Multicast in Networks of Broadcast Channels—Part II: Representation of Bounds on the Multicast Capacity Region

We apply the polymatroid broadcast model to inner and outer bounds on the multicast capacity region of networks of broadcast channels, which are a suitable network model to represent the wireless broadcast advantage in wireless networks. The considered model allows for a channel state such that the state sequence is noncausally known to all nodes to model for example medium access mechanisms. We establish that the cut-set outer bound and the noisy network coding inner bound with independent output quantization admit a formulation in the polymatroid broadcast model proposed in Part I for networks of broadcast channels with independent noise, i.e., the channel outputs are independent across all nodes given the channel inputs and the channel state. This applies in particular to networks of deterministic broadcast channels and networks of erasure broadcast channels with independent erasures given the channel state. The polymatroid broadcast structure inherent to these two bounds enables us to characterize the corresponding multicast rate regions by means of the weighted sum multicast rate maximization problem, which can be solved using the dual decomposition approach in Part I, which is based on the polymatroid broadcast model. For networks of erasure broadcast channels, we propose a simple erasure quantization strategy for noisy network coding based on this dual decomposition approach and submodular maximization. This approach achieves a sum rate performance that is close to the cut set outer bound, which is demonstrated in a bidirectional communication example.