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 $X$ 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 ${\displaystyle {\mathcal {C}}\subseteq {\mathcal {R}}}$ 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 ${\displaystyle H\subseteq X}$ 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.^[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 $O(\log n)$ 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 $\Sigma$ 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 $O(1)$ 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^[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 $O(1)$-approximate set cover in ${\displaystyle O(n\log ^{4}n)}$ time for range spaces induced by 2D disks.

See also

• Lebesgue covering dimension
• Carathéodory's extension theorem

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.
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
5. Arora, S.; Hazan, E.; Kale, S. (2012), "The Multiplicative Weights Update Method: a Meta-Algorithm and Applications", Theory of Computing
6. Brönnimann, H.; Goodrich, M. (1995), "Almost optimal set covers in finite VC-dimension", Discrete & Computational Geometry
7. Clarkson, Kenneth L. (1993-08-11). Dehne, Frank; Sack, Jörg-Rüdiger; Santoro, Nicola; et al., eds. Algorithms for polytope covering and approximation. Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 246–252. doi:10.1007/3-540-57155-8_252. ISBN 978-3-540-57155-1.