Online Class Cover Problem

In this paper, we study the online class cover problem where a (finite or infinite) family $\mathcal{F}$ of geometric objects and a set ${\mathcal{P}}_r$ 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 $\mathcal{F}$ that do not cover any points of ${\mathcal{P}}_r$. The objective of the problem is to place a minimum number of objects. When $\mathcal{F}$ consists of axis-parallel unit squares in ${\mathbb{R}}^2$, we prove that the competitive ratio of any deterministic online algorithm is $\mathrm{\Omega }(\log \mathrm{\mid }{\mathcal{P}}_r\mathrm{\mid })$, and also propose an $O(\log \mathrm{\mid }{\mathcal{P}}_r\mathrm{\mid })$-competitive deterministic algorithm for the problem.