Geometric Covering via Extraction Theorem
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.
Covering
Extraction theorem
Double-disks
Submodularity
Local search
Theory of computation~Computational geometry
Mathematics of computing~Approximation algorithms
7:1-7:20
Regular Paper
This work is supported by NSF grant CCF: AF-2311397 and by NSERC, Canada.
We are indebted to an anonymous reviewer who pointed out a simpler proof of the Extraction Theorem for unit disks.
Sayan
Bandyapadhyay
Sayan Bandyapadhyay
Department of Computer Science, Portland State University, OR, USA
https://orcid.org/0000-0001-8875-0102
Anil
Maheshwari
Anil Maheshwari
School of Computer Science, Carleton University, Ottawa, Canada
https://orcid.org/0000-0002-1274-4598
Sasanka
Roy
Sasanka Roy
ACMU, Indian Statistical Institute, Kolkata, India
Michiel
Smid
Michiel Smid
School of Computer Science, Carleton University, Ottawa, Canada
Kasturi
Varadarajan
Kasturi Varadarajan
Department of Computer Science, University of Iowa, IA, USA
10.4230/LIPIcs.ITCS.2024.7
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Sayan Bandyapadhyay, Anil Maheshwari, Sasanka Roy, Michiel Smid, and Kasturi Varadarajan
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