How to locate the minimum cut that represents the maximum flow capacity in a network graph
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The Maximum-Flow Minimum-Cut Theorem is a fundamental result in network flow theory, a branch of combinatorial optimisation and graph theory. It provides a mathematical proof of the relationship between the maximum flow in a flow network and the minimum cut in that network. The theorem states:
"In any flow network, the maximum amount of flow that can be sent from the source to the sink is equal to the minimum capacity of any cut that separates the source from the sink."
In other words, the maximum flow in a network is equal to the minimum capacity of any cut that divides the network into two disjoint sets, one containing the source and the other containing the sink. A "cut" in this context refers to a set of edges whose removal disconnects the source from the sink.
Mathematically, if F is the maximum flow, and C is the minimum capacity of any cut, the theorem can be expressed as:
F = C
The theorem has practical applications in various fields, including transportation and communication networks, where it helps in optimising the flow of resources or information through a network while considering capacity constraints. It is often used in algorithms like the Ford-Fulkerson algorithm to find the maximum flow in a network.
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