66,429 research outputs found
Minimum Cuts in Near-Linear Time
We significantly improve known time bounds for solving the minimum cut
problem on undirected graphs. We use a ``semi-duality'' between minimum cuts
and maximum spanning tree packings combined with our previously developed
random sampling techniques. We give a randomized algorithm that finds a minimum
cut in an m-edge, n-vertex graph with high probability in O(m log^3 n) time. We
also give a simpler randomized algorithm that finds all minimum cuts with high
probability in O(n^2 log n) time. This variant has an optimal RNC
parallelization. Both variants improve on the previous best time bound of O(n^2
log^3 n). Other applications of the tree-packing approach are new, nearly tight
bounds on the number of near minimum cuts a graph may have and a new data
structure for representing them in a space-efficient manner
Efficiently Finding Simple Schedules in Gaussian Half-Duplex Relay Line Networks
The problem of operating a Gaussian Half-Duplex (HD) relay network optimally
is challenging due to the exponential number of listen/transmit network states
that need to be considered. Recent results have shown that, for the class of
Gaussian HD networks with N relays, there always exists a simple schedule,
i.e., with at most N +1 active states, that is sufficient for approximate
(i.e., up to a constant gap) capacity characterization. This paper investigates
how to efficiently find such a simple schedule over line networks. Towards this
end, a polynomial-time algorithm is designed and proved to output a simple
schedule that achieves the approximate capacity. The key ingredient of the
algorithm is to leverage similarities between network states in HD and edge
coloring in a graph. It is also shown that the algorithm allows to derive a
closed-form expression for the approximate capacity of the Gaussian line
network that can be evaluated distributively and in linear time. Additionally,
it is shown using this closed-form that the problem of Half-Duplex routing is
NP-Hard.Comment: A short version of this paper was submitted to ISIT 201
Large Cuts with Local Algorithms on Triangle-Free Graphs
We study the problem of finding large cuts in -regular triangle-free
graphs. In prior work, Shearer (1992) gives a randomised algorithm that finds a
cut of expected size , where is the number of
edges. We give a simpler algorithm that does much better: it finds a cut of
expected size . As a corollary, this shows that in
any -regular triangle-free graph there exists a cut of at least this size.
Our algorithm can be interpreted as a very efficient randomised distributed
algorithm: each node needs to produce only one random bit, and the algorithm
runs in one synchronous communication round. This work is also a case study of
applying computational techniques in the design of distributed algorithms: our
algorithm was designed by a computer program that searched for optimal
algorithms for small values of .Comment: 1+17 pages, 8 figure
Max flow vitality in general and -planar graphs
The \emph{vitality} of an arc/node of a graph with respect to the maximum
flow between two fixed nodes and is defined as the reduction of the
maximum flow caused by the removal of that arc/node. In this paper we address
the issue of determining the vitality of arcs and/or nodes for the maximum flow
problem. We show how to compute the vitality of all arcs in a general
undirected graph by solving only max flow instances and, In
-planar graphs (directed or undirected) we show how to compute the vitality
of all arcs and all nodes in worst-case time. Moreover, after
determining the vitality of arcs and/or nodes, and given a planar embedding of
the graph, we can determine the vitality of a `contiguous' set of arcs/nodes in
time proportional to the size of the set.Comment: 12 pages, 3 figure
Degeneracy Algorithm for Random Magnets
It has been known for a long time that the ground state problem of random
magnets, e.g. random field Ising model (RFIM), can be mapped onto the
max-flow/min-cut problem of transportation networks. I build on this approach,
relying on the concept of residual graph, and design an algorithm that I prove
to be exact for finding all the minimum cuts, i.e. the ground state degeneracy
of these systems. I demonstrate that this algorithm is also relevant for the
study of the ground state properties of the dilute Ising antiferromagnet in a
constant field (DAFF) and interfaces in random bond magnets.Comment: 17 pages(Revtex), 8 Postscript figures(5color) to appear in Phys.
Rev. E 58, December 1st (1998
Electrical Flows, Laplacian Systems, and Faster Approximation of Maximum Flow in Undirected Graphs
We introduce a new approach to computing an approximately maximum s-t flow in
a capacitated, undirected graph. This flow is computed by solving a sequence of
electrical flow problems. Each electrical flow is given by the solution of a
system of linear equations in a Laplacian matrix, and thus may be approximately
computed in nearly-linear time.
Using this approach, we develop the fastest known algorithm for computing
approximately maximum s-t flows. For a graph having n vertices and m edges, our
algorithm computes a (1-\epsilon)-approximately maximum s-t flow in time
\tilde{O}(mn^{1/3} \epsilon^{-11/3}). A dual version of our approach computes a
(1+\epsilon)-approximately minimum s-t cut in time
\tilde{O}(m+n^{4/3}\eps^{-8/3}), which is the fastest known algorithm for this
problem as well. Previously, the best dependence on m and n was achieved by the
algorithm of Goldberg and Rao (J. ACM 1998), which can be used to compute
approximately maximum s-t flows in time \tilde{O}(m\sqrt{n}\epsilon^{-1}), and
approximately minimum s-t cuts in time \tilde{O}(m+n^{3/2}\epsilon^{-3})
All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs
For an undirected -vertex graph with non-negative edge-weights, we
consider the following type of query: given two vertices and in ,
what is the weight of a minimum -cut in ? We solve this problem in
preprocessing time for graphs of bounded genus, giving the first
sub-quadratic time algorithm for this class of graphs. Our result also improves
by a logarithmic factor a previous algorithm by Borradaile, Sankowski and
Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm
constructs a Gomory-Hu tree for the given graph, providing a data structure
with space that can answer minimum-cut queries in constant time. The
dependence on the genus of the input graph in our preprocessing time is
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