Entropy Functions on Two-Dimensional Faces of Polymatroidal Region of Degree Four: Part I: Problem Formulation and Graph-Coloring Approach

Abstract: Characterization of entropy functions is of fundamental importance in information theory. By imposing constraints on their Shannon outer bound, i.e., the polymatroidal region, one obtains the faces of the region and entropy functions on them with special structures. In this series of two papers, we characterize entropy functions on the 2-dimensional faces of the polymatroidal region of degree 4. In Part I, we formulate the problem, enumerate all 59 types of 2-dimensional faces of the region by an algorithm, and fully characterize entropy functions on 49 types of them. Among them, those non-trivial cases are mainly characterized by the graph-coloring technique. The entropy functions on the remaining 10 types of faces will be characterized in Part II, among which 8 types are fully characterized, and 2 types are partially characterized.