Evaluation of Marton´s inner bound for the general broadcast channel

The best known inner bound on the two-receiver general broadcast channel without a common message is due to Marton. This result was subsequently generalized in and to broadcast channels with a common message. However the latter region is not computable (except in certain special cases) as no bounds on the cardinality of its auxiliary random variables exist. Nor is it even clear that the inner bound is a closed set. The main obstacle in proving cardinality bounds is the fact that the Carathe¿odory theorem, the main known tool for proving cardinality bounds, does not yield a finite cardinality result. Our new tool is based on an identity that relates the second derivative of the Shannon entropy of a discrete random variable (under a certain perturbation) to the corresponding Fisher information. In order to go beyond the traditional Carathe¿odory type arguments, we identify certain properties that the auxiliary random variables corresponding to the extreme points of the inner bound satisfy. These properties are then used to establish cardinality bounds on the auxiliary random variables of the inner bound, thereby proving the computability of the region, and its closedness. Although existence of cardinality bounds renders Marton´s inner bound computable, it is still hard to evaluate the region. It is however shown that the computation can be significantly simplified if we further assume that Marton´s inner bound and the recent outer bound of Nair and El Gamal match at the given particular channel. In order to demonstrate this, we consider a large class of binary input broadcast channels and compute maximum of the sum rate of private messages assuming that the inner and the outer bound match at the given broadcast channel. We also show that the inner and the outer bound do not match for some broadcast channels, thus establishing a conjecture of.