# A Minimal Set of Shannon-type Inequalities for Functional Dependence Structures

## TL;DR

This paper characterizes a minimal set of Shannon-type inequalities considering functional dependence constraints, aiding in the analysis of network coding capacity and inequality redundancy.

## Contribution

It introduces a method to identify a minimal set of Shannon-type inequalities under functional dependence constraints, reducing redundancy.

## Key findings

- Identifies redundancies among elemental inequalities with functional dependencies
- Provides a framework for computing Shannon outer bounds in network coding
- Simplifies the analysis of feasible source rate regions

## Abstract

The minimal set of Shannon-type inequalities (referred to as elemental inequalities), plays a central role in determining whether a given inequality is Shannon-type. Often, there arises a situation where one needs to check whether a given inequality is a constrained Shannon-type inequality. Another important application of elemental inequalities is to formulate and compute the Shannon outer bound for multi-source multi-sink network coding capacity. Under this formulation, it is the region of feasible source rates subject to the elemental inequalities and network coding constraints that is of interest. Hence it is of fundamental interest to identify the redundancies induced amongst elemental inequalities when given a set of functional dependence constraints. In this paper, we characterize a minimal set of Shannon-type inequalities when functional dependence constraints are present.