Fast Static and Dynamic Approximation Algorithms for Geometric Optimization Problems: Piercing, Independent Set, Vertex Cover, and Matching

Abstract

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 $O(n^{d/2}\mathrm{polylog}n)$ 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 $O(\sqrt{n}\mathrm{polylog}n)$ 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 $(1+\epsilon )$-approximation algorithms for MVC and MCM with $O(\mathrm{polylog}n)$ amortized update time.

This paper includes the results on vertex cover and matching from our previous arXiv submission (arXiv:2402.07441), along with new results on piercing and independent sets. Abstract shortened to meet arXiv limit