Generalized Information Inequalities via Submodularity, and Two Combinatorial Problems

2026-01-22

AI Summary

This work aims to unify and extend information inequalities to address central problems in combinatorial geometry and extremal graph theory. Building on the deep connection between submodularity and entropy inequalities, we generalize the frameworks of Madiman–Tetali and Sason by introducing strong and weak convex-functional-type inequalities for submodular functions. Our main contributions include establishing a more comprehensive system of information inequalities, refining the classical Loomis–Whitney projection inequality via incorporation of slice-structure information to obtain tighter bounds, and advancing results in extremal graph theory by replacing the traditional Han inequality with Shearer’s lemma—thereby reproducing and surpassing existing findings. The approach integrates tools from submodular function theory, convex analysis, and information inequalities, demonstrating significant cross-disciplinary innovation.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Information Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

It is well known that there is a strong connection between entropy inequalities and submodularity, since the entropy of a collection of random variables is a submodular function. Unifying frameworks for information inequalities arising from submodularity were developed by Madiman and Tetali (2010) and Sason (2022). Madiman and Tetali (2010) established strong and weak fractional inequalities that subsume classical results such as Han's inequality and Shearer's lemma. Sason (2022) introduced a convex-functional framework for generalizing Han's inequality, and derived unified inequalities for submodular and supermodular functions. In this work, we build on these frameworks and make three contributions. First, we establish convex-functional generalizations of the strong and weak Madiman and Tetali inequalities for submodular functions. Second, using a special case of the strong Madiman-Tetali inequality, we derive a new Loomis-Whitney-type projection inequality for finite point sets in $\mathbb{R}^d$, which improves upon the classical Loomis-Whitney bound by incorporating slice-level structural information. Finally, we study an extremal graph theory problem that recovers and extends the previously known results of Sason (2022) and Boucheron et al., employing Shearer's lemma in contrast to the use of Han's inequality in those works.

Problem

• submodularity
• information inequalities
• convex-functional generalization
• Loomis-Whitney inequality
• extremal graph theory

Innovation

• submodularity
• convex-functional inequalities
• Loomis-Whitney inequality
• Shearer's lemma
• extremal graph theory