63,260 research outputs found
An Alternative Approach to Counting Minimum (s; t)-cuts in Planar Graphs
Finding and counting minimum cuts in graphs can be useful in image processing and segmentation and in networking systems such as computer or road networks. Researchers have previously developed polynomial-time algorithms to count minimum cuts in planar graphs which utilize the relationship between maximum network flows and minimum cuts.
This thesis presents a polynomial-time algorithm to count the number of minimum (s,t)-cuts in a planar graph without first finding a maximum flow. The presented algorithm is dependent on the finding that (s,t)-cuts in a planar graph correlate to certain cycles found in the dual of that graph, which can be efficiently counted
Shortest path and maximum flow problems in planar flow networks with additive gains and losses
In contrast to traditional flow networks, in additive flow networks, to every
edge e is assigned a gain factor g(e) which represents the loss or gain of the
flow while using edge e. Hence, if a flow f(e) enters the edge e and f(e) is
less than the designated capacity of e, then f(e) + g(e) = 0 units of flow
reach the end point of e, provided e is used, i.e., provided f(e) != 0. In this
report we study the maximum flow problem in additive flow networks, which we
prove to be NP-hard even when the underlying graphs of additive flow networks
are planar. We also investigate the shortest path problem, when to every edge e
is assigned a cost value for every unit flow entering edge e, which we show to
be NP-hard in the strong sense even when the additive flow networks are planar
NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs
In 1988, Vazirani gave an NC algorithm for computing the number of perfect
matchings in -minor-free graphs by building on Kasteleyn's scheme for
planar graphs, and stated that this "opens up the possibility of obtaining an
NC algorithm for finding a perfect matching in -free graphs." In this
paper, we finally settle this 30-year-old open problem. Building on recent NC
algorithms for planar and bounded-genus perfect matching by Anari and Vazirani
and later by Sankowski, we obtain NC algorithms for perfect matching in any
minor-closed graph family that forbids a one-crossing graph. This family
includes several well-studied graph families including the -minor-free
graphs and -minor-free graphs. Graphs in these families not only have
unbounded genus, but can have genus as high as . Our method applies as
well to several other problems related to perfect matching. In particular, we
obtain NC algorithms for the following problems in any family of graphs (or
networks) with a one-crossing forbidden minor:
Determining whether a given graph has a perfect matching and if so,
finding one.
Finding a minimum weight perfect matching in the graph, assuming
that the edge weights are polynomially bounded.
Finding a maximum -flow in the network, with arbitrary
capacities.
The main new idea enabling our results is the definition and use of
matching-mimicking networks, small replacement networks that behave the same,
with respect to matching problems involving a fixed set of terminals, as the
larger network they replace.Comment: 21 pages, 6 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
Transport Processes on Homogeneous Planar Graphs with Scale-Free Loops
We consider the role of network geometry in two types of diffusion processes:
transport of constant-density information packets with queuing on nodes, and
constant voltage-driven tunneling of electrons. The underlying network is a
homogeneous graph with scale-free distribution of loops, which is constrained
to a planar geometry and fixed node connectivity . We determine properties
of noise, flow and return-times statistics for both processes on this graph and
relate the observed differences to the microscopic process details. Our main
findings are: (i) Through the local interaction between packets queuing at the
same node, long-range correlations build up in traffic streams, which are
practically absent in the case of electron transport; (ii) Noise fluctuations
in the number of packets and in the number of tunnelings recorded at each node
appear to obey the scaling laws in two distinct universality classes; (iii) The
topological inhomogeneity of betweenness plays the key role in the occurrence
of broad distributions of return times and in the dynamic flow. The
maximum-flow spanning trees are characteristic for each process type.Comment: 14 pages, 5 figure
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