On the tightness of Marton's regions for semi-additive broadcast channels

We study cost constrained side-information channels, where the cost function depends on a state which is known only to the encoder. In the additive noise case, we bound the capacity loss due to not knowing the cost state at the decoder and show that it is small under various assumptions, and goes to zero in the limit of weak noise. This model plays an important role in the (non-degraded) broadcast channel. In the semi-additive noise case, we bound the gap between the best known single letter achievable region and the true capacity region, using tools developed for the first problem. In the limit of weak noise, we show that the bounds coincide, thus we get the complete characterization of the capacity region.