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

$\begin{array}{rl}\text{maximize}& \sum _{v:(s,v)\in E}{f}_{sv}\\ \begin{array}{r}\text{subject to}\\ \\ \\ \\ \end{array}& \begin{array}{rlrl}{f}_{uv}& \le c_{uv}& & \mathrm{\forall }(u,v)\in E\\ \sum _{u:(u,v)\in E}{f}_{uv}-\sum _{w:(v,w)\in E}{f}_{vw}& =0& & \mathrm{\forall }v\in V\setminus \{s,t\}\\ f_{uv}& \ge 0& & \mathrm{\forall }(u,v)\in E\end{array}\end{array}$