Online Class Cover Problem - De, Minati; Maheshwari, Anil; Mandal, Ratnadip (arXiv.org, 2024)
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 \(Ω(\log |{\cal P}_r|)\), and also propose an \(O(\log |{\cal P}_r|)\)-competitive deterministic algorithm for the problem.