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I thought I understood max flow perfectly until I got to the exam and we got this. I know how to compute a maximum flow by means of the Ford-Fulkerson algorithm, specify the residual network and augmenting path. But here they gave us a graph where some of the capacities were already full (if you can tell from the picture) and I couldn't find a new augmenting path. Is there another way to solve this? enter image description here

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A full edge, e.g. $a \rightarrow c$, has a residual capacity of $0$ in the residual network. So you can't make an augmenting path over that directed edge. However the reversed edge, $c \rightarrow a$ has a residual capacity of $5$ (since $c_{c \rightarrow a} = 0$ and $f_{c \rightarrow a} = -5$). Therefore you can create an augmenting path using the reversed edge. The residual network would be:

enter image description here

E.g. a possible augmenting path would be $s \rightarrow c \rightarrow a \rightarrow d \rightarrow t$. You can increase the flow along that path by 5 and get the following flow network:

enter image description here

This is also the maximal flow, since $\{s, c\} - \{a, b, d, t\}$ forms a saturated s-t-cut.

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