When the cut condition is enough
Amit Chakrabarti, Lisa Fleischer, Christophe Weibel · 2012
Let G=(V,E) be a supply graph and H=(V,F) a demand graph defined on the same set of vertices. An assignment of capacities to the edges of G and demands to the edges of H is said to satisfy the cut condition if for any cut in the graph, the total demand crossing the cut is no more than the total capacity crossing it. The pair (G,H) is called cut-sufficient if for any assignment of capacities and demands that satisfy the cut condition, there is a multiflow routing the demands defined on $H$ within the network with capacities defined on G.