Beyond the Butterfly - A Graph-Theoretic Characterization of the Feasibility of Network Coding with Two Simple Unicast Sessions

The problem of network coding with two simple unicast sessions is considered for general directed acyclic graphs. An explicit graph-theoretic characterization is provided for the feasibility of whether two symbols at different sources can be simultaneously transmitted to the designated sinks via network coding. The existence of a routing scheme is equivalent to finding edge-disjoint paths. Similarly, in this paper it is proven that the existence of a network coding scheme is equivalent to finding paths with controlled edge overlaps, and the characterization includes the well-studied butterfly graph as a special case. Various generalizations and implications are discussed based on the constructive nature of the flow-based conditions. For example, it is shown that a linear network coding scheme using only six paths is as effective as any non-linear network coding scheme.