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ConflictFree Coloring of Points with Respect to Rectangles and Approximation Algorithms for Discrete Independent Set
, 2012
"... In the conflictfree 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 orthog ..."
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In the conflictfree 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., axisparallel 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 conflictfree 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. 1
On ConflictFree MultiColoring?
"... Abstract A conflictfree coloring of a hypergraph H = (V, E), E ⊆ 2V, is a coloring of the vertices V such that every hyperedge E ∈ E contains a vertex of “unique ” color. Our goal is to minimize the total number of distinct colors. In its full generality, this problem is known as the conflictfree ..."
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Abstract A conflictfree coloring of a hypergraph H = (V, E), E ⊆ 2V, is a coloring of the vertices V such that every hyperedge E ∈ E contains a vertex of “unique ” color. Our goal is to minimize the total number of distinct colors. In its full generality, this problem is known as the conflictfree (hypergraph) coloring problem. It is known that Θ( m) colors might be needed in general. In this paper we study the relaxation of the problem where one is allowed to assign multiple colors to the same node. The goal here is to substantially reduce the total number of colors, while keeping the number of colors per node as small as possible. By a simple adaptation of a result by Pach and Tardos [2009] on the singlecolor version of the problem, one obtains that only O(log2m) colors in total are sufficient (on every instance) if each node is allowed to use up to O(logm) colors. By improving on the result of Pach and Tardos (under the assumption n m), we show that the same result can be achieved with O(logm·logn) colors in total, and eitherO(logm) orO(logn·log logm) ⊆ O(log2 n) colors per node. The latter coloring can be computed by a polynomialtime Las Vegas algorithm. 1
On Vertex Rankings of Graphs and its Relatives
"... A vertex ranking of a graph is an assignment of ranks (or colors) to the vertices of the graph, in such a way that any simple path connecting two vertices of equal rank, must contain a vertex of a higher rank. In this paper we study a relaxation of this notion, in which the requirement above should ..."
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A vertex ranking of a graph is an assignment of ranks (or colors) to the vertices of the graph, in such a way that any simple path connecting two vertices of equal rank, must contain a vertex of a higher rank. In this paper we study a relaxation of this notion, in which the requirement above should only hold for paths of some bounded length l for some fixed l. For instance, already the case l = 2 exhibits quite a different behavior than proper coloring. We prove upper and lower bounds on the minimum number of ranks required for several graph families, such as trees, planar graphs, graphs excluding a fixed minor and degenerate graphs. 1
Kinetic ConflictFree Coloring∗
"... A conflictfree coloring, or CFcoloring for short, of a set P of points in the plane with respect to disks is a coloring of the points of P with the following property: for any disk D containing at least one point of P there is a point p ∈ P ∩D so that no other point q ∈ P ∩D has the same color as ..."
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A conflictfree coloring, or CFcoloring for short, of a set P of points in the plane with respect to disks is a coloring of the points of P with the following property: for any disk D containing at least one point of P there is a point p ∈ P ∩D so that no other point q ∈ P ∩D has the same color as p. In this paper we study the problem of maintaining such a CFcoloring when the points in P move. We present two methods for this and evaluate the maximum number of colors used as well as the number of recolorings, both in theory and experimentally. 1