Optimal algorithms for approximate clustering
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Learnability and the Vapnik-Chervonenkis dimension
Journal of the ACM (JACM)
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
On arrangements of Jordan arcs with three intersections per pair
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
How to net a lot with little: small &egr;-nets for disks and halfspaces
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Almost tight bounds for &egr;-nets
Discrete & Computational Geometry
Reporting points in halfspaces
Computational Geometry: Theory and Applications
Efficient probabilistically checkable proofs and applications to approximations
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Fat Triangles Determine Linearly Many Holes
SIAM Journal on Computing
Approximations and optimal geometric divide-and-conquer
Selected papers of the 23rd annual ACM symposium on Theory of computing
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Computing Many Faces in Arrangements of Lines and Segments
SIAM Journal on Computing
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
On the union of k-curved objects
Computational Geometry: Theory and Applications
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Introduction to Algorithms
Lectures on Discrete Geometry
On the Boundary Complexity of the Union of Fat Triangles
SIAM Journal on Computing
Algorithms for Polytope Covering and Approximation
WADS '93 Proceedings of the Third Workshop on Algorithms and Data Structures
The Complexity of the Union of $(\alpha,\beta)$-Covered Objects
SIAM Journal on Computing
Improved Approximation Algorithms for Geometric Set Cover
Discrete & Computational Geometry
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Proceedings of the twenty-fourth annual symposium on Computational geometry
Improved Bounds on the Union Complexity of Fat Objects
Discrete & Computational Geometry
Efficient Colored Orthogonal Range Counting
SIAM Journal on Computing
Epsilon nets and union complexity
Proceedings of the twenty-fifth annual symposium on Computational geometry
Epsilon nets and union complexity
Proceedings of the twenty-fifth annual symposium on Computational geometry
Near-linear approximation algorithms for geometric hitting sets
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
Weighted geometric set cover via quasi-uniform sampling
Proceedings of the forty-second ACM symposium on Theory of computing
A note about weak ε-nets for axis-parallel boxes in d-space
Information Processing Letters
Exact and approximation algorithms for geometric and capacitated set cover problems
COCOON'10 Proceedings of the 16th annual international conference on Computing and combinatorics
Weighted geometric set multi-cover via quasi-uniform sampling
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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We show the existence of ε-nets of size O(1/ε log log 1/ε) for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane with "fat" triangular ranges, and for point sets in reals3 and axis-parallel boxes; these are the first known non-trivial bounds for these range spaces. Our technique also yields improved bounds on the size of ε-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of ε-nets of size O(1/ε log log log 1/ε) for the dual range space of "fat" regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Bronnimann and Goodrich, we obtain improved approximation factors (computable in randomized polynomial time) for the hitting set or the set cover problems associated with the corresponding range spaces.