Problem $5$: Consider a maximum flow instance on a network $N(V,A)$ with source $s$ and sink $t,$ and Problem $5$: Consider a maximam flow instance on a network $N(V,A)$ witi sonce $s$ and sink $t_1$ and

capacitios u $u_{a}0$ for ainA. Let $x$ be feasible tlow i$.$e$.$ satisfying

Sliow that

ubrace$({\displaystyle \underset{u=(u)}{\overset{?}{}}},sinA{x}_a$ubrace${)}_{\text{f}\text{l}\text{o}\text{w}\text{i}\text{n}\text{t}\text{o}s}-$ubrace$({\displaystyle \underset{\text{l}\text{o}\text{w}}{\overset{?}{}}}{\displaystyle \underset{\{\text{:}{s}_a)}{\overset{?}{}}},$uinAubrace${)}_{\text{f}\text{l}\text{o}\text{w}\text{o}\text{u}\text{t}ofs}=$ubrace$({\displaystyle \underset{a=(t)}{\overset{?}{}}},uinA{x}_{ai}$ubrace${)}_{\text{f}\text{l}\text{o}\text{w}\text{o}\text{u}\text{t}oft}-$ubrace$({\displaystyle \underset{u=(u)}{\overset{?}{}}},tinA{x}_a$ubrace${)}_{\text{f}\text{l}\text{o}\text{w}\text{i}\text{n}\text{t}\text{o}1}.$

That is$,$ the net flow that exits $s$ is exactly equal to fhe net flow entering t$.$

capacities $u_{a}0$ for ainA. Let $x$ be feasible flow, i$.$e$.,$ satisfying

${\displaystyle \underset{\text{f}\text{l}\text{o}\text{w}\text{i}\text{n}\text{t}\text{o}v}{\overset{{\displaystyle \underset{a=(u)}{\overset{?}{}}}}{}}},vinA{x}_a-$ubrace$({\displaystyle \underset{a=(v)}{\overset{?}{}}},uinA{x}_a$ubrace${)}_{\text{f}\text{l}\text{o}\text{w}\text{o}\text{u}\text{t}ofv}=0$ for all vin$\frac{V}{\frac{?}{?}}$

$0x_a{u}_a$ for all ainA

Show that

ubrace$({\displaystyle \underset{a=(u)}{\overset{?}{}}},sinA{x}_a$ubrace${)}_{\text{f}\text{l}\text{o}\text{w}\text{i}\text{n}\text{t}\text{o}s}-$ubrace$({\displaystyle \underset{a=(s)}{\overset{?}{}}},uinA{x}_a$ubrace${)}_{\text{f}\text{l}\text{o}\text{w}\text{o}\text{u}\text{t}ofs}=$ubrace$({\displaystyle \underset{a=(t)}{\overset{?}{}}},uinA{x}_a$ubrace${)}_{\text{f}\text{l}\text{o}\text{w}\text{o}\text{u}\text{t}oft}-$ubrace$({\displaystyle \underset{a=(u)}{\overset{?}{}}},tinA{x}_a$ubrace${)}_{\text{f}\text{l}\text{o}\text{w}\text{i}\text{n}\text{t}\text{o}t}.$

That is$,$ the net flow that exits $s$ is exactly equal to the net flow entering $t.$