In this chapter, we consider a simple model, based on a directed tree representation of the variables and constraints, called network matrices. We show how this model as well as its dual have integral optima when used as constraint matrices with integral right-hand sides. Finally, we show the applications of these models, especially in proving the integrality of the dual of the matroid intersection problem in Chapter 5, as well as the dual of the submodular flow problem in Chapter 7.

While our treatment of network matrices is based on its relations to uncrossed structures and their representations, they play a crucial role in the characterization of totally unimodular matrices, which are all constraint matrices that yield integral polytopes when used as constraint matrices with integral right-hand sides [121]. Note that total unimodularity of network matrices automatically implies integrality of the dual program when the right-hand sides of the dual are integral.

The integrality of the dual of the matroid intersection and submodular flow polyhedra can be alternately derived by showing the Total Dual Integrality of these systems [121]. Although our proof of these facts uses iterative rounding directly on the dual, there is a close connection between these two lines of proof since both use the underlying structure on span of the constraints defining the extreme points of the corresponding linear program.