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We investigate the two unicast flow problem over layered linear deterministicnetworks with arbitrary number of nodes. When the minimum cut value betweeneach source-destination pair is constrained to be 1, it is obvious that thetriangular rate region {(R_1,R_2):R_1,R_2> 0, R_1+R_2< 1} can be achieved, andthat one cannot achieve beyond the square rate region {(R_1,R_2):R_1,R_2> 0,R_1< 1,R_2< 1}. Analogous to the work by Wang and Shroff for wired networks, weprovide the necessary and sufficient conditions for the capacity region to bethe triangular region and the necessary and sufficient conditions for it to bethe square region. Moreover, we completely characterize the capacity region andconclude that there are exactly three more possible capacity regions of thisclass of networks, in contrast to the result in wired networks where only thetriangular and square rate regions are possible. Our achievability scheme isbased on linear coding over an extension field with at most four nodesperforming special linear coding operations, namely interference neutralizationand zero forcing, while all other nodes perform random linear coding.

Two Unicast Information Flows over Linear Deterministic Networks