Conference Paper
Online ConflictFree Colorings for Hypergraphs.
DOI: 10.1007/9783540734208_21 Conference: Automata, Languages and Programming, 34th International Colloquium, ICALP 2007, Wroclaw, Poland, July 913, 2007, Proceedings
Source: DBLP

Article: ConflictFree Coloring Made Stronger
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ABSTRACT: In FOCS 2002, Even et al. showed that any set of $n$ discs in the plane can be ConflictFree colored with a total of at most $O(\log n)$ colors. That is, it can be colored with $O(\log n)$ colors such that for any (covered) point $p$ there is some disc whose color is distinct from all other colors of discs containing $p$. They also showed that this bound is asymptotically tight. In this paper we prove the following stronger results: \begin{enumerate} \item [(i)] Any set of $n$ discs in the plane can be colored with a total of at most $O(k \log n)$ colors such that (a) for any point $p$ that is covered by at least $k$ discs, there are at least $k$ distinct discs each of which is colored by a color distinct from all other discs containing $p$ and (b) for any point $p$ covered by at most $k$ discs, all discs covering $p$ are colored distinctively. We call such a coloring a {\em $k$Strong ConflictFree} coloring. We extend this result to pseudodiscs and arbitrary regions with linear unioncomplexity. \item [(ii)] More generally, for families of $n$ simple closed Jordan regions with unioncomplexity bounded by $O(n^{1+\alpha})$, we prove that there exists a $k$Strong ConflictFree coloring with at most $O(k n^\alpha)$ colors. \item [(iii)] We prove that any set of $n$ axisparallel rectangles can be $k$Strong ConflictFree colored with at most $O(k \log^2 n)$ colors. \item [(iv)] We provide a general framework for $k$Strong ConflictFree coloring arbitrary hypergraphs. This framework relates the notion of $k$Strong ConflictFree coloring and the recently studied notion of $k$colorful coloring. \end{enumerate} All of our proofs are constructive. That is, there exist polynomial time algorithms for computing such colorings.06/2010;  [Show abstract] [Hide abstract]
ABSTRACT: We consider the kstrong conflictfree coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring has to be conflictfree, in the sense that in every interval I there are at least k colors each appearing exactly once in I. In this paper, we present a polynomial algorithm for the general problem; the algorithm has an approximation factor 52/k when k\geq2 and approximation factor 2 for k=1. In the special case the family contains all the possible intervals on the given set of points, we show that a 2 approximation algorithm exists, for any k\geq1.Algorithmica 05/2012; · 0.49 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Let $H=(V,E)$ be a hypergraph. A {\em conflictfree} coloring of $H$ is an assignment of colors to $V$ such that in each hyperedge $e \in E$ there is at least one uniquelycolored vertex. This notion is an extension of the classical graph coloring. Such colorings arise in the context of frequency assignment to cellular antennae, in battery consumption aspects of sensor networks, in RFID protocols and several other fields, and has been the focus of many recent research papers. In this paper, we survey this notion and its combinatorial and algorithmic aspects.Computing Research Repository  CORR. 05/2010;
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