1,347 research outputs found

    The VC-Dimension of Graphs with Respect to k-Connected Subgraphs

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    We study the VC-dimension of the set system on the vertex set of some graph which is induced by the family of its kk-connected subgraphs. In particular, we give tight upper and lower bounds for the VC-dimension. Moreover, we show that computing the VC-dimension is NP\mathsf{NP}-complete and that it remains NP\mathsf{NP}-complete for split graphs and for some subclasses of planar bipartite graphs in the cases k=1k = 1 and k=2k = 2. On the positive side, we observe it can be decided in linear time for graphs of bounded clique-width

    Star-graph expansions for bond-diluted Potts models

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    We derive high-temperature series expansions for the free energy and the susceptibility of random-bond qq-state Potts models on hypercubic lattices using a star-graph expansion technique. This method enables the exact calculation of quenched disorder averages for arbitrary uncorrelated coupling distributions. Moreover, we can keep the disorder strength pp as well as the dimension dd as symbolic parameters. By applying several series analysis techniques to the new series expansions, one can scan large regions of the (p,d)(p,d) parameter space for any value of qq. For the bond-diluted 4-state Potts model in three dimensions, which exhibits a rather strong first-order phase transition in the undiluted case, we present results for the transition temperature and the effective critical exponent γ\gamma as a function of pp as obtained from the analysis of susceptibility series up to order 18. A comparison with recent Monte Carlo data (Chatelain {\em et al.}, Phys. Rev. E64, 036120(2001)) shows signals for the softening to a second-order transition at finite disorder strength.Comment: 8 pages, 6 figure

    Quantization of gauge fields, graph polynomials and graph cohomology

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    We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular graphs, obtained from the two Symanzik polynomials. The transition to the full gauge theory amplitude is obtained by the use of a third, new, graph polynomial, the corolla polynomial. This implies effectively a covariant quantization without ghosts, where all the relevant signs of the ghost sector are incorporated in a double complex furnished by the corolla polynomial -we call it cycle homology- and by graph homology.Comment: 44p, many figures, to appea

    High-Temperature Series Expansions for Random Potts Models

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    We discuss recently generated high-temperature series expansions for the free energy and the susceptibility of random-bond q-state Potts models on hypercubic lattices. Using the star-graph expansion technique quenched disorder averages can be calculated exactly for arbitrary uncorrelated coupling distributions while keeping the disorder strength p as well as the dimension d as symbolic parameters. We present analyses of the new series for the susceptibility of the Ising (q=2) and 4-state Potts model in three dimensions up to order 19 and 18, respectively, and compare our findings with results from field-theoretical renormalization group studies and Monte Carlo simulations.Comment: 16 pages,cmp209.sty (included), 9 postscript figures, author information under http://www.physik.uni-leipzig.de/index.php?id=2

    Parallel Unsmoothed Aggregation Algebraic Multigrid Algorithms on GPUs

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    We design and implement a parallel algebraic multigrid method for isotropic graph Laplacian problems on multicore Graphical Processing Units (GPUs). The proposed AMG method is based on the aggregation framework. The setup phase of the algorithm uses a parallel maximal independent set algorithm in forming aggregates and the resulting coarse level hierarchy is then used in a K-cycle iteration solve phase with a 1\ell^1-Jacobi smoother. Numerical tests of a parallel implementation of the method for graphics processors are presented to demonstrate its effectiveness.Comment: 18 pages, 3 figure

    Distributed Dominating Set Approximations beyond Planar Graphs

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    The Minimum Dominating Set (MDS) problem is one of the most fundamental and challenging problems in distributed computing. While it is well-known that minimum dominating sets cannot be approximated locally on general graphs, over the last years, there has been much progress on computing local approximations on sparse graphs, and in particular planar graphs. In this paper we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs, and present (1) a local constant-time, constant-factor MDS approximation algorithm and (2) a local O(logn)\mathcal{O}(\log^*{n})-time approximation scheme. Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299
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