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We consider the problem of network coding across three unicast sessions overa directed acyclic graph, where each sender and the receiver is connected tothe network via a single edge of unit capacity. We consider a network model inwhich the middle of the network only performs random linear network coding, andrestrict our approaches to precoding-based linear schemes, where the sendersuse precoding matrices to encode source symbols. We adapt a precoding-basedinterference alignment technique, originally developed for the wirelessinterference channel, to construct a precoding-based linear scheme, which werefer to as as a {\em precoding-based network alignment scheme (PBNA)}. Aprimary difference between this setting and the wireless interference channelis that the network topology can introduce dependencies between elements of thetransfer matrix, which we refer to as coupling relations, and can potentiallyaffect the achievable rate of PBNA. We identify all possible such couplingrelations, and interpret these coupling relations in terms of network topologyand present polynomial-time algorithms to check the presence of these couplingrelations. Finally, we show that, depending on the coupling relations presentin the network, the optimal symmetric rate achieved by precoding-based linearscheme can take only three possible values, all of which can be achieved byPBNA.

Precoding-Based Network Alignment For Three Unicast Sessions