5,086 research outputs found

    Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements

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    Hyperplanes of the form x_j = x_i + c are called affinographic. For an affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we study the function f(M) that counts integral points in [1,M]^n that do not lie in any hyperplane of the arrangement. We show that f(M) is a piecewise polynomial function of positive integers M, composed of terms that appear gradually as M increases. Our approach is to convert the problem to one of counting integral proper colorations of a rooted integral gain graph. An application is to interval coloring in which the interval of available colors for vertex v_i has the form [(h_i)+1,M]. A related problem takes colors modulo M; the number of proper modular colorations is a different piecewise polynomial that for large M becomes the characteristic polynomial of the arrangement (by which means Athanasiadis previously obtained that polynomial). We also study this function for all positive moduli.Comment: 13 p

    A Time Hierarchy Theorem for the LOCAL Model

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    The celebrated Time Hierarchy Theorem for Turing machines states, informally, that more problems can be solved given more time. The extent to which a time hierarchy-type theorem holds in the distributed LOCAL model has been open for many years. It is consistent with previous results that all natural problems in the LOCAL model can be classified according to a small constant number of complexities, such as O(1),O(logn),O(logn),2O(logn)O(1),O(\log^* n), O(\log n), 2^{O(\sqrt{\log n})}, etc. In this paper we establish the first time hierarchy theorem for the LOCAL model and prove that several gaps exist in the LOCAL time hierarchy. 1. We define an infinite set of simple coloring problems called Hierarchical 2122\frac{1}{2}-Coloring}. A correctly colored graph can be confirmed by simply checking the neighborhood of each vertex, so this problem fits into the class of locally checkable labeling (LCL) problems. However, the complexity of the kk-level Hierarchical 2122\frac{1}{2}-Coloring problem is Θ(n1/k)\Theta(n^{1/k}), for kZ+k\in\mathbb{Z}^+. The upper and lower bounds hold for both general graphs and trees, and for both randomized and deterministic algorithms. 2. Consider any LCL problem on bounded degree trees. We prove an automatic-speedup theorem that states that any randomized no(1)n^{o(1)}-time algorithm solving the LCL can be transformed into a deterministic O(logn)O(\log n)-time algorithm. Together with a previous result, this establishes that on trees, there are no natural deterministic complexities in the ranges ω(logn)\omega(\log^* n)---o(logn)o(\log n) or ω(logn)\omega(\log n)---no(1)n^{o(1)}. 3. We expose a gap in the randomized time hierarchy on general graphs. Any randomized algorithm that solves an LCL problem in sublogarithmic time can be sped up to run in O(TLLL)O(T_{LLL}) time, which is the complexity of the distributed Lovasz local lemma problem, currently known to be Ω(loglogn)\Omega(\log\log n) and O(logn)O(\log n)

    Acyclic edge-coloring using entropy compression

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    An edge-coloring of a graph G is acyclic if it is a proper edge-coloring of G and every cycle contains at least three colors. We prove that every graph with maximum degree Delta has an acyclic edge-coloring with at most 4 Delta - 4 colors, improving the previous bound of 9.62 (Delta - 1). Our bound results from the analysis of a very simple randomised procedure using the so-called entropy compression method. We show that the expected running time of the procedure is O(mn Delta^2 log Delta), where n and m are the number of vertices and edges of G. Such a randomised procedure running in expected polynomial time was only known to exist in the case where at least 16 Delta colors were available. Our aim here is to make a pedagogic tutorial on how to use these ideas to analyse a broad range of graph coloring problems. As an application, also show that every graph with maximum degree Delta has a star coloring with 2 sqrt(2) Delta^{3/2} + Delta colors.Comment: 13 pages, revised versio

    Coloring triangle-free rectangle overlap graphs with O(loglogn)O(\log\log n) colors

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    Recently, it was proved that triangle-free intersection graphs of nn line segments in the plane can have chromatic number as large as Θ(loglogn)\Theta(\log\log n). Essentially the same construction produces Θ(loglogn)\Theta(\log\log n)-chromatic triangle-free intersection graphs of a variety of other geometric shapes---those belonging to any class of compact arc-connected sets in R2\mathbb{R}^2 closed under horizontal scaling, vertical scaling, and translation, except for axis-parallel rectangles. We show that this construction is asymptotically optimal for intersection graphs of boundaries of axis-parallel rectangles, which can be alternatively described as overlap graphs of axis-parallel rectangles. That is, we prove that triangle-free rectangle overlap graphs have chromatic number O(loglogn)O(\log\log n), improving on the previous bound of O(logn)O(\log n). To this end, we exploit a relationship between off-line coloring of rectangle overlap graphs and on-line coloring of interval overlap graphs. Our coloring method decomposes the graph into a bounded number of subgraphs with a tree-like structure that "encodes" strategies of the adversary in the on-line coloring problem. Then, these subgraphs are colored with O(loglogn)O(\log\log n) colors using a combination of techniques from on-line algorithms (first-fit) and data structure design (heavy-light decomposition).Comment: Minor revisio

    Pathwidth and nonrepetitive list coloring

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    A vertex coloring of a graph is nonrepetitive if there is no path in the graph whose first half receives the same sequence of colors as the second half. While every tree can be nonrepetitively colored with a bounded number of colors (4 colors is enough), Fiorenzi, Ochem, Ossona de Mendez, and Zhu recently showed that this does not extend to the list version of the problem, that is, for every 1\ell \geq 1 there is a tree that is not nonrepetitively \ell-choosable. In this paper we prove the following positive result, which complements the result of Fiorenzi et al.: There exists a function ff such that every tree of pathwidth kk is nonrepetitively f(k)f(k)-choosable. We also show that such a property is specific to trees by constructing a family of pathwidth-2 graphs that are not nonrepetitively \ell-choosable for any fixed \ell.Comment: v2: Minor changes made following helpful comments by the referee

    Locally identifying coloring in bounded expansion classes of graphs

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    A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors used by a locally identifying vertex-coloring. In this paper, we prove that for any graph class of bounded expansion, the lid-chromatic number is bounded. Classes of bounded expansion include minor closed classes of graphs. For these latter classes, we give an alternative proof to show that the lid-chromatic number is bounded. This leads to an explicit upper bound for the lid-chromatic number of planar graphs. This answers in a positive way a question of Esperet et al [L. Esperet, S. Gravier, M. Montassier, P. Ochem and A. Parreau. Locally identifying coloring of graphs. Electronic Journal of Combinatorics, 19(2), 2012.]

    Steinitz Theorems for Orthogonal Polyhedra

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    We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

    Joint Routing and STDMA-based Scheduling to Minimize Delays in Grid Wireless Sensor Networks

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    In this report, we study the issue of delay optimization and energy efficiency in grid wireless sensor networks (WSNs). We focus on STDMA (Spatial Reuse TDMA)) scheduling, where a predefined cycle is repeated, and where each node has fixed transmission opportunities during specific slots (defined by colors). We assume a STDMA algorithm that takes advantage of the regularity of grid topology to also provide a spatially periodic coloring ("tiling" of the same color pattern). In this setting, the key challenges are: 1) minimizing the average routing delay by ordering the slots in the cycle 2) being energy efficient. Our work follows two directions: first, the baseline performance is evaluated when nothing specific is done and the colors are randomly ordered in the STDMA cycle. Then, we propose a solution, ORCHID that deliberately constructs an efficient STDMA schedule. It proceeds in two steps. In the first step, ORCHID starts form a colored grid and builds a hierarchical routing based on these colors. In the second step, ORCHID builds a color ordering, by considering jointly both routing and scheduling so as to ensure that any node will reach a sink in a single STDMA cycle. We study the performance of these solutions by means of simulations and modeling. Results show the excellent performance of ORCHID in terms of delays and energy compared to a shortest path routing that uses the delay as a heuristic. We also present the adaptation of ORCHID to general networks under the SINR interference model
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