Online Class Cover Problem

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 $\cal F$ 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.