A Polymatroid Flow Model for Network Coded Multicast in Wireless Networks

We propose a new model for the wireless broadcast advantage based on a polymatroid structure. This model is a generalization of the predominating hypergraph model. The polymatroid structure yields a general max-flow min-cut characterization of multicast rate regions, which applies to a large variety of channel, physical layer, and medium access models. It includes the state-of-the-art hypergraph flow regions with lossless and lossy hyperarcs, i.e., Shannon rate models and packet erasure networks. Additionally, it generalizes to various other rate regions, e.g., the cut-set outer bounds for networks of a large variety of independent broadcast channels, including networks of independent Gaussian multiple-input multiple-output channels, and the capacity regions for networks of independent deterministic broadcast channels, which can in general not be modeled by the hypergraph flow model. We propose a dual decomposition approach for network utility optimization problems on the polymatroid broadcast flow region, which subsumes existing dual decomposition approaches based on lossless and lossy hypergraph flow regions. Our approach significantly simplifies the decomposition, especially for lossy hypergraph models in packet erasure networks, by fully exploiting the inherent polymatroid structure of the wireless broadcast. Additionally, it can be directly used to fully characterize and evaluate the cut-set bounds for networks of independent broadcast channels with polymatroid structure without previous knowledge about the relevant cuts.