1,601 research outputs found

    A Breezing Proof of the KMW Bound

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    In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW) proved a hardness result for several fundamental graph problems in the LOCAL model: For any (randomized) algorithm, there are input graphs with nn nodes and maximum degree Δ\Delta on which Ω(min{logn/loglogn,logΔ/loglogΔ})\Omega(\min\{\sqrt{\log n/\log \log n},\log \Delta/\log \log \Delta\}) (expected) communication rounds are required to obtain polylogarithmic approximations to a minimum vertex cover, minimum dominating set, or maximum matching. Via reduction, this hardness extends to symmetry breaking tasks like finding maximal independent sets or maximal matchings. Today, more than 1515 years later, there is still no proof of this result that is easy on the reader. Setting out to change this, in this work, we provide a fully self-contained and simple\mathit{simple} proof of the KMW lower bound. The key argument is algorithmic, and it relies on an invariant that can be readily verified from the generation rules of the lower bound graphs.Comment: 21 pages, 6 figure

    Planar Induced Subgraphs of Sparse Graphs

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    We show that every graph has an induced pseudoforest of at least nm/4.5n-m/4.5 vertices, an induced partial 2-tree of at least nm/5n-m/5 vertices, and an induced planar subgraph of at least nm/5.2174n-m/5.2174 vertices. These results are constructive, implying linear-time algorithms to find the respective induced subgraphs. We also show that the size of the largest KhK_h-minor-free graph in a given graph can sometimes be at most nm/6+o(m)n-m/6+o(m).Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph Algorithms and Application

    Characterization and Efficient Search of Non-Elementary Trapping Sets of LDPC Codes with Applications to Stopping Sets

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    In this paper, we propose a characterization for non-elementary trapping sets (NETSs) of low-density parity-check (LDPC) codes. The characterization is based on viewing a NETS as a hierarchy of embedded graphs starting from an ETS. The characterization corresponds to an efficient search algorithm that under certain conditions is exhaustive. As an application of the proposed characterization/search, we obtain lower and upper bounds on the stopping distance smins_{min} of LDPC codes. We examine a large number of regular and irregular LDPC codes, and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on, and in many cases the exact value of, smins_{min}. Finding smins_{min}, or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm
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