Almost Entropic Polymatroids

• Almost entropic polymatroids are defined as normalized, monotone, submodular set functions that can be approximated arbitrarily well by entropy vectors.
• The structural analysis involves a unique cone decomposition into tight and modular parts using convolution, which preserves almost entropicity during transformations.
• Recent research reveals complex phenomena including non-Shannon inequalities, duality failures, and detailed low-dimensional face classifications, underscoring the field’s rich geometric and computational challenges.

Searching arXiv for recent and foundational papers on almost entropic polymatroids, entropy regions, duality, and face characterizations. arXiv search query: "almost entropic polymatroids entropy region duality matroid" An almost entropic polymatroid is a polymatroid rank function lying in the closure $\overline{\Gamma_n^*}$ of the entropy region $\Gamma_n^*$ generated by discrete random variables. Equivalently, it is a normalized, monotone, submodular set function that can be approximated arbitrarily well by entropy vectors. The notion sits between exact entropicity and the full polymatroid cone $\Gamma_n$, the Shannon outer bound, and it is therefore the natural interface between information inequalities, matroid theory, and the geometry of entropy regions. Recent work studies almost entropic polymatroids both globally—through cone decompositions, non-Shannon inequalities, duality, and definability results—and locally, through exact classifications of entropic and almost entropic behavior on low-dimensional faces of $\Gamma_n$ (Matúš et al., 2013, He et al., 3 Feb 2026).

1. Definition and ambient geometric framework

Let $N_n=\{1,2,\dots,n\}$. For a random vector $(X_i)_{i\in N_n}$, the associated entropy function is

$\mathbf{h}(A)=H(X_A), \qquad A\subseteq N_n.$

All such entropy functions form the entropy region $\Gamma_n^*$. The Shannon outer bound is the polymatroid region $\Gamma_n$, defined by

$$\mathbf{h}(A)\ge 0,\qquad \mathbf{h}(A)\le \mathbf{h}(B)\ \text{if } A\subseteq B,\qquad \mathbf{h}(A)+\mathbf{h}(B)\ge \mathbf{h}(A\cap B)+\mathbf{h}(A\cup B).$$

A point of $\Gamma_n^*$0 is a polymatroid; if it is integer-valued and satisfies $\Gamma_n^*$1, it is a matroid rank function (He et al., 3 Feb 2026).

A polymatroid is entropic if it is realized exactly by Shannon entropies of discrete random variables. It is almost entropic if it lies in the closure $\Gamma_n^*$2. This closure is a closed convex cone, and its relative interior lies inside the entropy region. The inclusions

$\Gamma_n^*$3

are always valid; for $\Gamma_n^*$4, one has $\Gamma_n^*$5, while for $\Gamma_n^*$6 the inclusion is strict (Matúš et al., 2013, Mejia et al., 2015).

This distinction removes a common ambiguity. “Almost entropic” does not mean “realized exactly after rescaling by one convenient model”; it means “approximable by entropic polymatroids,” i.e. membership in a closure. That closure-based interpretation is what makes almost entropic polymatroids the correct domain for asymptotic information inequalities, limit constructions, and boundary phenomena.

2. Cone structure, convolution, and reduction to the tight part

A central structural result is that every polymatroid rank function $\Gamma_n^*$7 decomposes uniquely as

$\Gamma_n^*$8

where $\Gamma_n^*$9 is tight and $\Gamma_n$0 is modular. The formulas are

$\Gamma_n$1

$\Gamma_n$2

A polymatroid is tight if $\Gamma_n$3 for every $\Gamma_n$4. This yields a direct-sum decomposition of the cone of polymatroids into tight and modular parts, and the same separation persists at the level of almost entropic objects: if $\Gamma_n$5, then $\Gamma_n$6, while modular polymatroids are already entropic (Matúš et al., 2013).

The technical engine behind this reduction is convolution. For polymatroids $\Gamma_n$7 on $\Gamma_n$8,

$\Gamma_n$9

If $\Gamma_n$0 is modular, then $\Gamma_n$1 is again a polymatroid. Convolution with modular polymatroids is used to construct principal extensions, truncations, and contractions that preserve almost entropicity. This gives a controlled way to move through the cone while staying inside the almost entropic class (Matúš et al., 2013).

A further consequence is that

$\Gamma_n$2

Thus the genuinely difficult almost-entropic-but-non-entropic behavior is forced onto the boundary. This suggests that many of the subtle pathologies of entropy regions are boundary phenomena in the tight cone rather than bulk phenomena in the full polymatroid cone.

3. Non-Shannon inequalities and the complexity of the almost entropic region

The four-variable case already exhibits complexity beyond Shannon-type inequalities. Book extensions provide one systematic mechanism for generating valid inequalities. If $\Gamma_n$3 is a polymatroid on $\Gamma_n$4, an $\Gamma_n$5-page book extension $\Gamma_n$6 over the spine $\Gamma_n$7 is a polymatroid on

$\Gamma_n$8

such that the pages are totally independent over $\Gamma_n$9 and each restriction to $N_n=\{1,2,\dots,n\}$0 is isomorphic to $N_n=\{1,2,\dots,n\}$1. The associated $N_n=\{1,2,\dots,n\}$2-page book inequalities form a family $N_n=\{1,2,\dots,n\}$3, and the main theorem states that if a four-element polymatroid has an $N_n=\{1,2,\dots,n\}$4-page book extension over a two-element spine, then it satisfies all inequalities in $N_n=\{1,2,\dots,n\}$5 and their symmetry variants. Consequently every book inequality is a valid information inequality for entropic and almost entropic polymatroids (Csirmaz, 2013).

The two-page case recovers a classical bridge to Zhang–Yeung-type inequalities. For four variables, a 2-page book extension over a 2-element spine is characterized by the Zhang–Yeung inequality and its symmetry images. The broader book-inequality framework also recovers one of Matúš’s earlier infinite families and the Dougherty–Freiling–Zeger inequality family, thereby organizing several previously distinct non-Shannon constructions inside one extension-based formalism (Csirmaz, 2013).

The geometric complexity of the almost entropic region is sharper than mere non-polyhedrality. A standard theorem states that the almost-entropic region $N_n=\{1,2,\dots,n\}$6 is not semialgebraic; as a corollary, it is not polyhedral (Mejia et al., 2015). In this sense, the boundary cannot be captured by finitely many polynomial equalities and inequalities, let alone by finitely many linear inequalities. Combined with Matúš-type infinite families, this establishes that the four-variable almost entropic region has genuinely infinite descriptive complexity.

4. Low-dimensional face structure and explicit classifications

One line of work studies almost entropic behavior by intersecting $N_n=\{1,2,\dots,n\}$7 with selected faces and determining

$N_n=\{1,2,\dots,n\}$8

For $N_n=\{1,2,\dots,n\}$9, a complete enumeration procedure over pairs of extreme rays yields 59 distinct types of 2-dimensional faces. The cone $(X_i)_{i\in N_n}$0 has 28 facets and 41 extreme rays, falling into 11 types. Characterized families include all-entropic faces, faces reducible to known three-variable Matúš or Chen–Yeung faces, and several genuinely new four-variable cases characterized by graph-coloring arguments and explicit support constructions (Liu et al., 2023).

The face descriptions are highly rigid. Examples include the “purple” form

$(X_i)_{i\in N_n}$1

the “blue” quantization condition

$(X_i)_{i\in N_n}$2

a product-type quantization

$(X_i)_{i\in N_n}$3

and a partition-form condition

$(X_i)_{i\in N_n}$4

These formulas show that even on 2-dimensional slices, entropicity may be governed by discrete logarithmic constraints or piecewise-linear threshold behavior rather than by open convex sectors (Liu et al., 2023).

A more general classification is given for 2-dimensional faces of the form

$(X_i)_{i\in N_n}$5

where $(X_i)_{i\in N_n}$6 is a matroid extreme ray and $(X_i)_{i\in N_n}$7 is a rank-1 matroid. In coordinates

$(X_i)_{i\in N_n}$8

the entropic behavior falls into exactly four types: all-entropic, Matúš-type, Chen–Yeung-type, and non-entropic. The Matúš-type regime has the piecewise form

$(X_i)_{i\in N_n}$9

for some integer $\mathbf{h}(A)=H(X_A), \qquad A\subseteq N_n.$0, while points below the boundary satisfy

$\mathbf{h}(A)=H(X_A), \qquad A\subseteq N_n.$1

The Chen–Yeung-type regime is constrained to discrete vertical lines

$\mathbf{h}(A)=H(X_A), \qquad A\subseteq N_n.$2

The all-entropic regime allows all $\mathbf{h}(A)=H(X_A), \qquad A\subseteq N_n.$3, whereas the non-entropic regime contains no positive entropic point at all (He et al., 3 Feb 2026).

For faces of the form $\mathbf{h}(A)=H(X_A), \qquad A\subseteq N_n.$4, rank-1 $\mathbf{h}(A)=H(X_A), \qquad A\subseteq N_n.$5 gives the all-entropic case, whereas matroid extreme rays of rank at least $\mathbf{h}(A)=H(X_A), \qquad A\subseteq N_n.$6 yield Chen–Yeung-type behavior. For $\mathbf{h}(A)=H(X_A), \qquad A\subseteq N_n.$7, the face is Matúš-type precisely in a narrowly specified rank-2 circuit configuration and Chen–Yeung-type otherwise. The general theorem states that every 2-dimensional face of the form $\mathbf{h}(A)=H(X_A), \qquad A\subseteq N_n.$8 is one of the four types above (He et al., 3 Feb 2026).

5. Duality, balanced inequalities, and obstruction phenomena

Polymatroid duality is defined by

$\mathbf{h}(A)=H(X_A), \qquad A\subseteq N_n.$9

and transfers to entropy expressions through

$\Gamma_n^*$0

For information inequalities, duality is involutive modulo balancing: $\Gamma_n^*$1 is always balanced; if $\Gamma_n^*$2 is balanced, then $\Gamma_n^*$3; if not, $\Gamma_n^*$4 is the balanced version of $\Gamma_n^*$5. Shannon-type inequalities are stable under this duality operation, so the full polymatroid cone remains duality-closed (Kaced, 2016).

The almost entropic cone does not. For $\Gamma_n^*$6, the entropy region is not closed under duality. Equivalently, there exists an almost entropic polymatroid on five variables whose dual is not almost entropic. The proof uses the MMRV inequality on five variables, computes its formal dual, and evaluates the dual inequality on a carefully chosen binary distribution for which the dual expression becomes negative for sufficiently small $\Gamma_n^*$7. This resolves negatively an open question of Matúš from 1992 (Kaced, 2016).

This failure of duality closure is not merely a formal defect. It has consequences for matroid-based obstruction theory. Extension properties give necessary conditions for almost entropicity: the class of almost entropic polymatroids satisfies both the Ahlswede–Körner and copy lemma properties. Thus failure of these properties certifies non-almost-entropic behavior. This criterion yields explicit examples: the Vámos matroid is not almost entropic, $\Gamma_n^*$8 is not almost entropic, and the duals of several tic-tac-toe-configuration matroids are not almost entropic (Bamiloshin et al., 2023).

6. Representability classes, undecidability, and related directions

Almost entropicity occupies an intermediate position among several representation classes. Joint Shannon entropies define entropic polymatroids; limits of entropic polymatroids are almost entropic; linearly representable polymatroids are almost entropic; folded linear polymatroids are almost entropic; and, as stated in the extension-properties literature, algebraic matroids are almost entropic as well. At the same time, almost entropic is broader than algebraic or linear, and it is not closed under duality (Bamiloshin et al., 2023).

For matroids, entropic representability is the exact requirement that

$\Gamma_n^*$9

for some discrete random variables and $\Gamma_n$0. Almost entropicity is the approximation version: for every $\Gamma_n$1, one can approximate the rank function uniformly by a scaled entropy function. A parallel notion on the linear-algebraic side is almost multilinear representability, defined through approximation by scaled linear polymatroids (Kühne et al., 2022).

Two algorithmic boundaries are established. First, deciding whether a matroid is entropic is undecidable. Second, deciding whether a matroid is almost multilinear is undecidable. The reductions proceed through generalized Dowling geometries built from group presentations, translating group-theoretic word problems into entropy or approximate subspace realizability. The same framework yields undecidability of the conditional independence implication problem and implies that deciding whether an access structure admits an ideal secret sharing scheme is undecidable. By contrast, undecidability of almost entropic representability itself is not proved there; the paper identifies it as a natural extension and conjectures that similar methods should work (Kühne et al., 2022).

These developments suggest a broad picture. Almost entropic polymatroids are simultaneously geometric objects in a non-polyhedral, non-semialgebraic cone; local objects whose face intersections can often be classified exactly; and obstruction-sensitive objects constrained by extension properties, non-Shannon inequalities, and duality failure. The present theory therefore combines convex-geometric reduction, matroidal structure, information inequalities, and computational undecidability into a single research program centered on the closure of entropy vectors.