Polymatroids with Network Coding

TL;DR

This paper unifies network coding theories for correlated and independent sources, establishing necessary and sufficient conditions for reliable multicasting over noisy networks using polymatroidal structures.

Contribution

It provides a unified framework for network coding with correlated and independent sources, extending classical theorems to noisy networks with polymatroidal analysis.

Findings

Unified necessary and sufficient conditions for reliable multicasting

Polymatroidal structures are fundamental in network coding

Addresses both correlated and independent source transmission

Abstract

The problem of network coding for multicasting a single source to multiple sinks has first been studied by Ahlswede, Cai, Li and Yeung in 2000, in which they have established the celebrated max-flow mini-cut theorem on non-physical information flow over a network of independent channels. On the other hand, in 1980, Han has studied the case with correlated multiple sources and a single sink from the viewpoint of polymatroidal functions in which a necessary and sufficient condition has been demonstrated for reliable transmission over the network. This paper presents an attempt to unify both cases, which leads to establish a necessary and sufficient condition for reliable transmission over a noisy network for multicasting all the correlated multiple sources to all the multiple sinks. Furthermore, we address also the problem of transmitting "independent" sources over a multiple-access-type of network as well as over a broadcast-type of network, which reveals that the (co-) polymatroidal structures are intrinsically involved in these types of network coding.