eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-01-24
7:1
7:20
10.4230/LIPIcs.ITCS.2024.7
article
Geometric Covering via Extraction Theorem
Bandyapadhyay, Sayan
1
https://orcid.org/0000-0001-8875-0102
Maheshwari, Anil
2
https://orcid.org/0000-0002-1274-4598
Roy, Sasanka
3
Smid, Michiel
2
Varadarajan, Kasturi
4
Department of Computer Science, Portland State University, OR, USA
School of Computer Science, Carleton University, Ottawa, Canada
ACMU, Indian Statistical Institute, Kolkata, India
Department of Computer Science, University of Iowa, IA, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol287-itcs2024/LIPIcs.ITCS.2024.7/LIPIcs.ITCS.2024.7.pdf
Covering
Extraction theorem
Double-disks
Submodularity
Local search