1,547 research outputs found

    Conflict-Free Coloring of Intersection Graphs of Geometric Objects

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    In FOCS'2002, Even et al. introduced and studied the notion of conflict-free colorings of geometrically defined hypergraphs. They motivated it by frequency assignment problems in cellular networks. This notion has been extensively studied since then. A conflict-free coloring of a graph is a coloring of its vertices such that the neighborhood (pointed or closed) of each vertex contains a vertex whose color differs from the colors of all other vertices in that neighborhood. In this paper we study conflict-colorings of intersection graphs of geometric objects. We show that any intersection graph of n pseudo-discs in the plane admits a conflict-free coloring with O(\log n) colors, with respect to both closed and pointed neighborhoods. We also show that the latter bound is asymptotically sharp. Using our methods, we also obtain a strengthening of the two main results of Even et al. which we believe is of independent interest. In particular, in view of the original motivation to study such colorings, this strengthening suggests further applications to frequency assignment in wireless networks. Finally, we present bounds on the number of colors needed for conflict-free colorings of other classes of intersection graphs, including intersection graphs of axis-parallel rectangles and of \rho-fat objects in the plane.Comment: 18 page

    Conflict-Free Coloring of Intersection Graphs

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    A conflict-free k-coloring of a graph G=(V,E) assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v\u27s neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well studied in graph theory. Here we study the conflict-free coloring of geometric intersection graphs. We demonstrate that the intersection graph of n geometric objects without fatness properties and size restrictions may have conflict-free chromatic number in Omega(log n/log log n) and in Omega(sqrt{log n}) for disks or squares of different sizes; it is known for general graphs that the worst case is in Theta(log^2 n). For unit-disk intersection graphs, we prove that it is NP-complete to decide the existence of a conflict-free coloring with one color; we also show that six colors always suffice, using an algorithm that colors unit disk graphs of restricted height with two colors. We conjecture that four colors are sufficient, which we prove for unit squares instead of unit disks. For interval graphs, we establish a tight worst-case bound of two

    Conflict-Free Coloring Made Stronger

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    In FOCS 2002, Even et al. showed that any set of nn discs in the plane can be Conflict-Free colored with a total of at most O(logn)O(\log n) colors. That is, it can be colored with O(logn)O(\log n) colors such that for any (covered) point pp there is some disc whose color is distinct from all other colors of discs containing pp. 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 nn discs in the plane can be colored with a total of at most O(klogn)O(k \log n) colors such that (a) for any point pp that is covered by at least kk discs, there are at least kk distinct discs each of which is colored by a color distinct from all other discs containing pp and (b) for any point pp covered by at most kk discs, all discs covering pp are colored distinctively. We call such a coloring a {\em kk-Strong Conflict-Free} coloring. We extend this result to pseudo-discs and arbitrary regions with linear union-complexity. \item [(ii)] More generally, for families of nn simple closed Jordan regions with union-complexity bounded by O(n1+α)O(n^{1+\alpha}), we prove that there exists a kk-Strong Conflict-Free coloring with at most O(knα)O(k n^\alpha) colors. \item [(iii)] We prove that any set of nn axis-parallel rectangles can be kk-Strong Conflict-Free colored with at most O(klog2n)O(k \log^2 n) colors. \item [(iv)] We provide a general framework for kk-Strong Conflict-Free coloring arbitrary hypergraphs. This framework relates the notion of kk-Strong Conflict-Free coloring and the recently studied notion of kk-colorful coloring. \end{enumerate} All of our proofs are constructive. That is, there exist polynomial time algorithms for computing such colorings

    Conflict-Free Coloring of Planar Graphs

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    A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general} planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on Discrete Mathematics) of extended abstract that appears in Proceeedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 1951-196

    Nice labeling problem for event structures: a counterexample

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    In this note, we present a counterexample to a conjecture of Rozoy and Thiagarajan from 1991 (called also the nice labeling problem) asserting that any (coherent) event structure with finite degree admits a labeling with a finite number of labels, or equivalently, that there exists a function f:NNf: \mathbb{N} \mapsto \mathbb{N} such that an event structure with degree n\le n admits a labeling with at most f(n)f(n) labels. Our counterexample is based on the Burling's construction from 1965 of 3-dimensional box hypergraphs with clique number 2 and arbitrarily large chromatic numbers and the bijection between domains of event structures and median graphs established by Barth\'elemy and Constantin in 1993

    Online and quasi-online colorings of wedges and intervals

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    We consider proper online colorings of hypergraphs defined by geometric regions. We prove that there is an online coloring algorithm that colors NN intervals of the real line using Θ(logN/k)\Theta(\log N/k) colors such that for every point pp, contained in at least kk intervals, not all the intervals containing pp have the same color. We also prove the corresponding result about online coloring a family of wedges (quadrants) in the plane that are the translates of a given fixed wedge. These results contrast the results of the first and third author showing that in the quasi-online setting 12 colors are enough to color wedges (independent of NN and kk). We also consider quasi-online coloring of intervals. In all cases we present efficient coloring algorithms
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