On the feasibility of precoding-based network alignment for three unicast sessions

We consider the problem of network coding across three unicast sessions over a directed acyclic graph, when each session has min-cut one. Previous work by Das et al. adapted a precoding-based interference alignment technique, originally developed for the wireless interference channel, specifically to this problem. We refer to this approach as precoding-based network alignment (PBNA). Similar to the wireless setting, PBNA asymptotically achieves half the minimum cut; different from the wireless setting, its feasibility depends on the graph structure. Das et al. provided a set of feasibility conditions for PBNA with respect to a particular precoding matrix. However, the set consisted of an infinite number of conditions, which is impossible to check in practice. Furthermore, the conditions were purely algebraic, without interpretation with regards to the graph structure. In this paper, we first prove that the set of conditions provided by Das. et al are also necessary for the feasibility of PBNA with respect to any precoding matrix. Then, using two graph-related properties and a degree-counting technique, we reduce the set to just four conditions. This reduction enables an efficient algorithm for checking the feasibility of PBNA on a given graph.