The max flow is a fundamental graph optimization problem in computer science and graph theory. It involves finding the maximum amount of flow that can travel through a directed, weighted graph (network) from a source node (\(s\)) to a sink node (\(t\)) without exceeding the capacity of any edge.

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Max-Flow Min-Cut Theorem

The max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink is equal to the total capacity of the minimum cut (the smallest total edge capacity that disconnects the source from the sink). This theorem, introduced by Ford and Fulkerson in 1956, identifies the bottleneck of a network, where the max flow equals the capacity of this minimum cut.