Generalized Information Inequalities via Submodularity, and Two Combinatorial 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 $\mathbb{R}<sup>d$, 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.

• The paper provides generalized inequalities for convex functions of submodular set functions, extending earlier results by Madiman and Tetali, with applications in extremal combinatorics and probability
• These generalized inequalities lead to a refined Loomis–Whitney-type projection inequality to improve estimates of the cardinality of the set and yield tighter entropy-based bounds by incorporating localized structure.
• For extremal graph theory, a new edge-counting bound for relatively confusability graphs recovers and extends prior work, with computational insights into node-adjacency under partial noise corruption.

Overview

This paper, by Jakhar, Kurri, Chillara, and Prabhakaran (IIIT Hyderabad and TIFR), develops convex-functional generalizations of the Madiman–Tetali inequalities for submodular functions and applies them to two combinatorial problems: a strengthened Loomis–Whitney-type projection inequality for finite point sets in $\mathbb{R}^d$, and an extremal graph theory problem motivated by confusability graphs of noisy channels. The unifying theme is that entropy is submodular, so inequalities proved for arbitrary submodular set functions specialize to information-theoretic statements, and conversely entropy methods yield combinatorial counting bounds.

The paper builds on two prior frameworks. Madiman and Tetali established strong and weak fractional inequalities for submodular functions that subsume Han's inequality, Shearer's lemma, and Fujishige's inequalities; Sason introduced a convex-functional approach in which a convex function $g$ is applied to normalized submodular values. The present work merges these lines: it applies $g$ to normalized values appearing in the strong form of the Madiman–Tetali inequality, which conditions on additional structure (the sets $<S$ and $S^c \setminus >S$), yielding strictly tighter bounds than Sason's versions in certain regimes.

Convex-functional generalizations

Let $\gamma$ be a fractional partition over a family $\mathcal{F}$ of subsets of $[1:n]$, let $f$ be submodular with $f(\phi)=0$, and let $g$0 be monotonically non-decreasing. The main theorem states:

• If $g$1 is convex, then

$g$2

• If $g$3 is concave, then

$g$4

Both assertions also hold with $g$5 supermodular and $g$6 monotonically non-increasing. Analogous statements hold for fractional coverings and packings when $g$7 is non-decreasing in $g$8. The proof is elementary: normalize the strong Madiman–Tetali bound, note that the weights $g$9 sum to one because $g$0 is a fractional partition, and apply Jensen's inequality.

A weakened corollary drops conditioning on the upper side ($g$1 instead of $g$2) and adds conditioning on the lower side ($g$3), recovering exactly Sason's earlier inequality in the uniform-family case while the strong version remains strictly stronger. Notably, the corollary also supplies necessary and sufficient equality conditions: for strictly increasing $g$4, equality holds if and only if $g$5 is linear on an interval containing the relevant normalized values and $g$6 is modular. This extends the equality analysis of Jakhar et al. for the weak Madiman–Tetali inequalities to the convex-functional setting.

Two consequences deserve emphasis. First, taking $g$7 and $g$8 recovers an entropy power-type inequality for joint distributions due to Madiman and Tetali, while the strong form yields the strictly sharper bound

$g$9

Second, the paper observes a structural limitation: unlike the linear Madiman–Tetali inequalities, whose upper- and lower-bound gaps satisfy an exact duality under complementation of the family, the convex-functional inequalities do not in general admit such a duality because $<S$0 may be nonlinear.

A strong Loomis–Whitney-type inequality

For a finite point set $<S$1 of size $<S$2, the classical Loomis–Whitney inequality gives $<S$3, where $<S$4, etc., are projection cardinalities. Radhakrishnan's entropy proof uses Han's inequality. The paper instead applies the strong form of the Madiman–Tetali inequality, which permits conditioning on coordinates already revealed, and obtains

$<S$5

where $<S$6 is the maximum number of distinct $<S$7 pairs among points sharing a single $<S$8-coordinate. Since $<S$9, this always improves the classical bound whenever slice-level structure is available. A worked example shows the improvement concretely: for a six-point configuration with $S^c \setminus >S$0, $S^c \setminus >S$1, $S^c \setminus >S$2, the classical bound is 100 while the new bound is 80. The argument generalizes to $S^c \setminus >S$3 dimensions as

$S^c \setminus >S$4

and, because the strong Han inequality holds for any ordering of the ground set, analogous inequalities arise from choosing any coordinate as the "sliced" one. The implication is that projection inequalities can be tightened using information about coordinate slices rather than global projections alone — though the price is that the bound requires knowledge of slice cardinalities, not just projections.

Extremal graph problem via Shearer's lemma

The second application concerns graphs $S^c \setminus >S$5 with vertex set $S^c \setminus >S$6 where two vertices are adjacent iff their difference pattern $S^c \setminus >S$7 belongs to an arbitrary prescribed family $S^c \setminus >S$8. This generalizes both Boucheron et al. (where $S^c \setminus >S$9 consists of singletons) and Sason (where $\gamma$0 contains all sets of size at most $\gamma$1), and admits a channel-coding interpretation: vertices are binary codewords and edges indicate possible confusability through a noisy channel. Crucially, confusability here depends on the full difference pattern rather than only Hamming distance, which is precisely why Shearer's lemma — valid for arbitrary families — is needed, whereas Han's inequality sufficed only for the equal-size families treated previously.

Writing $\gamma$2 for the size-$\gamma$3 members of $\gamma$4, $\gamma$5, and defining integers $\gamma$6 and $\gamma$7 that capture, respectively, how many vertices agree with $\gamma$8 outside $\gamma$9 while the flipped vector $\mathcal{F}$0 is or is not in $\mathcal{F}$1, the paper proves

$\mathcal{F}$2

where $\mathcal{F}$3 is the maximum integer such that every index appears in at least $\mathcal{F}$4 complements of members of $\mathcal{F}$5. The proof lower-bounds $\mathcal{F}$6 in terms of edge counts using the conditional probability bounds $\mathcal{F}$7 and $\mathcal{F}$8, and upper-bounds the same quantity via Shearer's lemma applied to the complement family $\mathcal{F}$9; the paper notes explicitly that Han's inequality is inapplicable here since $[1:n]$0 does not contain all subsets of a fixed size. This bound recovers the results of Boucheron et al. ($[1:n]$1) and of Sason as special cases, and a worked example with a mixed-size family yields $[1:n]$2.

Limitations and open questions

Several caveats are stated or implicit in the paper. The fractional covering/packing variants of the convex-functional inequalities require the monotonicity assumption that $[1:n]$3 be non-decreasing in $[1:n]$4, which restricts the class of admissible submodular functions. The absence of a gap duality between the upper and lower convex-functional bounds means the two sides cannot be related by complementation as in the linear theory. The strong Loomis–Whitney inequality requires slice-level information ($[1:n]$5), which may not be available when only projection counts are known, and the paper does not address whether the improvement can be tight or characterize when it is attained. For the extremal graph bound, the parameters $[1:n]$6 and $[1:n]$7 depend on the internal structure of $[1:n]$8; the examples evaluate them only at their minimum admissible values, and no general method for computing tight values is given. Whether the convex-functional framework admits equality characterizations beyond the modular/linear case analyzed in the corollary remains open.

Conclusion

The paper contributes a clean synthesis of the Madiman–Tetali and Sason frameworks: applying convex functions to normalized values within the strong fractional inequalities yields bounds that dominate existing ones, with full equality conditions in the weak case. Its combinatorial payoffs are a slice-refined Loomis–Whitney inequality in arbitrary dimension and a unified edge-counting bound for pattern-confusability graphs on the hypercube that strictly generalizes prior results by replacing Han's inequality with Shearer's lemma. The main open issues are the loss of gap duality under nonlinear $[1:n]$9, the monotonicity requirement for covering/packing variants, and the computation of tight structural parameters in the graph application.