The network flow with the upper and lower bounds of __ graph theory

Transferred from http://blog.csdn.net/regina8023/article/details/45815023 It 's written in front .

Network flow with upper and lower bounds is limited by the flow of the edge, which must be within the range of [Down,up].

In fact, the ordinary network flow is a special network with the upper and lower bounds of the flow, but each side of the flow limit to [0,cap]. Classification

There are two kinds of network flows with upper and lower bounds:

Have sink
All points are required to meet the flow balance

No sink
In addition to the source point meeting point to meet the flow balance (source point only outflow, meeting point only inflow) no Yuanhui feasible flow

We can think in order to meet the lower limit of traffic, you can directly set the flow of the edge to Up−down.

But after we figure out the maximum flow of the graph, plus the lower down, it makes certain points that don't meet the flow balance.

How to solve it.

We can create additional sources and additional sinks to supplement and absorb these flows.

Specifically, this is:

From the theory of graph theory Hubertau

So for a side u->v, we have three edges in the new diagram: U->v cap=up−down s->v Cap=down u->t