Let H = (V, E) be a hypergraph. A conflict-free coloring of H is an assignment of colors to V such that in each hyperedge e ∈ E there is at least one uniquely-colored 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.

A coloring of the vertices of a hypergraph H is called conflict-free if each hyperedge E of H contains a vertex of “unique ” color that does not get repeated in E. The smallest number of colors required for such a coloring is called the conflict-free chromatic number of H, and is denoted by χCF(H). This parameter was first introduced by Even et al. (FOCS 2002) in a geometric setting, in connection with frequency assignment problems for cellular networks. Here we analyze this notion for general hypergraphs. It is shown that χCF(H) ≤ 1/2 + √ 2m + 1/4, for every hypergraph with m edges, and that this bound is tight. Better bounds of the order of m 1/t log m are proved under the assumption that the size of every edge of H is at least 2t − 1, for some t ≥ 3. Using Lovász’s Local Lemma, the same result holds for hypergraphs, in which the size of every edge is at least 2t − 1 and every edge intersects at most m others. We give efficient polynomial time algorithms to obtain such colorings. Our machinery can also be applied to the hypergraphs induced by the neighborhoods of the vertices of a graph. It turns out that in this case we need much fewer colors. For example, it is shown that the vertices of any graph G with maximum degree ∆ can be colored with log 2+ǫ ∆ colors, so that the neighborhood of every vertex contains a point of “unique ” color. We give an efficient deterministic algorithm to find such a coloring, based on a randomized algorithmic version of the Lovász Local Lemma, suggested by Beck, Molloy and Reed. To achieve this, we need (1) to correct a small error in the Molloy-Reed approach; (2) to restate and reprove their result in a deterministic form.

Abstract. A conflict-free coloring for a given set of disks is a coloring of the disks such that for any point p on the plane there is a disk among the disks covering p having a color different from that of the rest of the disks that covers p. In the dynamic offline setting, a sequence of disks is given, we have to color the disks one-by-one according to the order of the sequence and maintain the conflict-free property at any time for the disks that are colored. This paper focuses on unit disks, i.e., disks with radius one. We give an algorithm that colors a sequence of n unit disks in the dynamic offline setting using O(log n) colors. The algorithm is asymptotically optimal because Ω(log n) colors is necessary to color some set of n unit disks for any value of n [9]. 1

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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(n0.368) with respect to orthogonal ranges in two dimensions (i.e., axis-parallel 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 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(n0.368)-approximation algorithm for this problem with respect to orthogonal ranges in the plane. 1

conflict-free colorings of point sets and simple regions