## Conflict-Free Colorings of Simple Geometric Regions with Applications to Frequency Assignment in Cellular Networks (2002)

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Citations: | 42 - 7 self |

### BibTeX

@MISC{Even02conflict-freecolorings,

author = {Guy Even and Zvi Lotker and Dana Ron and Shakhar Smorodinsky},

title = {Conflict-Free Colorings of Simple Geometric Regions with Applications to Frequency Assignment in Cellular Networks},

year = {2002}

}

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### Abstract

Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem that we call Minimum Conflict-Free Coloring (Min-CF-Coloring). In its general form, the input of the Min-CF-coloring problem is a set system (X, S), where each S 2 S is a subset of X . The output is a coloring of the sets in S that satisfies the following constraint: for every x 2 X there exists a color i and a unique set S 2 S, such that x 2 S and (S) = i. The goal is to minimize the number of colors used by the coloring .

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Citation Context ...garding the envelope of R ′ is that if every boundary segment is given a symbol that corresponds to the rectangle it belongs to, then the sequence of symbols is a Davenport–Schinzel sequence DS(n, 2) =-=[SA95]-=-. Namely, no two consecutive symbols are equal, and there is no alternating subsequence of length 4 (i.e., no “...a...b...a...b...” for every pair of symbols a ̸= b). As a consequence, if two rectangl... |

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Citation Context ...is problem is not easier than vertex coloring in graphs and is even equally hard to approximate. An adaptation of the NP-completeness proof of minimum coloring of intersection graphs of unit disks by =-=[CCJ90]-=- proves that even CF-coloring of unit disks (or unit squares) in the plane is NP-complete. Since this proof is based on a reduction from coloring planar graphs, it follows that approximating the minim... |

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Citation Context ... above, min-CF-coloring of general set systems is not easier (even to approximate) than vertex-coloring in graphs. The latter problem is of course known to be NP-hard, and is hard even to approximate =-=[FK98]-=-. The problem remains hard for the special case of unit disks (and squares), and it is even NP-hard to achieve an approximation ratio of 4 3 − ε for every ε>0(byan adaptation of [CCJ90]). Marathe et a... |

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Citation Context ...ed to the problem of minimizing the number of time slots required to broadcast information in a single-hop radio network. In view of this relation, it has been observed by Bar-Yehuda ([B01], based on =-=[BGI92]-=-) that every set system (X, S) can be CFmulticolored using O(log |X|·log |S|) colors. Mathematical optimization techniques have been used to solve a family of frequency assignment problems that arise ... |

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Citation Context ...of intersection graphs of unit disks. They presented an approximation algorithm with an approximation ratio of 3. Motivated by channel assignment problems in radio networks, Krumke, Marathe, and Ravi =-=[KMR01]-=- presented a 2-approximation algorithm for theCONFLICT-FREE COLORING OF SIMPLE GEOMETRIC REGIONS 99 distance-2 coloring problem in families of graphs that generalize intersection graphs of disks. A n... |

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Citation Context ...problem is related to the problem of minimizing the number of time slots required to broadcast information in a single-hop radio network. In view of this relation, it has been observed by Bar-Yehuda (=-=[B01]-=-, based on [BGI92]) that every set system (X, S) can be CFmulticolored using O(log |X|·log |S|) colors. Mathematical optimization techniques have been used to solve a family of frequency assignment pr... |

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Citation Context ...deled in this fashion because CF-coloring allows for conflicts between base-stations, provided that another base-station serves the “area of conflict.” Even models that use nonbinary constraints (see =-=[DBJC98]-=-) do not capture CF-coloring. We note that the above models take into account interferences between close frequencies, while we have ignored this issue for the sake of simplicity. We can, however, inc... |

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Citation Context ...Let R⊆2X denote the set of ranges obtained by intersecting X with all scaled translations of C. Then there exists a CF-coloring of X with respect to R using O(log |X|) colors. Recently, Pach and Tóth =-=[PT03]-=- proved that Ω(log |X|) colors are required for CF-coloring every set X of points in the plane with respect to disks. 1.2.2. CF-coloring of chains. A chain S is a collection of subsets, each assigned ... |