Is there any variation of the Ford-Fulkerson that adds an extra dimension of "weight" to the edges?
By that, I mean some edges are more ideal than others, and while all the possibilities are there, it will prioritize the ideal edges over the lesser ideal edges.
There are two common generalisations that I know of to add weights.
Min cost flow
Suppose you had a weight for each edge and wanted to compute the flow that satisfied the constraints and had minimum cost. (Cost = sum of weight * units flowing along associated edge)
This problem is called minimum cost flow.
An implementation can be found in networkx called min-cost-flow.
Here is a good topcoder tutorial on a primal-dual approach.
My favorite algorithm for this was actually invented by Fulkerson himself in 1961 and is called the out of kilter algorithm.
Max flow min cost
Suppose you definitely wanted the maximum flow, but wanted to choose the flow with least weight.
This differs from the first interpretation, in that a min-cost-flow may not give the maximum possible flow. For example, suppose we have a single edge start->end with the constraint that the flow is in the range 0 to 1, and a weight of 1.
The min-cost-flow will be a flow of 0, while the max-flow-min-cost will have a flow of 1.
An implementation for this can be found in networkx called max_flow_min_cost.
来源：https://stackoverflow.com/questions/16762231/ford-fulkerson-algorithm-with-weighted-edges