18","proceeding":null,"authors":["Sándor P. Fekete","Utkarsh Gupta","Phillip Keldenich","Christian Scheffer","Sahil Shah"],"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 $\\lambda\\geq 1$, the critical covering area $A^*(\\lambda)$ is the minimum value for which any set of disks with total area at least $A^*(\\lambda)$ can cover a rectangle of dimensions $\\lambda\\times 1$. We show that there is a threshold value $\\lambda_2 = \\sqrt{\\sqrt{7}/2 - 1/4} \\approx 1.035797\\ldots$, such that for $\\lambda<\\lambda_2$ the critical covering area $A^*(\\lambda)$ is $A^*(\\lambda)=3\\pi\\left(\\frac{\\lambda^2}{16} +\\frac{5}{32} + \\frac{9}{256\\lambda^2}\\right)$, and for $\\lambda\\geq \\lambda_2$, the critical area is $A^*(\\lambda)=\\pi(\\lambda^2+2)/4$; these values are tight. For the special case $\\lambda=1$, i.e., for covering a unit square, the critical covering area is $\\frac{195\\pi}{256}\\approx 2.39301\\ldots$. The proof uses a careful combination of manual and automatic analysis, demonstrating the power of the employed interval arithmetic technique.","url_abs":"http://arxiv.org/abs/2003.08236v1","url_pdf":"http://arxiv.org/pdf/2003.08236v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"worst-case-optimal-covering-of-rectangles-by","repo_url":"https://github.com/phillip-keldenich/circlecover","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}