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$\Sigma$ File:Aa4567890.png

The maximum flow problem can be described as the following linear program.

${\begin{aligned}{\text{maximize}}\quad &\sum _{v:(s,v)\in E}f_{sv}\\{\begin{aligned}{\text{subject to}}\\\\\\\\\end{aligned}}\quad &{\begin{aligned}f_{uv}&\leq c_{uv}&\quad &\forall (u,v)\in E\\\sum _{u:(u,v)\in E}f_{uv}-\sum _{w:(v,w)\in E}f_{vw}&=0&\quad &\forall v\in V\setminus \{s,t\}\\f_{uv}&\geq 0&\quad &\forall (u,v)\in E\end{aligned}}\end{aligned}}$