Online Class Cover Problem

In this paper, we study the online class cover problem where a (finite or infinite) family \(\cal F\) of geometric objects and a set \({\cal P}_r\) of red points in \(\mathbb{R}^d\) are given a prior, and blue points from \(\mathbb{R}^d\) arrives one after another. Upon the arrival of a blue point,...

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Bibliographic Details
Published in:	arXiv.org
Main Authors:	De, Minati, Maheshwari, Anil, Mandal, Ratnadip
Format:	Paper
Language:	English
Published:	Ithaca Cornell University Library, arXiv.org 03.07.2024
Subjects:	Algorithms Rectangles
ISSN:	2331-8422
Online Access:	Get full text
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Summary:	In this paper, we study the online class cover problem where a (finite or infinite) family \(\cal F\) of geometric objects and a set \({\cal P}_r\) of red points in \(\mathbb{R}^d\) are given a prior, and blue points from \(\mathbb{R}^d\) arrives one after another. Upon the arrival of a blue point, the online algorithm must make an irreversible decision to cover it with objects from \(\cal F\) that do not cover any points of \({\cal P}_r\). The objective of the problem is to place a minimum number of objects. When \(\cal F\) consists of axis-parallel unit squares in \(\mathbb{R}^2\), we prove that the competitive ratio of any deterministic online algorithm is \(\Omega(\log \|{\cal P}_r\|)\), and also propose an \(O(\log \|{\cal P}_r\|)\)-competitive deterministic algorithm for the problem.
Bibliography:	SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50
ISSN:	2331-8422
DOI:	10.48550/arxiv.2308.07020