New Lower Bound and Algorithms for Online Geometric Hitting Set Problem

TL;DR

This paper advances the understanding of online geometric hitting set problems by establishing new lower bounds and developing algorithms with improved competitive ratios for various geometric objects and point sets.

Contribution

It introduces new lower bounds and near-optimal algorithms for online hitting set problems involving hypercubes, fat objects, and regular polygons in different dimensions.

Findings

Lower bound of Ω(d log M) for hitting hypercubes in Z^d

Randomized algorithm with O(d^2 log M) competitive ratio

Deterministic algorithm with improved bounds for fat objects and polygons

Abstract

The hitting set problem is one of the fundamental problems in combinatorial optimization and is well-studied in offline setup. We consider the online hitting set problem, where only the set of points is known in advance, and objects are introduced one by one. Our objective is to maintain a minimum-sized hitting set by making irrevocable decisions. Here, we present the study of two variants of the online hitting set problem depending on the point set. In the first variant, we consider the point set to be the entire $\mathbb{Z}^d$, while in the second variant, we consider the point set to be a finite subset of $\mathbb{R}^2$. If you use points in $\mathbb{Z}^d$ to hit homothetic hypercubes in $\mathbb{R}^d$ with side lengths in $[1,M]$, we show that the competitive ratio of any algorithm is $\Omega(d\log M)$, whether it is deterministic or random. This improves the recently known deterministic lower bound of $\Omega(\log M)$ by a factor of $d$. Then, we present an almost tight randomized algorithm with a competitive ratio $O(d^2\log M)$ that significantly improves the best-known competitive ratio of $25^d\log M$. Next, we propose a simple deterministic ${\lfloor\frac{2}{\alpha}+2\rfloor^d}(\lfloor\log_{2}M\rfloor+1)$ competitive algorithm to hit similarly sized {$\alpha$-fat objects} in $\mathbb{R}^d$ having diameters in the range $[1, M]$ using points in $\mathbb{Z}^d$. This improves the current best-known upper bound by a factor of at least $5^d$. Finally, we consider the hitting set problem when the point set consists of $n$ points in $\mathbb{R}^2$, and the objects are homothetic regular $k$-gons having diameter in the range $[1, M]$. We present an $O(\log n\log M)$ competitive randomized algorithm for that. Whereas no result was known even for squares. In particular, our results answer some of the open questions raised by Khan et al. (SoCG'23) and Alefkhani et al. (WAOA'23).