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

TL;DR

This paper introduces faster static and dynamic approximation algorithms for key geometric optimization problems like piercing, independent set, vertex cover, and matching, improving efficiency and extending dynamic capabilities.

Contribution

The paper presents near-linear time approximation algorithms and fully dynamic data structures for several geometric problems, significantly improving previous results in speed and dynamic updates.

Findings

O(log log n)-approximation for MPS with near-linear time

Fully dynamic algorithms for MIS, MVC, and MCM with polylogarithmic update time

Improved algorithms for rectangles and fat objects in geometric spaces

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 $O(n^{d/2}\mathop{\rm polylog} n)$ 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.