3,743 research outputs found

    Extensions of Fractional Precolorings show Discontinuous Behavior

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    We study the following problem: given a real number k and integer d, what is the smallest epsilon such that any fractional (k+epsilon)-precoloring of vertices at pairwise distances at least d of a fractionally k-colorable graph can be extended to a fractional (k+epsilon)-coloring of the whole graph? The exact values of epsilon were known for k=2 and k\ge3 and any d. We determine the exact values of epsilon for k \in (2,3) if d=4, and k \in [2.5,3) if d=6, and give upper bounds for k \in (2,3) if d=5,7, and k \in (2,2.5) if d=6. Surprisingly, epsilon viewed as a function of k is discontinuous for all those values of d

    Fractional coloring of triangle-free planar graphs

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    We prove that every planar triangle-free graph on nn vertices has fractional chromatic number at most 31n+1/33-\frac{1}{n+1/3}

    Local Graph Coloring and Index Coding

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    We present a novel upper bound for the optimal index coding rate. Our bound uses a graph theoretic quantity called the local chromatic number. We show how a good local coloring can be used to create a good index code. The local coloring is used as an alignment guide to assign index coding vectors from a general position MDS code. We further show that a natural LP relaxation yields an even stronger index code. Our bounds provably outperform the state of the art on index coding but at most by a constant factor.Comment: 14 Pages, 3 Figures; A conference version submitted to ISIT 2013; typos correcte

    Distributed local approximation algorithms for maximum matching in graphs and hypergraphs

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    We describe approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in a hypergraph of rank rr. Our main result is a deterministic algorithm to generate a matching which is an O(r)O(r)-approximation to the maximum weight matching, running in O~(rlogΔ+log2Δ+logn)\tilde O(r \log \Delta + \log^2 \Delta + \log^* n) rounds. (Here, the O~()\tilde O() notations hides polyloglog Δ\text{polyloglog } \Delta and polylog r\text{polylog } r factors). This is based on a number of new derandomization techniques extending methods of Ghaffari, Harris & Kuhn (2017). As a main application, we obtain nearly-optimal algorithms for the long-studied problem of maximum-weight graph matching. Specifically, we get a (1+ϵ)(1+\epsilon) approximation algorithm using O~(logΔ/ϵ3+polylog(1/ϵ,loglogn))\tilde O(\log \Delta / \epsilon^3 + \text{polylog}(1/\epsilon, \log \log n)) randomized time and O~(log2Δ/ϵ4+logn/ϵ)\tilde O(\log^2 \Delta / \epsilon^4 + \log^*n / \epsilon) deterministic time. The second application is a faster algorithm for hypergraph maximal matching, a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of local graph algorithms. This gives an algorithm for (2Δ1)(2 \Delta - 1)-edge-list coloring in O~(log2Δlogn)\tilde O(\log^2 \Delta \log n) rounds deterministically or O~((loglogn)3)\tilde O( (\log \log n)^3 ) rounds randomly. Another consequence (with additional optimizations) is an algorithm which generates an edge-orientation with out-degree at most (1+ϵ)λ\lceil (1+\epsilon) \lambda \rceil for a graph of arboricity λ\lambda; for fixed ϵ\epsilon this runs in O~(log6n)\tilde O(\log^6 n) rounds deterministically or O~(log3n)\tilde O(\log^3 n ) rounds randomly

    TDMA is Optimal for All-unicast DoF Region of TIM if and only if Topology is Chordal Bipartite

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    The main result of this work is that an orthogonal access scheme such as TDMA achieves the all-unicast degrees of freedom (DoF) region of the topological interference management (TIM) problem if and only if the network topology graph is chordal bipartite, i.e., every cycle that can contain a chord, does contain a chord. The all-unicast DoF region includes the DoF region for any arbitrary choice of a unicast message set, so e.g., the results of Maleki and Jafar on the optimality of orthogonal access for the sum-DoF of one-dimensional convex networks are recovered as a special case. The result is also established for the corresponding topological representation of the index coding problem
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