Online Geometric Covering and Piercing

TL;DR

This paper introduces a deterministic online algorithm for the geometric piercing set problem for similarly sized fat objects in Euclidean space, providing bounds on competitive ratios and establishing lower bounds for various shapes.

Contribution

It presents the first deterministic online algorithm with competitive ratio bounds for piercing similarly sized convex objects in higher dimensions.

Findings

Deterministic algorithm for hypercubes with ratio at most $3^d\lceil ext{log}_2 k ceil + 2^d$.

Lower bounds on competitive ratios for $ ext{α}$-fat objects in 2D and hypercubes in higher dimensions.

Upper bound on the competitive ratio for the online unit covering problem with convex objects.

Abstract

We consider the online version of the piercing set problem, where geometric objects arrive one by one, and the online algorithm must maintain a valid piercing set for the already arrived objects by making irrevocable decisions. It is easy to observe that any deterministic algorithm solving this problem for intervals in $\mathbb{R}$ has a competitive ratio of at least $\Omega(n)$. This paper considers the piercing set problem for similarly sized objects. We propose a deterministic online algorithm for similarly sized fat objects in $\mathbb{R}^d$. For homothetic hypercubes in $\mathbb{R}^d$ with side length in the range $[1,k]$, we propose a deterministic algorithm having a competitive ratio of at most~$3^d\lceil\log_2 k\rceil+2^d$. In the end, we show deterministic lower bounds of the competitive ratio for similarly sized $\alpha$-fat objects in $\mathbb{R}^2$ and homothetic hypercubes in $\mathbb{R}^d$. Note that piercing translated copies of a convex object is equivalent to the unit covering problem, which is well-studied in the online setup. Surprisingly, no upper bound of the competitive ratio was known for the unit covering problem when the corresponding object is anything other than a ball or a hypercube. Our result yields an upper bound of the competitive ratio for the unit covering problem when the corresponding object is any convex object in $\mathbb{R}^d$.