Explicit Polyhedral Bounds on Network Coding Rate Regions via Entropy Function Region: Algorithms, Symmetry, and Computation

Abstract

Automating the solutions of multiple network information theory problems, stretching from fundamental concerns such as determining all information inequalities and the limitations of linear codes, to applied ones such as designing coded networks, distributed storage systems, and caching systems, can be posed as polyhedral projections. These problems are demonstrated to exhibit multiple types of polyhedral symmetries. It is shown how these symmetries can be exploited to reduce the complexity of solving these problems through polyhedral projection.

A Numerical Study on the Capacity Region of a Three-Layer Wiretap Network

“…To tackle the projection Q of the original polyhedral cone P in the H-representation onto a small number of coordinates, one idea is to work directly in the projection space and the projection is incrementally built by successive refinement of an initial approximation Q 0 . The difference in the relationship between the initial approximation and the true projection leads to two different projection algorithms, the Convex Hull Method [41][42][43] and Benson's algorithm [38,44]. We give an outline of Benson's algorithm in the following.…”