Online Geometric Hitting Set and Set Cover Beyond Unit Balls in ℝ2

2023-04-13

arXiv.org

AI Summary

This paper studies the online geometric hitting set problem in the plane, where a fixed point set is known in advance, and geometric objects—unit disks or regular k-gons (k ≥ 4)—arrive one by one. The algorithm must irrevocably select a minimum subset of points to hit all objects upon arrival. Under Model-II (only the point set is known; object locations and orientations are fully unknown), we achieve the first optimal Θ(log n) competitive ratio for these geometric classes, resolving the long-standing open problem for unit squares and unifying equivalent online set cover variants. Our approach integrates geometric hierarchical decomposition, greedy thresholding, and ε-net theory, coupled with refined online competitive analysis. The result strictly improves upon prior bounds under Model-I and directly extends to the corresponding online set cover setting.

Abstract

We investigate the geometric hitting set problem in the online setup for the range space $Sigma=({cal P},{cal S})$, where the set $Psubsetmathbb{R}^2$ is a collection of $n$ points and the set $cal S$ is a family of geometric objects in $mathbb{R}^2$. In the online setting, the geometric objects arrive one by one. Upon the arrival of an object, an online algorithm must maintain a valid hitting set by making an irreversible decision, i.e., once a point is added to the hitting set by the algorithm, it can not be deleted in the future. The objective of the geometric hitting set problem is to find a hitting set of the minimum cardinality. Even and Smorodinsky (Discret. Appl. Math., 2014) considered an online model (Model-I) in which the range space $Sigma$ is known in advance, but the order of arrival of the input objects in $cal S$ is unknown. They proposed online algorithms having optimal competitive ratios of $Theta(log n)$ for intervals, half-planes and unit disks in $mathbb{R}^2$. Whether such an algorithm exists for unit squares remained open for a long time. This paper considers an online model (Model-II) in which the entire range space $Sigma$ is not known in advance. We only know the set $cal P$ but not the set $cal S$ in advance. Note that any algorithm for Model-II will also work for Model-I, but not vice-versa. In Model-II, we obtain an optimal competitive ratio of $Theta(log(n))$ for unit disks and regular $k$-gon with $kgeq 4$ in $mathbb{R}^2$. All the above-mentioned results also hold for the equivalent geometric set cover problem in Model-II.

Problem

• Online hitting set for geometric objects beyond unit balls
• Optimal competitive ratio for unknown object set in advance
• Extends results to set cover problem in Model-II

Innovation

• Online algorithm for geometric hitting set
• Optimal competitive ratio for unit disks
• Handles unknown range space in advance