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# Geometric Covering via Extraction Theorem

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LIPIcs.ITCS.2024.7.pdf
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## Acknowledgements

We are indebted to an anonymous reviewer who pointed out a simpler proof of the Extraction Theorem for unit disks.

## Cite As

Sayan Bandyapadhyay, Anil Maheshwari, Sasanka Roy, Michiel Smid, and Kasturi Varadarajan. Geometric Covering via Extraction Theorem. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.7

## Abstract

In this work, we address the following question. Suppose we are given a set D of positive-weighted disks and a set T of n points in the plane, such that each point of T is contained in at least two disks of D. Then is there always a subset S of D such that the union of the disks in S contains all the points of T and the total weight of the disks of D that are not in S is at least a constant fraction of the total weight of the disks in D? In our work, we prove the Extraction Theorem that answers this question in the affirmative. Our constructive proof heavily exploits the geometry of disks, and in the process, we make interesting connections between our work and the literature on local search for geometric optimization problems. The Extraction Theorem helps to design the first polynomial-time O(1)-approximations for two important geometric covering problems involving disks.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
• Mathematics of computing → Approximation algorithms
##### Keywords
• Covering
• Extraction theorem
• Double-disks
• Submodularity
• Local search

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