"Network Flow" maximum flow: the flow of demand, the flow of the lower bound of the sideband

1) The circulation of the point belt demand:

New frame Features: There are multiple supply points (d (v) <0), which are called source points, and there are multiple demand points (d (v) >0), which are called meeting points. It still satisfies the capacity conditions in the traditional maximum flow (0 <= F (e) <= cap (e)) and demand conditions (f_in (v)-f_out (v) = d (v)).

Problem to solve: because there are multiple source and sink points, you no longer consider maximizing the problem, but consider a viable flow (feasibility) that does not meet the capacity conditions and requirements conditions.

The method of judging feasibility is to convert the feasible flow problem with demand {D (v)} into the problem of finding the maximum s-t flow in another network. Another network is constructed in the following way :

Here's an example:

To find the maximum s-t flow in G ', then the maximum flow value is how much to prove that there is a viable flow in the original G ... The answer is P272, theorem 7.50:

a viable flow with {D (v)} exists in G, when and only if the maximum s*-t* flow of G ' has a value of D (where D is the and of all requirements, and is also all supply and).

2) The flow of points with demand and the lower bound of the sideband:

New frame Features: There are multiple supply points (d (v) <0), known as the source point, there are multiple demand points (d (v) >0), which are called sinks, and at the same time, each side e has the minimum flow low (e) Requirements (that is, some edges must be used, And still meet the capacity conditions in the traditional maximum flow (low (E) <= f (e) <= cap (e)) and demand conditions (f_in (v)-f_out (v) = d (v)).

Problem to solve: because there are multiple source and sink points, you no longer consider maximizing the problem, but consider a viable flow (feasibility) that does not meet the capacity conditions and requirements conditions.

the method of judging the feasibility is to convert the feasible flow problem of the dot band requirement {d (v)} and the lower bound {low (E)} to the problem of determining the feasible flow in another network with only dot requirement {d (v)}, and then to find the maximum s-t flow in the third network. Another network is constructed in the following way :
For all edges with a lower volume low (e), the capacity becomes 0 <= F (e) <= cap (e)-Low (e), which is equivalent to providing low (e) of so much traffic in advance, so that the tail node V of E should also be able to provide a corresponding reduction in the flow of low (e); The Head node v should be based on the original flow requirements, plus the low (e) imposed on it. may not be described clearly, look at the picture:

The conversion to the nether is equivalent to v to W with a low (e) flow already flowing past, that is not to add a capacity of low (e) Reverse edge.
Answer: No. The reverse side is equivalent to giving the opportunity to regret, adding a reverse edge, after which you can push back the low (e) flow that has already passed, which is not what we want to see.