Broadcast Channel Coding: Algorithmic Aspects and Non-Signaling Assistance

We address the problem of coding for classical broadcast channels, which entails maximizing the success probability that can be achieved by sending a fixed number of messages over a broadcast channel. For point-to-point channels, Barman and Fawzi found in~\cite{BF18} a $(1-e^{-1})$-approximation algorithm running in polynomial time, and showed that it is \textrm{NP}-hard to achieve a strictly better approximation ratio. Furthermore, these algorithmic results were at the core of the limitations they established on the power of non-signaling assistance for point-to-point channels. It is natural to ask if similar results hold for broadcast channels, exploiting links between approximation algorithms of the channel coding problem and the non-signaling assisted capacity region. In this work, we make several contributions on algorithmic aspects and non-signaling assisted capacity regions of broadcast channels. For the class of deterministic broadcast channels, we describe a $(1-e^{-1})^2$-approximation algorithm running in polynomial time, and we show that the capacity region for that class is the same with or without non-signaling assistance. Finally, we show that in the value query model, we cannot achieve a better approximation ratio than $\Omega\left(\frac{1}{\sqrt{m}}\right)$ in polynomial time for the general broadcast channel coding problem, with $m$ the size of one of the outputs of the channel.