Fast Static and Dynamic Approximation Algorithms for Geometric Optimization Problems: Piercing, Independent Set, Vertex Cover, and Matching
Sujoy Bhore, Timothy M. Chan
TL;DR
The paper addresses four fundamental geometric optimization problems—Minimum Piercing Set, Maximum Independent Set, Minimum Vertex Cover, and Maximum-Cardinality Matching—through near-linear-time static and dynamic approximation algorithms. It introduces two unifying approaches: a rounding/divide-and-conquer framework for MPS and MIS that yields fast static and dynamic results, and a MWU/kernelization-based framework for MVC and MCM that enables efficient dynamic maintenance via small cores and phase-based rebuilds. The results include near-linear static algorithms and fully dynamic $(1+ε)$-approximations for disks, rectangles, and fat objects across constant dimensions, with polylog update times in many cases and extensions to bipartite settings. These techniques bridge classic combinatorial optimization with dynamic geometric data structures, offering practical tools for large-scale, evolving geometric datasets and informing avenues for future improvements and broader object families.
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 $O(n^{1+δ})$ time for an arbitrarily small constant $δ>0$. 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 $O(n^δ)$ 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 $O(n^{1+δ})$ 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 $O(n^δ)$ 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 $O(n^δ)$ 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.