Conflict-Free Coloring and its Applications

Smorodinsky, Shakhar

doi:10.1007/978-3-642-41498-5_12

Cited by 58 publications

(60 citation statements)

References 62 publications

(124 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, the coloring of geometric shapes in the plane is related to the problems of cover-decomposability, conflict-free colorings and ǫ-nets; these problems have applications in sensor networks and frequency assignment as well as other areas. For surveys on these and related problems see [19,26].…”

Section: Introductionmentioning

confidence: 99%

An Abstract Approach to Polychromatic Coloring: Shallow Hitting Sets in ABA-free Hypergraphs and Pseudohalfplanes

Keszegh

Pálvölgyi

2016

Lecture Notes in Computer Science

View full text Add to dashboard Cite

The goal of this paper is to give a new, abstract approach to cover-decomposition and polychromatic colorings using hypergraphs on ordered vertex sets. We introduce an abstract version of a framework by Smorodinsky and Yuditsky, used for polychromatic coloring halfplanes, and apply it to so-called ABA-free hypergraphs, which are a generalization of interval graphs. Using our methods, we prove that (2k − 1)-uniform ABA-free hypergraphs have a polychromatic k-coloring, a problem posed by the second author. We also prove the same for hypergraphs defined on a point set by pseudohalfplanes. These results are best possible. We could only prove slightly weaker results for dual hypergraphs defined by pseudohalfplanes, and for hypergraphs defined by pseudohalfspheres. We also introduce another new notion that seems to be important for investigating polychromatic colorings and ǫ-nets, shallow hitting sets. We show that all the above hypergraphs have shallow hitting sets, if their hyperedges are containment-free.

show abstract

Section: Introductionmentioning

confidence: 99%

An Abstract Approach to Polychromatic Coloring: Shallow Hitting Sets in ABA-free Hypergraphs and Pseudohalfplanes

Keszegh

Pálvölgyi

2016

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…For a recent survey on the problem and its applications, we refer to [16]. The conflict-free coloring problem can be formally described as follows.…”

Section: Introductionmentioning

confidence: 99%

Conflict-Free Coloring for Rectangle Ranges Using O(n .382) Colors

Ajwani

Elbassioni

Govindarajan

et al. 2012

Discrete Comput Geom

View full text Add to dashboard Cite

Given a set of points P ⊆ R 2 , a conflict-free coloring of P is an assignment of colors to points of P , such that each non-empty axis-parallel rectangle T in the plane contains a point whose color is distinct from all other points in P ∩ T. This notion has been the subject of recent interest, and is motivated by frequency assignment in wireless cellular networks: one naturally would like to minimize the number of frequencies (colors) assigned to bases stations (points), such that within any range (for instance, rectangle), there is no interference. We show that any set of n points in R 2 can be conflict-free colored with˜Owith˜ with˜O(n .382+) colors in expected polynomial time, for any arbitrarily small > 0. This improves upon the previously known bound of O(p n log log n/ log n).

show abstract

“…Many results on CF-coloring are known in the literature [25]. For example, there is an O(log n) upper bound in the dual for disks w.r.t.…”

Section: Main Results On Points Wrt Rectangles In Two Dimensionsmentioning

confidence: 99%

Conflict-free coloring of points with respect to rectangles and approximation algorithms for discrete independent set

Chan

2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

View full text Add to dashboard Cite

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(n 0.368 ) with respect to orthogonal ranges in two dimensions (i.e., axis-parallel rectangles), which is the first improvement over the previous bound O(n 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(n 0.368 )-approximation algorithm for this problem with respect to orthogonal ranges in the plane.

show abstract

Conflict-Free Coloring and its Applications

Cited by 58 publications

References 62 publications

An Abstract Approach to Polychromatic Coloring: Shallow Hitting Sets in ABA-free Hypergraphs and Pseudohalfplanes

An Abstract Approach to Polychromatic Coloring: Shallow Hitting Sets in ABA-free Hypergraphs and Pseudohalfplanes

Conflict-Free Coloring for Rectangle Ranges Using O(n .382) Colors

Conflict-free coloring of points with respect to rectangles and approximation algorithms for discrete independent set

Contact Info

Product

Resources

About