1,347 research outputs found
The VC-Dimension of Graphs with Respect to k-Connected Subgraphs
We study the VC-dimension of the set system on the vertex set of some graph
which is induced by the family of its -connected subgraphs. In particular,
we give tight upper and lower bounds for the VC-dimension. Moreover, we show
that computing the VC-dimension is -complete and that it remains
-complete for split graphs and for some subclasses of planar
bipartite graphs in the cases and . 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
We derive high-temperature series expansions for the free energy and the
susceptibility of random-bond -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 as well as the
dimension as symbolic parameters. By applying several series analysis
techniques to the new series expansions, one can scan large regions of the
parameter space for any value of . 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 as a function of
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
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
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
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 -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
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 -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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