  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.
