Information Inequalities via Submodularity and a Problem in Extremal Graph Theory

TL;DR

This paper develops a unified method for deriving information inequalities using submodularity, introduces new inequalities, and applies them to extremal graph theory, leading to refined bounds from an information-theoretic perspective.

Contribution

It presents a unified approach for deriving inequalities for submodular functions, introduces new information inequalities, and applies these to extremal graph theory problems.

Findings

New information inequalities derived, including a generalized Han's inequality.

Unified framework simplifies existing inequalities and reproduces known results.

Application to graph theory yields refined bounds using information-theoretic methods.

Abstract

The present paper offers, in its first part, a unified approach for the derivation of families of inequalities for set functions which satisfy sub/supermodularity properties. It applies this approach for the derivation of information inequalities with Shannon information measures. Connections of the considered approach to a generalized version of Shearer's lemma, and other related results in the literature are considered. Some of the derived information inequalities are new, and also known results (such as a generalized version of Han's inequality) are reproduced in a simple and unified way. In its second part, this paper applies the generalized Han's inequality to analyze a problem in extremal graph theory. This problem is motivated and analyzed from the perspective of information theory, and the analysis leads to generalized and refined bounds. The two parts of this paper are meant to be independently accessible to the reader.