14,083 research outputs found
Tree-width of hypergraphs and surface duality
In Graph Minors III, Robertson and Seymour write: "It seems that the
tree-width of a planar graph and the tree-width of its geometric dual are
approximately equal - indeed, we have convinced ourselves that they differ by
at most one". They never gave a proof of this. In this paper, we prove a
generalisation of this statement to embedding of hypergraphs on general
surfaces, and we prove that our bound is tight
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
Approximating the Spectrum of a Graph
The spectrum of a network or graph with adjacency matrix ,
consists of the eigenvalues of the normalized Laplacian . This set of eigenvalues encapsulates many aspects of the structure
of the graph, including the extent to which the graph posses community
structures at multiple scales. We study the problem of approximating the
spectrum , of in the regime where the graph is too
large to explicitly calculate the spectrum. We present a sublinear time
algorithm that, given the ability to query a random node in the graph and
select a random neighbor of a given node, computes a succinct representation of
an approximation , such that . Our algorithm has query complexity and running time ,
independent of the size of the graph, . We demonstrate the practical
viability of our algorithm on 15 different real-world graphs from the Stanford
Large Network Dataset Collection, including social networks, academic
collaboration graphs, and road networks. For the smallest of these graphs, we
are able to validate the accuracy of our algorithm by explicitly calculating
the true spectrum; for the larger graphs, such a calculation is computationally
prohibitive.
In addition we study the implications of our algorithm to property testing in
the bounded degree graph model
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
Floridian high-voltage power-grid network partitioning and cluster optimization using simulated annealing
Many partitioning methods may be used to partition a network into smaller
clusters while minimizing the number of cuts needed. However, other
considerations must also be taken into account when a network represents a real
system such as a power grid. In this paper we use a simulated annealing Monte
Carlo (MC) method to optimize initial clusters on the Florida high-voltage
power-grid network that were formed by associating each load with its "closest"
generator. The clusters are optimized to maximize internal connectivity within
the individual clusters and minimize the power deficiency or surplus that
clusters may otherwise have.Comment: 9 pages, 3 figures, University of Georgia 24th Annual CSP Worksho
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