697 research outputs found
On the minimum bisection of random regular graphs
In this paper we give new asymptotically almost sure lower and upper bounds
on the bisection width of random regular graphs. The main contribution is a
new lower bound on the bisection width of , based on a first moment
method together with a structural decomposition of the graph, thereby improving
a 27 year old result of Kostochka and Melnikov. We also give a complementary
upper bound of , combining known spectral ideas with original
combinatorial insights. Developping further this approach, with the help of
Monte Carlo simulations, we obtain a non-rigorous upper bound of .Comment: 48 pages, 20 figure
Linear orderings of random geometric graphs (extended abstract)
In random geometric graphs, vertices are randomly distributed on [0,1]^2 and pairs of vertices are connected by edges
whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a
certain measure is minimized. In this paper, we study several layout problems on random geometric graphs: Bandwidth,
Minimum Linear Arrangement, Minimum Cut, Minimum Sum Cut, Vertex Separation and Bisection. We first prove that
some of these problems remain \NP-complete even for geometric graphs. Afterwards, we compute lower bounds that hold
with high probability on random geometric graphs. Finally, we characterize the probabilistic behavior of the lexicographic
ordering for our layout problems on the class of random geometric graphs.Postprint (published version
The Peculiar Phase Structure of Random Graph Bisection
The mincut graph bisection problem involves partitioning the n vertices of a
graph into disjoint subsets, each containing exactly n/2 vertices, while
minimizing the number of "cut" edges with an endpoint in each subset. When
considered over sparse random graphs, the phase structure of the graph
bisection problem displays certain familiar properties, but also some
surprises. It is known that when the mean degree is below the critical value of
2 log 2, the cutsize is zero with high probability. We study how the minimum
cutsize increases with mean degree above this critical threshold, finding a new
analytical upper bound that improves considerably upon previous bounds.
Combined with recent results on expander graphs, our bound suggests the unusual
scenario that random graph bisection is replica symmetric up to and beyond the
critical threshold, with a replica symmetry breaking transition possibly taking
place above the threshold. An intriguing algorithmic consequence is that
although the problem is NP-hard, we can find near-optimal cutsizes (whose ratio
to the optimal value approaches 1 asymptotically) in polynomial time for
typical instances near the phase transition.Comment: substantially revised section 2, changed figures 3, 4 and 6, made
minor stylistic changes and added reference
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
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