Two unicast information flows over linear deterministic networks

We investigate the two unicast flow problem over layered linear deterministic networks with arbitrary number of nodes. When the minimum cut value between each source-destination pair is constrained to be 1, it is obvious that the triangular rate region {(R₁, R₂) : R₁, R₂ ≥ 0, R₁ + R₂ ≤ 1} can be achieved, and that one cannot achieve beyond the square rate region {(R₁, R₂) : R₁, R₂ ≥ 0, R₁ ≤ 1, R₂ ≤ 1}. Analogous to the work by Wang and Shroff for wired networks [1], we provide the necessary and sufficient conditions for the capacity region to be the triangular region and the necessary and sufficient conditions for it to be the square region. Moreover, we completely characterize the capacity region and conclude that there are exactly three more possible capacity regions of this class of networks, in contrast to the result in wired networks where only two rate regions are possible. Our achievability scheme is based on linear coding over an extension field with at most four nodes performing special linear coding operations, namely interference neutralization and zero forcing, while all other nodes perform random linear coding.