Analysis of precoding-based intersession network coding and the corresponding 3-unicast interference alignment scheme

Recently, a new precoding-based intersession network coding (NC) scheme has been proposed, which applies the interference alignment technique, originally devised for wireless interference channels, to the 3-unicast problem of directed acyclic networks. Motivated by the graph-theoretic characterizations of classic linear NC results, this paper investigates several key relationships between the point-to-point network channel gains and the underlying graph structure. Such relationships are critical when characterizing graph- theoretically the feasibility of precoding-based solutions. One example of the applications of our results is to answer (at least partially) the conjectures of the 3-unicast interference alignment technique and the corresponding graph-theoretic characterization conditions.