Network flow - minimum flow through edge In my max flow network, I would like to have an edge with upper bound of the flow (a.k.a. edge capacity) $c_{max}$. However, I would also like to add a lower bound for the flow through the edge, $c_{min}$. 

I need to be sure that the minimum flow satisfies not only all the upper bounds (capacities), but also all the lower bounds.

Is it even possible? And if yes, which changes do I need to make to my flow network so that the max flow algorithm takes these lower bounds into account?
Yes, it is possible to solve such problems efficiently.  The minimum-cost circulation problem is a generalization of the standard network flow problem, which allows you to set both lower bounds and upper bounds on the flow through each edge.  (You can set all costs equal to 1.)  There are polynomial-time algorithms to solve instances of the minimum-cost circulation problem.

I don't know if there's a simple way to do with a standard algorithm for network flow.