5,800 research outputs found

    Conflict-Free Coloring of Points and Simple Regions in the Plane

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    We study conflict-free colorings, where the underlying set systems arise in geometry. Our main result is a general framework for conflict-free coloring of regions with low union complexity. A coloring of regions is conflict-free if for any covered point in the plane, there exists a region that covers it with a unique color (i.e., no other region covering this point has the same color). For example, we show that we can conflict-free color any family of n pseudo-discs with O(log n) color

    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

    Coloring half-planes and bottomless rectangles

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    We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this problem is that of the axis-parallel rectangles. We completely solve the problem for a special case of them, for bottomless rectangles. We also give an almost complete answer for half-planes and pose several open problems. Moreover we give efficient coloring algorithms

    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 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
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