15,511 research outputs found

    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

    On Derandomizing Local Distributed Algorithms

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    The gap between the known randomized and deterministic local distributed algorithms underlies arguably the most fundamental and central open question in distributed graph algorithms. In this paper, we develop a generic and clean recipe for derandomizing LOCAL algorithms. We also exhibit how this simple recipe leads to significant improvements on a number of problem. Two main results are: - An improved distributed hypergraph maximal matching algorithm, improving on Fischer, Ghaffari, and Kuhn [FOCS'17], and giving improved algorithms for edge-coloring, maximum matching approximation, and low out-degree edge orientation. The first gives an improved algorithm for Open Problem 11.4 of the book of Barenboim and Elkin, and the last gives the first positive resolution of their Open Problem 11.10. - An improved distributed algorithm for the Lov\'{a}sz Local Lemma, which gets closer to a conjecture of Chang and Pettie [FOCS'17], and moreover leads to improved distributed algorithms for problems such as defective coloring and kk-SAT.Comment: 37 page
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