On the inner and outer bounds for 2-receiver discrete memoryless broadcast channels

TL;DR

This paper compares and simplifies the inner and outer bounds for the capacity region of two-receiver discrete memoryless broadcast channels, providing a computable outer bound and a potential example where bounds differ.

Contribution

It proves the equivalence of certain outer bounds, expresses the bounds in a computable form, and presents a channel example suggesting the bounds may differ.

Findings

Inner and outer bounds coincide for known capacity channels

A computable outer bound with bounded auxiliary variables is derived

An example channel indicates the bounds could be different under a conjecture

Abstract

We study the best known general inner bound[MAR '79] and outer bound[N-EG'07] for the capacity region of the two user discrete memory less channel. We prove that a seemingly stronger outer bound is identical to a weaker form of the outer bound that was also presented in [N-EG'07]. We are able to further express the best outer bound in a form that is computable, i.e. there are bounds on the cardinalities of the auxiliary random variables. The inner and outer bounds coincide for all channels for which the capacity region is known and it is not known whether the regions described by these bounds are same or different. We present a channel, where assuming a certain conjecture backed by simulations and partial theoretical results, one can show that the bounds are different.