Optimal algorithms for approximate clustering
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Applications of random sampling in computational geometry, II
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Reporting points in halfspaces
Computational Geometry: Theory and Applications
Randomized algorithms
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
On the hardness of approximating minimization problems
Journal of the ACM (JACM)
Improved bounds on the sample complexity of learning
Journal of Computer and System Sciences
Approximation algorithms
Algorithms for Polytope Covering and Approximation
WADS '93 Proceedings of the Third Workshop on Algorithms and Data Structures
Using the Pseudo-Dimension to Analyze Approximation Algorithms for Integer Programming
WADS '01 Proceedings of the 7th International Workshop on Algorithms and Data Structures
Improved Approximation Algorithms for Geometric Set Cover
Discrete & Computational Geometry
Maximizing the guarded boundary of an Art Gallery is APX-complete
Computational Geometry: Theory and Applications
Coverage problems in wireless sensor networks: designs and analysis
International Journal of Sensor Networks
Epsilon nets and union complexity
Proceedings of the twenty-fifth annual symposium on Computational geometry
On the set multi-cover problem in geometric settings
Proceedings of the twenty-fifth annual symposium on Computational geometry
Hitting sets when the VC-dimension is small
Information Processing Letters
Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes
SIAM Journal on Computing
Relative (p,ε)-Approximations in Geometry
Discrete & Computational Geometry
Geometric Approximation Algorithms
Geometric Approximation Algorithms
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We consider the set multicover problem in geometric settings. Given a set of points P and a collection of geometric shapes (or sets) F, we wish to find a minimum cardinality subset of F such that each point p ∈ P is covered by (contained in) at least d(p) sets. Here, d(p) is an integer demand (requirement) for p. When the demands d(p) = 1 for all p, this is the standard set cover problem. The set cover problem in geometric settings admits an approximation ratio that is better than that for the general version. In this article, we show that similar improvements can be obtained for the multicover problem as well. In particular, we obtain an O(log opt) approximation for set systems of bounded VC-dimension, and an O(1) approximation for covering points by half-spaces in three dimensions and for some other classes of shapes.