The circulation problem and its variants are a generalisation of network flow problems, with the added constraint of a lower bound on edge flows, and with flow conservation also being required for the source and sink (i.e. there are no special nodes).
In the minimum cost variant of the problem, minimize In a multi-commodity circulation problem, you also need to keep track of the flow of the individual commodities: There is also a lower bound on each flow of commodity.
[1] For the case of multiple commodities, the problem is NP-complete for integer flows.
[2] For fractional flows, it is solvable in polynomial time, as one can formulate the problem as a linear program.
Below are given some problems, and how to solve them with the general circulation setup given above.