A Maximum Flow Min cut theorem for Optimizing Network

A MAXIMUM FLOW MIN CUT THEOREM FOR OPTIMIZING NETWORK

Shethwala Ridhvesh

OUTLINESIntroductionMax flow theoremFord-Fulkerson algorithmMin cut theoremConclusionReferences

MAX FLOW THEOREM-> A directed, weighted graph is called a (flow) network.-> Each edge has a weight and direction.-> We assume there exists a source and a sink.

The flow over a network is a function f: E -> R, assigning values to each of the edges in the network which are nonnegative and less than the capacity of that edge. For each intermediate vertex, the outflow and inflow must be equal.

The value of this flow is the total amount leaving the source (and thus entering the sink).

FORD-FULKERSON MAX FLOWThe Ford-Fulkerson algorithm for finding the maximum flow:

a. Construct the Residual Graphb. Find a path from the source to the sink with

strictly positive flow.c. If this path exists, update flow to include it. Go

to Step a.d. Else, the flow is maximal.e. The (s,t)-cut has as S all vertices reachable from

the source, and T as V - S.

FORD-FULKERSON MAX FLOW

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This is the original network, and the original residual network.

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Find any s-t path in G(x)

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FORD-FULKERSON MAX FLOW

Find any s-t path

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Find any s-t path

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Find any s-t path

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Find any s-t path

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There is no s-t path in the residual network. This flow is optimal

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These are the nodes that are reachable from node s.

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Here is the optimal flow

MIN CUT A cut is a partition of the vertices into disjoint

subsets S and T. In a flow network, the source is located in S, and the sink is located in T.

The cut-set of a cut is the set of edges that begin in S and end in T.

The capacity of a cut is sum of the weights of the edges beginning in S and ending in T.

MIN CUT

MIN CUT

Max flow in network

MIN CUT

APPLICATIONS - Traffic problem on road

- Data Mining- Distributed Computing- Image processing- Project selection- Bipartite Matching

CONCLUSION Using this Max-flow min-cut theorem we can

maximize the flow in network and can use the maximum capacity of route for optimizing network.

REFERENCES Ford, Jr., L. R., and D. R. Fulkerson. “Maximal Flow

Through a Network.” Canadian Journal of Mathematics 8 (1956): 399-404. Canadian Mathematical Society. Web. 2 June,2010

Ellis L. Johnson, Committee Chair, George L. Nemhauser: Shortest paths and multicommoditynetwork flow,2003

FORD.L.R. AND D. R. FULKERSON 1956. Maximal Flow Through a Network. Can. J. Math. 8,399-404.

Cormen, Thomas H. Introduction to Algorithms. 2nd ed. Cambridge, Massachusetts: MIT, 2001

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