# Near-Linear Algorithms for Geometric Hitting Sets and Set Covers

## Abstract

Given a finite range space $$\Sigma =(\mathsf {X},\mathcal {R})$$, with $$N= |\mathsf {X}| + |\mathcal {R}|$$, 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 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 $$O(N\mathrm {polylog}(N))$$ expected-time randomized O(1)-approximation algorithms for both hitting set and set cover if $$\mathsf {X}$$ is a set of points and $$\mathcal {R}$$ a set of disks in $$\mathbb {R}^2$$.

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1. 1.

The VC-dimension of a range space $$\Sigma = (\mathsf {X}, \mathcal {R})$$ is the size of the largest subset $$A \subseteq X$$ such that $$|\{A\cap R \mid R \in \mathcal {R}\}| = 2^{|A|}$$.

2. 2.

Throughout this paper, we use $$\log x$$ to denote $$\log _2 x$$.

3. 3.

Using a sweep-plane technique, the running time of the algorithm for (P5) can be improved by a logarithmic factor. However, since this step is not the bottleneck, we use the above approach, which is simpler.

4. 4.

Recall that k is an integer satisfying $$k/2 < \kappa \le k$$.

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## Acknowledgements

The authors thank Kamesh Munagala for useful discussions and the two reviewers for their helpful comments.

## Author information

Correspondence to Jiangwei Pan.