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

The problem of automating rate region converse proofs in multi-source network coding is considered, and an algorithm for explicitly computing polyhedral outer bounds on the rate regions of multi-source network coding instances using known bounds on the entropy function region is proposed. This algorithm is based on the Convex Hull Method, which is an algorithm to compute projections of polyhedra. An interpretation of the network symmetry group as a group of symmetries of a polyhedron is discussed, which in turn enables the use of well-known techniques for exploiting symmetry in polyhedral computation to reduce the complexity of calculating the rate region bounds. A variant of the convex hull method that is capable of exploiting the knowledge of the network symmetry group is discussed. The techniques described in this work are implemented in a software package, called the Information Theoretic Converse Prover. The computational performance of convex hull method for explicitly computing rate region outer bounds with varying knowledge of symmetry is discussed, followed by several examples of computation with the Information Theoretic Converse Prover. Index Terms network coding, symmetry, polyhedral projection, computer-assisted proofs