Near-Linear Algorithms for Geometric Hitting Sets and Set Covers - Agarwal, Pankaj K.; Pan, Jiangwei (Discrete & computational geometry, 2020)
Given a finite range space Σ = ( X , R ) , with N = | X | + | R | , we present two simple algorithms, based on the multiplicative-weight method, for computing a small-size hitting set or set cover of Σ . The first algorithm is a simpler variant of the Brönnimann–Goodrich algorithm but more efficient to implement, and the second algorithm can be viewed as solving a two-player zero-sum game. These algorithms, in conjunction with some standard geometric data structures, lead to near-linear algorithms for computing a small-size hitting set or set cover for a number of geometric range spaces. For example, they lead to O ( N polylog ( N ) ) expected-time randomized O (1)-approximation algorithms for both hitting set and set cover if X is a set of points and R a set of disks in R 2 .