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Generalized Information Inequalities via Submodularity, and Two Combinatorial Problems

Gunank Jakhar, Gowtham R. Kurri, Suryajith Chillara, Vinod M. Prabhakaran

TL;DR

The paper investigates how submodularity governs information inequalities and extends this framework via convex-functional methods. It provides convex-functional generalizations of Madiman–Tetali's strong and weak inequalities for submodular functions, enabling non-linear transformations that preserve information-structure insights. A Loomis–Whitney–type projection bound that incorporates slice-level information is derived, strengthening the classical geometric bound. Finally, an extremal graph theory problem is analyzed through Shearer's lemma, yielding bounds that generalize prior results and demonstrating the broad applicability to combinatorial counting and geometric problems.

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.

Generalized Information Inequalities via Submodularity, and Two Combinatorial Problems

TL;DR

The paper investigates how submodularity governs information inequalities and extends this framework via convex-functional methods. It provides convex-functional generalizations of Madiman–Tetali's strong and weak inequalities for submodular functions, enabling non-linear transformations that preserve information-structure insights. A Loomis–Whitney–type projection bound that incorporates slice-level information is derived, strengthening the classical geometric bound. Finally, an extremal graph theory problem is analyzed through Shearer's lemma, yielding bounds that generalize prior results and demonstrating the broad applicability to combinatorial counting and geometric problems.

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 , 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.
Paper Structure (13 sections, 7 theorems, 53 equations, 1 figure)

This paper contains 13 sections, 7 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized Information Inequalities via Submodularity
  4. Two Combinatorial Problems
  5. A Strong Loomis-Whitney-Type Inequality
  6. An Extremal Graph Theory Problem
  7. Proof of Theorem \ref{['SasonImprov']}
  8. Fractional covering/packing version for Theorem \ref{['SasonImprov']}
  9. Proof of Corollary \ref{['WeakSasonImprov']}
  10. Proof of Theorem \ref{['StrongLW']}
  11. Proof of Theorem \ref{['StrongLW-ddim']}
  12. Proof of Theorem \ref{['ExtremalGraph']}
  13. Details of Remark \ref{['BLMSasonRemark']}

Key Result

Theorem 1

Let $f: 2^{[1:n]} \rightarrow \mathbb{R}$ be any submodular function with $f(\phi) = 0$. Let $\gamma:\mathcal{F}\rightarrow \mathbb{R}_+$ be any fractional partition with respect to a family $\mathcal{F}$ of subsets of $[1:n]$. Then the following statements hold: The fractional partition $\gamma$ in the lower and upper bounds can be replaced by fractional packing $\beta$ and fractional covering $

Figures (1)

  • Figure 1: Illustration of the comparison between the classical Loomis–Whitney bound and the strong Loomis–Whitney–type bound in $\mathbb{R}^3$. The yellow points represent the set $S$. The red, blue, and green points denote the projections of $S$ onto the $xy$-, $yz$-, and $zx$-planes, respectively. The slice at $x=2$ has the largest number of distinct $yz$-projections, leading to a tighter bound than the Loomis-Whitney bound.

Theorems & Definitions (23)

  • Definition 1: Sub/Supermodular, and Modular Functions Fujishige05
  • Definition 2: Fractional Partition, Covering, Packing
  • Theorem 1: MadimanT10
  • Theorem 2
  • Remark 1
  • Remark 2
  • proof : Proof Sketch of Theorem \ref{['SasonImprov']}
  • Corollary 1
  • Remark 3
  • Remark 4
  • ...and 13 more