Geometric set cover problem

The geometric set cover problem is the special case of the set cover problem in geometric settings. The input is a range space $$\Sigma = (X, \mathcal{R})$$ where $$X$$ is a universe of points in $$\mathbb{R}^d$$ and $$\mathcal{R}$$ is a family of subsets of $$X$$ called ranges, defined by the intersection of $$X$$ and geometric shapes such as disks and axis-parallel rectangles. The goal is to select a minimum-size subset $$\mathcal{C} \subseteq \mathcal{R}$$ of ranges such that every point in the universe $$X$$ is covered by some range in $$\mathcal{C}$$.

Given the same range space $$\Sigma$$, a closely related problem is the geometric hitting set problem, where the goal is to select a minimum-size subset $$H \subseteq X$$ of points such that every range of $$\mathcal{R}$$ has nonempty intersection with $$H$$, i.e., is hit by $$H$$.

In the one-dimensional case, where $$X$$ contains points on the real line and $$\mathcal{R}$$ is defined by intervals, both the geometric set cover and hitting set problems can be solved in polynomial time using a simple greedy algorithm. However, in higher dimensions, they are known to be NP-complete even for simple shapes, i.e., when $$\mathcal{R}$$ is induced by unit disks or unit squares. The discrete unit disc cover problem is a geometric version of the general set cover problem which is NP-hard.

Many approximation algorithms have been devised for these problems. Due to the geometric nature, the approximation ratios for these problems can be much better than the general set cover/hitting set problems. Moreover, these approximate solutions can even be computed in near-linear time.

Approximation algorithms
The greedy algorithm for the general set cover problem gives $$O(\log n)$$ approximation, where $$n = \max\{|X|, |\mathcal{R}|\}$$. This approximation is known to be tight up to constant factor. However, in geometric settings, better approximations can be obtained. Using a multiplicative weight algorithm, Brönnimann and Goodrich showed that an $$O(\log \mathsf{OPT})$$-approximate set cover/hitting set for a range space $$\Sigma$$ with constant VC-dimension can be computed in polynomial time, where $$\mathsf{OPT} \le n$$ denotes the size of the optimal solution. The approximation ratio can be further improved to $$O(\log \log \mathsf{OPT})$$ or $$O(1)$$ when $$\mathcal{R}$$ is induced by axis-parallel rectangles or disks in $$\mathbb{R}^2$$, respectively.

Near-linear-time algorithms
Based on the iterative-reweighting technique of Clarkson and Brönnimann and Goodrich, Agarwal and Pan gave algorithms that computes an approximate set cover/hitting set of a geometric range space in $$O(n~\mathrm{polylog}(n))$$ time. For example, their algorithms computes an $$O(\log \log \mathsf{OPT})$$-approximate hitting set in $$O(n \log^3n\log\log\log \mathsf{OPT})$$ time for range spaces induced by 2D axis-parallel rectangles; and it computes an $$O(1)$$-approximate set cover in $$O(n \log^4n)$$ time for range spaces induced by 2D disks.