VVProblem $2$

$($Flow variations$)$ Let $G=\{V,(vec(E)),c_e,w_v,s,t\}$ be a flow network. To each vertex vin$\frac{V}{\frac{?}{?}}\{s,t\},$ we associate a

weight, $w_v>0.$ We want to compute a flow $f$ of maximum value satisfying the following extra constraint:

AAvin$\frac{V}{\frac{?}{?}}\{s,t\}$ the flow entering $v$ must be at most $w_v.$

Reduce this variation of the maximum flow problem to an input that can be solved running the Ford$-$Fulkerson

Algorithm. Also, briefly justify why your reduction is indeed an optimal solution to the given problem.

We expect: a detailed explanation of the transformation of the given input to a network that will serve as

input to Ford$-$Fulkerson Algorithm, as well as how you recover a solution to the given problem from the

max flow returned by Ford$-$Fulkerson Algorithm.