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 where is a universe of points in and is a family of subsets of called ranges, defined by the intersection of and geometric shapes such as disks and axis-parallel rectangles. The goal is to select a minimum-size subset of ranges such that every point in the universe is covered by some range in .
Given the same range space , a closely related problem is the geometric hitting set problem, where the goal is to select a minimum-size subset of points such that every range of has nonempty intersection with , i.e., is hit by .
In the one-dimensional case, where contains points on the real line and 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 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.
The greedy algorithm for the general set cover problem gives approximation, where . 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 -approximate set cover/hitting set for a range space with constant VC-dimension can be computed in polynomial time, where denotes the size of the optimal solution. The approximation ratio can be further improved to or when is induced by axis-parallel rectangles or disks in , respectively.
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 time. For example, their algorithms computes an -approximate hitting set in time for range spaces induced by 2D axis-parallel rectangles; and it computes an -approximate set cover in time for range spaces induced by 2D disks.
- 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
- https://cs.uwaterloo.ca/~alopez-o/files/OtDUDCP_2011.pdf On the Discrete Unit Disk Cover Problem
- 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.
- 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
- 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
- Brönnimann, H.; Goodrich, M. (1995), "Almost optimal set covers in finite VC-dimension", Discrete & Computational Geometry, 14 (4): 463–479, doi:10.1007/bf02570718
- 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.