Problem $3(15$pts$)$

Solve the following Max$-$Flow problems using the Max$-$Flow algorithm discussed in class and in the discussion sections $($the Verstraete lecture notes give in Section $8\mathrm{.}4$ a slightly more detailed implementation which we however did not discuss in class; the worked out example $24$ may still potentially help your understanding, though$)$; below the networks have just one number on each edge, namely the capacity.

To get full points, you need to submit the resulting max$-$flow and its value together with a cut that certifies optimality of the flow $($i$.$e$.,$ whose capacity is equal to the value of the flow you have found$)$; no need to submit the steps of the algorithm:

$($a$)$

$($b$)$

$($c$)$