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

Chan, Timothy M.

doi:10.1145/2261250.2261293

Cited by 11 publications

(10 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The spirit of our technique is in some sense in a similar flavor to Chan's algorithm [17] which computes a conflict-free coloring of d-dimensional points (w.r.t. rectangle ranges) using O(n 1−0.632/(2 d−3 −0.368) ) colors.…”

Section: Our Results and Techniquesmentioning

confidence: 99%

Geometric Pricing: How Low Dimensionality Helps in Approximability

Chalermsook,

Elbassioni,

Nanongkai

et al. 2012

Preprint

View full text Add to dashboard Cite

Consider the following toy problem. There are m rectangles and n points on the plane. Each rectangle R is a consumer with budget B R , who is interested in purchasing the cheapest item (point) inside R, given that she has enough budget. Our job is to price the items to maximize the revenue. This problem can also be defined on higher dimensions. We call this problem the geometric pricing problem.In high dimensions, the above problem is equivalent to the unlimited-supply profit-maximizing pricing problem, which has been studied extensively in approximation algorithms and algorithmic game theory communities. Previous studies suggest that the latter problem is too general to obtain a sub-linear approximation ratio (in terms of the number of items) even when the consumers are restricted to have very simple purchase strategies.

show abstract

Section: Our Results and Techniquesmentioning

confidence: 99%

Geometric Pricing: How Low Dimensionality Helps in Approximability

Chalermsook,

Elbassioni,

Nanongkai

et al. 2012

Preprint

View full text Add to dashboard Cite

show abstract

“…Considering the family of axis-parallel rectangles, recall that the Delaunaygraph in this case is not necessarily planar. Estimating the chromatic number of the Delaunay-graph in this case is a famous open problem with a large gap between the best known lower bound of Ω( log n log log n ) [6] and the best known upper bound of O(n 0.368 ) [5] (where n is the size of S). There is however a proper O(log n)-coloring with respect to rectangles that contain at least 3 points from S [2].…”

Section: Related Workmentioning

confidence: 99%

Coloring Delaunay-Edges and their Generalizations

Ackerman¹,

Keszegh²,

Pálvölgyi³

2018

Preprint

View full text Add to dashboard Cite

We consider geometric hypergraphs whose vertex set is a finite set of points (e.g., in the plane), and whose hyperedges are the intersections of this set with a family of geometric regions (e.g., axis-parallel rectangles). A typical coloring problem for such geometric hypergraphs asks, given an integer k, for the existence of an integer m = m(k), such that every set of points can be k-colored such that every hyperedge of size at least m contains points of different (or all k) colors. We generalize this notion by introducing coloring of (unordered) t-tuples of points such that every hyperedge that contains enough points contains t-tuples of different (or all) colors. In particular, we consider all t-tuples and t-tuples that are themselves hyperedges. The latter, with t = 2, is equivalent to coloring the edges of the so-called Delaunay-graph.In this paper we study colorings of Delaunay-edges with respect to halfplanes, pseudo-disks, axis-parallel and bottomless rectangles, and also discuss colorings of t-tuples of geometric and abstract hypergraphs, and connections between the standard coloring of vertices and coloring of t-tuples of vertices.

show abstract

“…Since then, the notion of CF-coloring has attracted significant attention from researchers, and its different aspects and variants have been studied in dozens of papers. See, e.g., [2,6,9,10,20,25,43] for a sample of various aspects of CF-coloring, and the survey [54] and the references therein for more on CF-colorings and their applied and theoretical aspects.…”

Section: Introduction 1backgroundmentioning

confidence: 99%

Conflict-Free Coloring of String Graphs

Keller

Rok

Smorodinsky

2020

Discrete Comput Geom

View full text Add to dashboard Cite

Conflict-free coloring (in short, CF-coloring) of a graph G = (V, E) is a coloring of V such that the neighborhood of each vertex contains a vertex whose color differs from the color of any other vertex in that neighborhood. Bounds on CF-chromatic numbers have been studied both for general graphs and for intersection graphs of geometric shapes. In this paper we obtain such bounds for several classes of string graphs, i.e., intersection graphs of curves in the plane:(i) We provide a general upper bound of O(χ(G) 2 log n) on the CF-chromatic number of any string graph G with n vertices in terms of the classical chromatic number χ(G). This result stands in contrast to general graphs where the CF-chromatic number can be Ω( √ n) already for bipartite graphs.(ii) For some central classes of string graphs, the CF-chromatic number is as large as Θ( √ n),which was shown to be the upper bound for any graph even in the non-geometric context. For several such classes (e.g., intersection graphs of frames) we prove a tight bound of Θ(log n) with respect to the relaxed notion of k-CF-coloring (in which the punctured neighborhood of each vertex contains a color that appears at most k times), for a small constant k.(iii) We obtain a general upper bound on the k-CF-chromatic number of arbitrary hypergraphs: Any hypergraph with m hyperedges can be k-CF colored withÕ(m 1 k+1 ) colors. This bound, which extends a bound of Pach and Tardos (2009), is tight for some string graphs, up to a logarithmic factor.(iv) Our fourth result concerns circle graphs in which coloring problems are motivated by VLSI designs. We prove a tight bound of Θ(log n) on the CF-chromatic number of circle graphs, and an upper bound of O(log 3 n) for a wider class of string graphs that contains circle graphs, namely, intersection graphs of grounded L-shapes.

show abstract

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

Cited by 11 publications

References 36 publications

Geometric Pricing: How Low Dimensionality Helps in Approximability

Geometric Pricing: How Low Dimensionality Helps in Approximability

Coloring Delaunay-Edges and their Generalizations

Conflict-Free Coloring of String Graphs

Contact Info

Product

Resources

About