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, 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 the minimum number of objects. When $\cal F$ consists of all possible translates of a square in $\mathbb{R}^2$, we prove that the competitive ratio of any deterministic online algorithm is $\Omega(\log |{\cal P}_r|)$. On the other hand, when the objects are all possible translates of a rectangle in $\mathbb{R}^2$, we propose an $O(\log |{\cal P}_r|)$-competitive deterministic algorithm for the problem.Comment: 27 pages, 23 figure