Broadcast Channel Coding: Algorithmic Aspects and Non-Signaling Assistance

• Published in: IEEE transactions on information theory Vol. 70; no. 11; pp. 7563 - 7580
• Main Authors: Fawzi, Omar, Ferme, Paul
• Format: Journal Article
• Language: English
• Published: IEEE 01.11.2024; Institute of Electrical and Electronics Engineers
• Subjects: Approximation algorithms; Broadcast channel; Channel coding; Computer Science; Correlation; Decoding; non-signaling; nonlocal; Polynomials; Quantum entanglement; Random variables; Non-signaling; Channel Coding; Approximation algorithms; Broadcast channel
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2024.3410047

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 a <inline-formula> <tex-math notation="LaTeX">(1-e^{-1}) </tex-math></inline-formula>-approximation algorithm running in polynomial time, and showed that it is 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 <inline-formula> <tex-math notation="LaTeX">(1-e^{-1})^{2} </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">\Omega \left ({{\frac {1}{\sqrt {m}}}\right) </tex-math></inline-formula> in polynomial time for the general broadcast channel coding problem, with m the size of one of the outputs of the channel.