An Edge Reduction Lemma for linear network coding and an application to two-unicast networks

In this paper, we study linear network coding over a wireline network of orthogonal capacitated links represented by a directed acyclic graph. Applying the algebraic framework for linear network coding over Koetter-Medard, we recover an Edge Reduction Lemma - a fundamental connection between edge deletion and the network transfer matrix in the context of the algebraic framework. We study the two-unicast network capacity problem where there are two independent sources and two independent destinations. Using the Edge Reduction Lemma, we make a connection between the linear transfer matrices in the two-unicast setting and the Generalized Network Sharing edge cut bound. Finally, using random linear network coding, we also derive an achievable rate region for the two-unicast problem that is computable purely from the various min-cuts in the graph.