Past exam of the probability, statistics and options I course of the University of Oxford / 2007 / S / 2 / a / Solution

flow conservation determines the unspecified initial arcs:Every entry respects capacity, and the initial value is four. In the residual network take the augmenting pathThe step $4\to 3$ is the reverse residual arc of the used arc $3\to 4$. Its use cancels some existing flow rather than sending new physical flow on a nonexistent original arc. The residual capacities along the path are $2,1,2,2,1$, so augment by their minimum, one.

The resulting original-arc flows areTheir value is five. The cut separating the sink from all other nodes has capacity $c_{46}+c_{56}=3+2=5$. Every feasible flow is bounded by this cut capacity, so the value-five flow is optimal.