Max flow - Min cut theorem

Question

I understand the Ford-Fulkerson's method for finding the max flow, but I'm having trouble understanding how min-cut gives the value of max flow.

Max flow - min cut theorem states that the maximum flow passing from source to sink is equal to the value of min cut.

Min-cut in CLRS is defined as :

A min cut of a network is a cut whose capacity is minimum over all cuts of the network.

If the capacity is minimum, it means that there exist augmenting paths with higher capacities, then how come paths with lower capacity yeild max flow? By capacity does the author mean residual capacity ? Because then it all makes sense.

Answer 1

If I understood the question correctly, there is some misunderstanding. An augmenting path cannot have a higher capacity than the capacity of the minimal cut. Intuitively, for any fixed network flow and any fixed cut, the flow must transport its value from the cut's partition which contains the source to the partition which contains the terminal; however, this value cannot be larger than the cut's capacity.

To phrase it more roughly, the value of any flow cannot be larger than the "bottleneck" of the network, which is the minimimum possible capacity of a cut.