Extremal Solutions for Network Flow with Differential Constraints

In network flow problems, there is a well-known one-to-one relationship between extreme points of the feasibility region and trees in the associated undirected graph. The same is true for the dual differential problem. In this paper, we study a problem variant with both differential constraints and constraints on flow conservation at every node, which we call differential flow. This variant is motivated by an application in the expansion planning of energy networks. We show that all extreme points in the differential flow polytope still directly correspond to graph-theoretical structures in the underlying network, namely a generalization of spanning trees. The reverse is generally also true except in very special cases where the network parameters satisfy a set of particular equations. We furthermore show that these exceptional cases can never occur in cactus graphs and present additional, sufficient criteria for when the one-to-one correspondence between extreme points and graph-theoretical structures holds. Finally, we show that it is generally NP-hard to decide for a specific network whether the graph-theoretical characterization holds for all extreme points.