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

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

In the one-dimensional case, where ${\displaystyle X}$ contains points on the real line and ${\displaystyle {\mathcal {R}}}$ 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 ${\displaystyle {\mathcal {R}}}$ is induced by unit disks or unit squares.[1] The discrete unit disc cover problem is a geometric version of the general set cover problem which is NP-hard.[2]

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.[3]

## Approximation algorithms

The greedy algorithm for the general set cover problem gives ${\displaystyle O(\log n)}$ approximation, where ${\displaystyle n=\max\{|X|,|{\mathcal {R}}|\}}$. This approximation is known to be tight up to constant factor.[4] However, in geometric settings, better approximations can be obtained. Using a multiplicative weight algorithm,[5] Brönnimann and Goodrich[6] showed that an ${\displaystyle O(\log {\mathsf {OPT}})}$-approximate set cover/hitting set for a range space ${\displaystyle \Sigma }$ with constant VC-dimension can be computed in polynomial time, where ${\displaystyle {\mathsf {OPT}}\leq n}$ denotes the size of the optimal solution. The approximation ratio can be further improved to ${\displaystyle O(\log \log {\mathsf {OPT}})}$ or ${\displaystyle O(1)}$ when ${\displaystyle {\mathcal {R}}}$ is induced by axis-parallel rectangles or disks in ${\displaystyle \mathbb {R} ^{2}}$, respectively.

## Near-linear-time algorithms

Based on the iterative-reweighting technique of Clarkson[7] and Brönnimann and Goodrich,[6] Agarwal and Pan[3] gave algorithms that computes an approximate set cover/hitting set of a geometric range space in ${\displaystyle O(n~\mathrm {polylog} (n))}$ time. For example, their algorithms computes an ${\displaystyle O(\log \log {\mathsf {OPT}})}$-approximate hitting set in ${\displaystyle O(n\log ^{3}n\log \log \log {\mathsf {OPT}})}$ time for range spaces induced by 2D axis-parallel rectangles; and it computes an ${\displaystyle O(1)}$-approximate set cover in ${\displaystyle O(n\log ^{4}n)}$ time for range spaces induced by 2D disks.

## References

1. Fowler, R.J.; Paterson, M.S.; Tanimoto, S.L. (1981), "Optimal packing and covering in the plane are NP-complete", Inf. Process. Lett., 12 (3): 133–137, doi:10.1016/0020-0190(81)90111-3
2. https://cs.uwaterloo.ca/~alopez-o/files/OtDUDCP_2011.pdf On the Discrete Unit Disk Cover Problem
3. Agarwal, Pankaj K.; Pan, Jiangwei (2014). "Near-Linear Algorithms for Geometric Hitting Sets and Set Covers". Proceedings of the thirtieth annual symposium on Computational Geometry.
4. Feige, Uriel (1998), "A threshold of ln n for approximating set cover", Journal of the ACM, 45 (4): 634–652, CiteSeerX 10.1.1.70.5014, doi:10.1145/285055.285059
5. Arora, S.; Hazan, E.; Kale, S. (2012), "The Multiplicative Weights Update Method: a Meta-Algorithm and Applications", Theory of Computing, 8: 121–164, doi:10.4086/toc.2012.v008a006
6. Brönnimann, H.; Goodrich, M. (1995), "Almost optimal set covers in finite VC-dimension", Discrete & Computational Geometry, 14 (4): 463–479, doi:10.1007/bf02570718
7. Clarkson, Kenneth L. (1993-08-11). "Algorithms for polytope covering and approximation". In Dehne, Frank; Sack, Jörg-Rüdiger; Santoro, Nicola; et al. (eds.). Algorithms and Data Structures. Lecture Notes in Computer Science. 709. Springer Berlin Heidelberg. pp. 246–252. doi:10.1007/3-540-57155-8_252. ISBN 978-3-540-57155-1.
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