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).

翻译：击中集问题是组合优化中的基本问题之一，在离线设定下已得到充分研究。我们考虑在线击中集问题，其中仅点集预先已知，而对象逐个引入。我们的目标是通过做出不可撤销的决策来维护一个最小规模的击中集。在此，我们根据点集的不同，提出了对在线击中集问题两种变体的研究。在第一种变体中，我们考虑点集为整个 $\mathbb{Z}^d$；而在第二种变体中，我们考虑点集为 $\mathbb{R}^2$ 的一个有限子集。若使用 $\mathbb{Z}^d$ 中的点来击中 $\mathbb{R}^d$ 中边长为 $[1,M]$ 区间的相似超立方体，我们证明任何算法的竞争比均为 $\Omega(d\log M)$，无论该算法是确定性的还是随机的。这通过因子 $d$ 改进了最近已知的确定性下界 $\Omega(\log M)$。接着，我们提出了一种几乎紧致的随机算法，其竞争比为 $O(d^2\log M)$，显著改进了目前已知的最佳竞争比 $25^d\log M$。随后，我们提出了一种简单的确定性 ${\lfloor\frac{2}{\alpha}+2\rfloor^d}(\lfloor\log_{2}M\rfloor+1)$ 竞争算法，用于使用 $\mathbb{Z}^d$ 中的点来击中 $\mathbb{R}^d$ 中直径在 $[1, M]$ 范围内、大小相似的 {$\alpha$-fat 对象}。这至少以 $5^d$ 的因子改进了当前已知的最佳上界。最后，我们考虑当点集由 $\mathbb{R}^2$ 中的 $n$ 个点组成，且对象是直径为 $[1, M]$ 范围的相似正 $k$ 边形时的击中集问题。我们为此提出了一种 $O(\log n\log M)$ 竞争的随机算法。而此前即使对于正方形也未有结果。特别地，我们的结果回答了 Khan 等人 (SoCG'23) 和 Alefkhani 等人 (WAOA'23) 提出的一些开放性问题。