1,257 research outputs found
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
On the complexity of computing the -restricted edge-connectivity of a graph
The \emph{-restricted edge-connectivity} of a graph , denoted by
, is defined as the minimum size of an edge set whose removal
leaves exactly two connected components each containing at least vertices.
This graph invariant, which can be seen as a generalization of a minimum
edge-cut, has been extensively studied from a combinatorial point of view.
However, very little is known about the complexity of computing .
Very recently, in the parameterized complexity community the notion of
\emph{good edge separation} of a graph has been defined, which happens to be
essentially the same as the -restricted edge-connectivity. Motivated by the
relevance of this invariant from both combinatorial and algorithmic points of
view, in this article we initiate a systematic study of its computational
complexity, with special emphasis on its parameterized complexity for several
choices of the parameters. We provide a number of NP-hardness and W[1]-hardness
results, as well as FPT-algorithms.Comment: 16 pages, 4 figure
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
Vertex Sparsifiers: New Results from Old Techniques
Given a capacitated graph and a set of terminals ,
how should we produce a graph only on the terminals so that every
(multicommodity) flow between the terminals in could be supported in
with low congestion, and vice versa? (Such a graph is called a
flow-sparsifier for .) What if we want to be a "simple" graph? What if
we allow to be a convex combination of simple graphs?
Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC
2010], we give efficient algorithms for constructing: (a) a flow-sparsifier
that maintains congestion up to a factor of , where , (b) a convex combination of trees over the terminals that maintains
congestion up to a factor of , and (c) for a planar graph , a
convex combination of planar graphs that maintains congestion up to a constant
factor. This requires us to give a new algorithm for the 0-extension problem,
the first one in which the preimages of each terminal are connected in .
Moreover, this result extends to minor-closed families of graphs.
Our improved bounds immediately imply improved approximation guarantees for
several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on
Approximation Algorithms for Combinatorial Optimization Problems (APPROX),
2010. Final version to appear in SIAM J. Computin
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