Online Hitting Sets for Disks of Bounded Radii

TL;DR

This paper introduces algorithms for the online minimum hitting set problem in geometric spaces, achieving logarithmic competitive ratios for disks with bounded radii and extending to convex bodies and integer lattice subsets.

Contribution

It provides the first logarithmic-competitive algorithms for online hitting sets for disks of bounded radii and generalizes to convex bodies and integer lattice subsets with the lowest point property.

Findings

Achieves an $O(\log M \log n)$-competitive algorithm for disks of radii in $[1,M]$

Extends results to positive homothets of convex bodies with scaling factors in $[1,M]$

Develops an $O(\log N)$-competitive algorithm for integer lattice subsets with the lowest point property.

Abstract

We present algorithms for the online minimum hitting set problem in geometric range spaces: given a set $P$ of $n$ points in the plane and a sequence of geometric objects that arrive one-by-one, we need to maintain a hitting set at all times by making irrevocable decisions. For disks of radii in the interval $[1,M]$, we present an $O(\log M \log n)$-competitive algorithm. This result generalizes from disks to positive homothets of any convex body in the plane with scaling factors in the interval $[1,M]$. As a main technical tool, we reduce the problem to the online hitting set problem for a finite subset of integer points and geometric objects with the lowest point property, introduced in this paper, which behave similarly to bottomless rectangles. Specifically, for a given $N>1$, we present an $O(\log N)$-competitive algorithm for the variant where $P$ is a subset of an $N\times N$ section of the integer lattice, and the geometric objects have the lowest point property.