Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 20H / a

The max-flow min-cut theorem states that in a finite flow network, the maximum value of a feasible source-to-sink flow equals the minimum capacity of a source-to-sink cut.

First consider any feasible flow $f$ and any cut $(A,V\setminus A)$ with $S\in A$ and $T\notin A$. Summing flow conservation over the vertices in $A$ cancels all contributions from edges internal to $A$ and givesThus every flow value is at most every cut capacity.

A maximum flow exists because the feasible flows form a nonempty compact subset of a finite-dimensional Euclidean space and the flow value is continuous. Let $f$ be maximum and form its residual network. If there were an augmenting path from $S$ to $T$, increasing $f$ by the path's positive bottleneck capacity would contradict maximality. Let $A$ be the set of vertices reachable from $S$ in the residual network. Then $T\notin A$. Every original edge from $A$ to its complement is saturated, while every original edge entering $A$ carries zero flow; otherwise the corresponding forward or reverse residual edge would make its other endpoint reachable. ConsequentlyThe general upper bound is attained by this flow and cut, proving the theorem.