  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.
