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
Coloring graphs with sparse neighborhoods
Journal of Combinatorial Theory Series B
Lectures on Discrete Geometry
Using the Pseudo-Dimension to Analyze Approximation Algorithms for Integer Programming
WADS '01 Proceedings of the 7th International Workshop on Algorithms and Data Structures
Polynomial-time approximation schemes for packing and piercing fat objects
Journal of Algorithms
Conflict-Free Coloring of Points and Simple Regions in the Plane
Discrete & Computational Geometry
Improved Approximation Algorithms for Geometric Set Cover
Discrete & Computational Geometry
Conflict-free coloring for rectangle ranges using O(n.382) colors
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles
Random Structures & Algorithms - Proceedings of the Thirteenth International Conference “Random Structures and Algorithms” held May 28–June 1, 2007, Tel Aviv, Israel
Maximum independent set of rectangles
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Coloring Geometric Range Spaces
Discrete & Computational Geometry
Approximation algorithms for maximum independent set of pseudo-disks
Proceedings of the twenty-fifth annual symposium on Computational geometry
Hitting sets when the VC-dimension is small
Information Processing Letters
Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Conflict-Free colorings of rectangles ranges
STACS'06 Proceedings of the 23rd Annual conference on Theoretical Aspects of Computer Science
Computing the independence number of intersection graphs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Geometric packing under non-uniform constraints
Proceedings of the twenty-eighth annual symposium on Computational geometry
On coloring points with respect to rectangles
Journal of Combinatorial Theory Series A
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In the conflict-free coloring problem, for a given range space, we want to bound the minimum value F(n) such that every set P of n points can be colored with F(n) colors with the property that every nonempty range contains a unique color. We prove a new upper bound O(n0.368) with respect to orthogonal ranges in two dimensions (i.e., axis-parallel rectangles), which is the first improvement over the previous bound O(n0.382) by Ajwani, Elbassioni, Govindarajan, and Ray [SPAA'07]. This result leads to an O(n1-0.632/2d-2) upper bound with respect to orthogonal ranges (boxes) in dimension d, and also an O(n1-0.632/(2d-3-0.368)) upper bound with respect to dominance ranges (orthants) in dimension d ≥ 4. We also observe that combinatorial results on conflict-free coloring can be applied to the analysis of approximation algorithms for discrete versions of geometric independent set problems. Here, given a set P of (weighted) points and a set S of ranges, we want to select the largest(-weight) subset Q ⊂ P with the property that every range of S contains at most one point of Q. We obtain, for example, a randomized O(n0.368)-approximation algorithm for this problem with respect to orthogonal ranges in the plane.