Extremal solutions for Network Flow with Differential Constraints -- A Generalization of Spanning Trees

TL;DR

This paper extends the classical relationship between extreme points of network flow feasibility regions and spanning trees to a more complex setting involving differential constraints, with applications in energy network expansion planning.

Contribution

It generalizes the extremal solutions characterization to problems with simultaneous differential and network flow constraints, identifying conditions where classical graph-theoretic structures still apply.

Findings

Extreme points correspond to graph structures even with differential constraints

Characterization of graphs where exceptions to the classical relationship do not occur

Criteria for parameter values that prevent exceptional cases

Abstract

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 problems where the constraints of both problems appear simultaneously, a variant which is motivated by an application in the expansion planning of energy networks. We show that all extreme points still directly correspond to graph-theoretical structures in the underlying network. The reverse is generally also true in all but certain exceptional cases. We furthermore characterize graphs in which these exceptional cases never occur and present additional criteria for when those cases do not occur due to parameter values.