Near-Linear Algorithms for Geometric Hitting Sets and Set Covers

Given a finite range space Sigma = (X, R), with N = vertical bar X vertical bar+vertical bar R vertical bar, we present two simple algorithms, based on the multiplicative-weight method, for computing a small-size hitting set or set cover of Sigma. The first algorithm is a simpler variant of the BronnimannGoodrich 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( Npolylog( 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.