Network Coding Bounds and Submodularity

Abstract

This chapter applies the multicast models for wireless networks, in particular, the proposed generalized submodular cut model and the proposed penalized polymatroid broadcast model, to general outer bounds, inner bounds, and approximations of the multicast capacity regions of discrete memoryless networks. It focuses on the class of independent noise discrete memoryless networks, i.e., the outputs are independent given the inputs. Determining the multicast capacity region of discrete memoryless networks, which includes the capacity (region) of point-to-point communication, bidirectional communication, and multiple access communication in relay networks, has been a longstanding open problem. The multicast capacity region has only recently been determined for some special cases, namely, graphical networks, hypergraphical networks, deterministic linear finite field networks, networks of independent deterministic broadcast channels, and networks of independent erasure broadcast channels with erasure location side-information at the destinations. For general networks, one has to resort to studying inner and outer bounds on the multicast capacity region. The analysis in this chapter focuses on four multicast rate regions related to the multicast capacity region, namely, the cut-set outer bound, the independent input approximation of the cut-set outer bound, the noisy network coding inner bound, and the elementary hypergraph decomposition inner bound. The goal of this chapter is to identify conditions under which these bounds admit representations by means of multicast rate regions that are generated by submodular cut rate regions, polymatroid broadcast rate regions, or hyperarc rate regions. In particular, this chapter reveals an intricate relationship between submodularity and independent noise networks, based on the submodularity of the entropy function on sets of random variables.