Is there a canonical network for network information theory?

Abstract

In recent years, work has begun to emerge demonstrating intriguing relationships between seemingly disparate information theoretic problems. For example, recent results establish powerful ties between solutions for networks of memoryless channels and networks of noiseless links (network coding networks), between network coding networks in which every internal node can code and a particular subset of network coding networks in which only a single internal node can code (index coding networks), and between multiple multicast demands on memoryless networks and multiple unicast demands on memoryless networks. While the results vary widely, together, they hint at the potential for a unifying theory. In this work, we consider one possible framework for such a theory. Inspired by ideas from the field of computational complexity theory, the proposed framework generalizes definitions and techniques for reduction, completeness, and approximation to the information theoretic domain. One possible outcome from such a theory is a taxonomy of information theoretic problems where problems in the same taxonomic class share similar properties in terms of their code designs, capacities, or other forms of solution. Another potential outcome is the identification of small classes of network information theoretic problems whose solutions, were they available, would solve all information theoretic problems in a much larger class. A third potential outcome is the development of techniques by which approximate solution for one family of network information theoretic problems can be obtained from precise or approximate solution of another family of networks.