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

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(n¹+^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 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 R² and achieved only O(√n polylog n) amortized expected update time. • For n axis-aligned rectangles in R², we give an O(1)-approximation algorithm for MIS that runs in O(n¹+^δ) time. Our result significantly improves the running time of the celebrated algorithm by Mitchell (FOCS 2021) (which was about O(n²¹)), 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. Previously, Bhore, Nöllenburg, Tóth, and Wulms (SoCG 2024) obtained efficient dynamic O(1)-approximation algorithms only for disks in R² and only in the unweighted setting. • For n axis-aligned rectangles in R², we give a dynamic (₂³ + ε)-approximation algorithm for MVC with O(polylog n) amortized update time for any constant ε > 0. Our static result improves the running time of Bar-Yehuda, Hermelin, and Rawitz (2011). For disks in R² or hypercubes in any constant dimension, we give the first fully dynamic (1 + ε)-approximation algorithm for MVC with O(polylog n) amortized update time. • For (monochromatic or bichromatic) disks in R² or hypercubes in any constant dimension, we give the first fully dynamic (1 + ε)-approximation algorithm for MCM with O(polylog n) amortized update time.