70,016 research outputs found
Practical Minimum Cut Algorithms
The minimum cut problem for an undirected edge-weighted graph asks us to
divide its set of nodes into two blocks while minimizing the weight sum of the
cut edges. Here, we introduce a linear-time algorithm to compute near-minimum
cuts. Our algorithm is based on cluster contraction using label propagation and
Padberg and Rinaldi's contraction heuristics [SIAM Review, 1991]. We give both
sequential and shared-memory parallel implementations of our algorithm.
Extensive experiments on both real-world and generated instances show that our
algorithm finds the optimal cut on nearly all instances significantly faster
than other state-of-the-art algorithms while our error rate is lower than that
of other heuristic algorithms. In addition, our parallel algorithm shows good
scalability
Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids
The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero
(with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27].
Analogous theorems hold for the flow polynomial of bridgeless graphs and for
the characteristic polynomial of loopless matroids. Here we exhibit all these
results as special cases of more general theorems on real zero-free regions of
the multivariate Tutte polynomial Z_G(q,v). The proofs are quite simple, and
employ deletion-contraction together with parallel and series reduction. In
particular, they shed light on the origin of the curious number 32/27.Comment: LaTeX2e, 49 pages, includes 5 Postscript figure
Multilevel mesh partitioning for optimising domain shape
Multilevel algorithms are a successful class of optimisation techniques which address the mesh partitioning problem for mapping meshes onto parallel computers. They usually combine a graph contraction algorithm together with a local optimisation method which refines the partition at each graph level. To date these algorithms have been used almost exclusively to minimise the cut-edge weight in the graph with the aim of minimising the parallel communication overhead. However it has been shown that for certain classes of problem, the convergence of the underlying solution algorithm is strongly influenced by the shape or aspect ratio of the subdomains. In this paper therefore, we modify the multilevel algorithms in order to optimise a cost function based on aspect ratio. Several variants of the algorithms are tested and shown to provide excellent results
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