Selecting Points that are Heavily Covered by Pseudo-Circles, Spheres or Rectangles
Combinatorics, Probability and Computing
Low-Dimensional Linear Programming with Violations
SIAM Journal on Computing
Decomposition of multiple coverings into many parts
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Restricted strip covering and the sensor cover problem
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
On The Chromatic Number of Geometric Hypergraphs
SIAM Journal on Discrete Mathematics
CJCDGCGT'05 Proceedings of the 7th China-Japan conference on Discrete geometry, combinatorics and graph theory
Decomposition of multiple coverings into many parts
Computational Geometry: Theory and Applications
Decomposition of multiple coverings into more parts
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Notes: Coloring axis-parallel rectangles
Journal of Combinatorial Theory Series A
Conflict-Free coloring made stronger
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
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Given a set of points in R2 or R3, we aim to color them such that every region of a certain family (for instance disks) containing at least a certain number of points contains points of many different colors. Using k colors, it is not always possible to ensure that every region containing k points contains all k colors. Thus, we introduce two relaxations: either we allow the number of colors to increase to c(k), or we require that the number of points in each region increases to p(k). We give upper bounds on c(k) and p(k) for halfspaces, disks, and pseudodisks. We also consider the dual question, where we want to color regions instead of points. This is related to previous results of Pach, Tardos and Tóth on decompositions of coverings.