Flow with Lower Bounds

Some problems require minimum flow on certain edges. Edge $(u,v)$ must carry at least $l(u,v)$ flow and at most $c(u,v)$ flow. Convert the problem: subtract $l(u,v)$ from the capacity and add $l(u,v)$ to a demand at $u$ and supply at $v$.

Then solve the modified max flow problem. If a feasible flow exists, you can recover it from the converted solution. This technique is advanced but comes up in problems like circulation and scheduling with constraints.