Communicating the sum of sources in a 3-sources/3-terminals network; revisited

We consider the problem of multicasting sums over directed acyclic networks with unit capacity edges. A set of source nodes s_i observe independent unit-entropy source processes X_i and want to communicate Σ X_i to a set of terminals t_j. Previous work on this problem has established necessary and sufficient conditions on the s_i -t_j connectivity in the case when there are two sources or two terminals (Ramamoorthy ´08), and in the case of three sources and three terminals (Langberg-Ramamoorthy ´09). In particular the latter result establishes that each terminal can recover the sum if there are two edge disjoint paths between each s_i-t_j pair. In this work, we provide a new and significantly simpler proof of this result, and introduce techniques that may be of independent interest in other network coding problems.