New Lower Bound and Algorithms for Online Geometric Hitting Set Problem

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 $\mathbb{R}^2$. For hitting similarly sized {$\alpha$-fat objects} in $\mathbb{R}^d$ with diameters in the range $[1, M]$ using points in $\mathbb{Z}^d$, we propose a deterministic algorithm having a competitive ratio of at most ${\lfloor\frac{2}{\alpha}+2\rfloor^d}$ $\left(\lfloor\log_{2}M\rfloor+1\right)$. This improves the current best-known upper bound due to Alefkhani et al. [WAOA'23]. Then, for homothetic hypercubes in $\mathbb{R}^d$ with side lengths in the range $[1, M]$ using points in $\mathbb{Z}^d$, we propose a randomized algorithm having a competitive ratio of $O(d^2\log M)$. To complement this result, we show that no randomized algorithm can have a competitive ratio better than $\Omega(d\log M)$. This improves the current best-known (deterministic) upper and lower bound of $25^d\log M$ and $\Omega(\log M)$, respectively, due to Alefkhani et al. [WAOA'23]. Next, 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. In particular, for a fixed $M$ this result partially answers an open question for squares proposed by Khan et al. [SoCG'23] and Alefkhani et al. [WAOA'23].