Online Class Cover Problem
Minati De, Anil Maheshwari, and Ratnadip Mandal

TL;DR
This paper introduces the online class cover problem involving covering blue points with geometric objects while avoiding red points, providing bounds on algorithm competitiveness and an efficient solution for axis-parallel squares.
Contribution
It formulates the online class cover problem, establishes a logarithmic lower bound on competitive ratio, and presents an optimal logarithmic-competitive algorithm for axis-parallel squares.
Findings
Deterministic algorithms have a competitive ratio of at least logarithmic in the number of red points.
An $O(\log |{ m P}_r|)$-competitive algorithm is proposed.
The problem is fundamental for online geometric covering with constraints.
Abstract
In this paper, we study the online class cover problem where a (finite or infinite) family of geometric objects and a set of red points in are given a prior, and blue points from 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 that do not cover any points of . The objective of the problem is to place a minimum number of objects. When consists of axis-parallel unit squares in , we prove that the competitive ratio of any deterministic online algorithm is , and also propose an -competitive deterministic algorithm for the problem.
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Taxonomy
TopicsOptimization and Search Problems · Auction Theory and Applications · Complexity and Algorithms in Graphs
