Information Inequalities via Submodularity and a Problem in Extremal Graph Theory

Area of Science:

• Information Theory
• Graph Theory
• Set Functions

Background:

• Sub/supermodular set functions are fundamental in various mathematical fields.
• Information inequalities play a crucial role in quantifying information flow and uncertainty.
• Extremal graph theory problems often benefit from combinatorial and information-theoretic approaches.

Purpose of the Study:

• To develop a unified framework for deriving inequalities of set functions with sub/supermodularity properties.
• To derive novel information inequalities using Shannon information measures.
• To apply these inequalities to analyze and refine bounds in extremal graph theory.

Main Methods:

• A unified approach for deriving families of inequalities for set functions.
• Application of Shannon information measures to derive information inequalities.
• Utilizing a generalized Han's inequality for analysis in extremal graph theory.

Main Results:

• A unified derivation of inequalities for sub/supermodular set functions.
• New information inequalities and reproduction of known results like generalized Han's inequality.
• Generalized and refined bounds for an extremal graph theory problem using information theory.

Conclusions:

• The unified approach provides a systematic method for deriving set function inequalities.
• The derived information inequalities offer new insights and simplify existing results.
• Information-theoretic analysis yields improved bounds in extremal graph theory.