Submodular structure and optimal quantization in Gaussian multiple access relay networks

We discuss noisy network coding bounds on the capacity region of Gaussian multiple access relay networks. These networks feature a single destination node and multiple nodes that can be either sources or relays or both. We show that simple inner and outer bounds on the capacity region obtained from noisy network coding exhibit a submodular structure and differ only in a constant, which is independent of the channel parameters and grows linearly only in the number of dedicated relay nodes. This tightens previous results where the gap grows linear in the total network size. Furthermore, the combination of submodularity with convexity of the bound expressions with respect to the optimal quantization noise parameters leads to an efficient characterization of a noisy network coding achievable rate region for multiple access relay networks via Lagrangian duality.