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arXiv 2409.11166cs.CG
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在线几何命中集问题的新下界与算法

New Lower Bound and Algorithms for Online Geometric Hitting Set Problem

Minati De, Ratnadip Mandal, Satyam Singh

备注：
30 pages and 10 figures. arXiv admin note: text overlap with arXiv:2304.06780
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AI总结：

本文研究在线几何命中集问题，针对整数格点和平面有限点集两种情形，给出了改进的竞争比下界及多个更优的确定性与随机化算法，回答了若干开放问题。

摘要：

命中集问题是组合优化中的基本问题之一，在离线情形下已得到充分研究。我们考虑在线命中集问题，其中仅点集事先已知，而对象逐个出现。我们的目标是通过做出不可撤销的决策来维持一个最小规模的命中集。本文根据点集的不同，研究在线命中集问题的两种变体。在第一种变体中，我们考虑点集为整个 $\mathbb{Z}^d$；在第二种变体中，我们考虑点集为 $\mathbb{R}^2$ 的一个有限子集。如果使用 $\mathbb{Z}^d$ 中的点去命中 $\mathbb{R}^d$ 中边长在 $[1,M]$ 范围内的位似超立方体，我们证明任何算法的竞争比都是 $\Omega(d\log M)$，无论它是确定性的还是随机的。这将最近已知的确定性下界 $\Omega(\log M)$ 改进了 $d$ 倍。随后，我们提出一个几乎紧的随机化算法，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 $25^d\log M$。接着，我们提出一个简单的确定性 ${\lfloor\frac{2}{\alpha}+2\rfloor^d}(\lfloor\log_{2}M\rfloor+1)$ 竞争算法，使用 $\mathbb{Z}^d$ 中的点去命中 $\mathbb{R}^d$ 中直径在 $[1, M]$ 范围内、大小相近的 {$\alpha$-fat objects}。这将当前已知的最佳上界至少改进了 $5^d$ 倍。最后，我们考虑点集由 $\mathbb{R}^2$ 中 $n$ 个点组成、对象为直径在 $[1, M]$ 范围内的位似正 $k$ 边形时的命中集问题。我们为此提出一个 $O(\log n\log M)$ 竞争的随机化算法。而此前即使对于正方形也没有已知结果。特别地，我们的结果回答了 Khan 等人（SoCG'23）和 Alefkhani 等人（WAOA'23）提出的一些开放问题。

英文摘要：

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 $Ω(d\log M)$, whether it is deterministic or random. This improves the recently known deterministic lower bound of $Ω(\log M)$ 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 ${\lfloor\frac{2}α+2\rfloor^d}(\lfloor\log_{2}M\rfloor+1)$ competitive algorithm to hit similarly sized {$α$-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).

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