We develop simple and general techniques to obtain faster (near-linear time) static approximation algorithms, as well as efficient dynamic data structures, for four fundamental geometric optimization problems: minimum piercing set (MPS), maximum independent set (MIS), minimum vertex cover (MVC), and maximum-cardinality matching (MCM). Highlights of our results include the following: * For $n$ axis-aligned boxes in any constant dimension $d$, we give an $O(\log \log n)$-approximation algorithm for MPS that runs in $O(n^{1+\delta})$ time for an arbitrarily small constant $\delta>0$. This significantly improves the previous $O(\log\log n)$-approximation algorithm by Agarwal, Har-Peled, Raychaudhury, and Sintos (SODA~2024), which ran in $O(n^{d/2}\mathop{\rm polylog} n)$ time. * Furthermore, we show that our algorithm can be made fully dynamic with $O(n^{\delta})$ amortized update time. Previously, Agarwal et al.~(SODA~2024) obtained dynamic results only in $\mathbb{R}^2$ and achieved only $O(\sqrt{n}\mathop{\rm polylog} n)$ amortized expected update time. * For $n$ axis-aligned rectangles in $\mathbb{R}^2$, we give an $O(1)$-approximation algorithm for MIS that runs in $O(n^{1+\delta})$ time. Our result significantly improves the running time of the celebrated algorithm by Mitchell (FOCS~2021) (which was about $O(n^{21})$), and answers one of his open questions. Our algorithm can also be made fully dynamic with $O(n^{\delta})$ amortized update time. * For $n$ (unweighted or weighted) fat objects in any constant dimension, we give a dynamic $O(1)$-approximation algorithm for MIS with $O(n^{\delta})$ amortized update time. * For disks in $\mathbb{R}^2$ or hypercubes in any constant dimension, we give the first fully dynamic $(1+\varepsilon)$-approximation algorithms for MVC and MCM with $O(\mathop{\rm polylog}n)$ amortized update time.

翻译：针对四个基础几何优化问题——最小覆盖集（MPS）、最大独立集（MIS）、最小顶点覆盖（MVC）和最大基数匹配（MCM），我们开发了简洁通用的技术，以获得更快的（近线性时间）静态近似算法以及高效的动态数据结构。我们的主要成果包括：* 对于任意常数维度 $d$ 中的 $n$ 个轴对齐包围盒，我们提出了一种 $O(\log \log n)$ 近似度的 MPS 算法，其运行时间为 $O(n^{1+\delta})$，其中 $\delta>0$ 为任意小常数。这显著改进了 Agarwal、Har-Peled、Raychaudhury 和 Sintos（SODA~2024）先前提出的 $O(\log\log n)$ 近似算法，该算法的运行时间为 $O(n^{d/2}\mathop{\rm polylog} n)$。* 此外，我们证明该算法可实现完全动态化，且具有 $O(n^{\delta})$ 的摊销更新时间。此前，Agarwal 等人（SODA~2024）仅在 $\mathbb{R}^2$ 中获得了动态结果，且仅实现了 $O(\sqrt{n}\mathop{\rm polylog} n)$ 的摊销期望更新时间。* 对于 $\mathbb{R}^2$ 中的 $n$ 个轴对齐矩形，我们提出了一种 $O(1)$ 近似度的 MIS 算法，其运行时间为 $O(n^{1+\delta})$。我们的结果显著改进了 Mitchell（FOCS~2021）著名算法的运行时间（原约为 $O(n^{21})$），并解答了他提出的一个开放性问题。该算法同样可实现完全动态化，且具有 $O(n^{\delta})$ 的摊销更新时间。* 对于任意常数维度中的 $n$ 个（未加权或加权）胖体对象，我们提出了一种动态的 $O(1)$ 近似度 MIS 算法，其摊销更新时间为 $O(n^{\delta})$。* 对于 $\mathbb{R}^2$ 中的圆盘或任意常数维度中的超立方体，我们首次提出了完全动态的 $(1+\varepsilon)$ 近似度 MVC 和 MCM 算法，其摊销更新时间为 $O(\mathop{\rm polylog}n)$。