Entropy Functions on Two-Dimensional Faces of Polymatroid Region with One Extreme Ray Containing Rank-One Matroid

Characterization of entropy functions is of fundamental importance in information theory. By imposing constraints on their Shannon outer bound, i.e., the polymatroidal region, one obtains the faces of the region and entropy functions on them with special structures. In this paper, we characterize entropy functions on 2-dimensional faces of polymatroid region of degree n with one extreme ray containing rank-1 matroid. We classify all such 2-dimensional faces with another extreme ray containing a matroid into four types.

💡 Research Summary

The paper tackles the long‑standing problem of characterizing entropy functions by focusing on two‑dimensional faces of the polymatroid region Γₙ, the Shannon outer bound for entropy vectors. The authors consider faces that are spanned by two extreme rays, one of which contains a rank‑one matroid (i.e., a uniform matroid Uₙ¹,ⁿ′ that consists of a set of parallel elements and the remaining elements as loops). The second extreme ray is required to contain an arbitrary matroid M. Such a face is denoted (M, Uₙ¹,ⁿ′) and can be parametrized by a pair of non‑negative scalars (a, b) so that the rank function on the face is h = a·r_M + b·r_U, where r_M and r_U are the rank functions of M and the rank‑one matroid respectively.

A key technical tool introduced is the p‑characteristic set χ_M = { v ∈ ℤ, v ≥ 2 | log v·r_M ∈ Γₙ* }, which captures the integer values for which a scaled version of the matroid rank function is entropic. When M is a connected matroid, χ_M is non‑empty and governs the admissible values of a in the (a, b) parametrization.

The authors classify every such two‑dimensional face into exactly four mutually exclusive types, based on the entropic status of points (a, b) with a, b > 0:

1. All‑entropic – every point in the positive quadrant is entropic.
2. Matúš‑type – for some v ∈ χ_M, the region log(v − 1) < a < log v together with a + b ≥ log v consists of entropic points, while the region a + b < log⌈e^a⌉ is non‑entropic.
3. Chen‑Yuang‑type – entropic points occur only on the vertical line a = log v (v ∈ χ_M) with any b > 0; all other points are non‑entropic.
4. Non‑entropic – no point with a, b > 0 is entropic; this happens precisely when M itself is non‑entropic.

The paper proves that the non‑entropic case can arise only if the matroid M on the second extreme ray is already non‑entropic (Lemma 3). For faces of the form (M, Uₙ¹,¹), Theorem 7 shows that they are always Chen‑Yuang‑type: any entropic point forces a = log v for some integer v, and the admissible v’s are exactly the elements of χ_M. The proof relies on restricting the rank function to a circuit of M, which reduces the problem to the well‑studied uniform matroid U_{k‑1,k} and invokes known results on its entropy region.

The analysis of faces (M, Uₙ¹,²) is more intricate. Lemma 8 establishes that two distinct extreme‑ray matroids cannot have one’s circuit family contained in the other’s, which guarantees that (M, Uₙ¹,²) is indeed a two‑dimensional face (Proposition 9). Theorem 11 provides a complete dichotomy: if M has total rank 2 and contains a three‑element circuit {1, 2, i} such that every other element is either parallel to i or a loop, then the face is Matúš‑type; otherwise it falls into the Chen‑Yuang‑type. In the Matúš‑type situation the same logarithmic interval conditions as in definition 2 apply, while in the Chen‑Yuang‑type case the only entropic points lie on the line a = log v.

Concrete examples illustrate the classification. The classic Matúš face (U_{2,3}, U_{3}^{1,2}) and the Chen‑Yuang face (U_{2,3}, U_{3}^{1,1}) are recovered as special cases. Additional faces such as (U_{2,4}, U_{4}^{1,3}) and (U_{2,4}, U_{4}^{1,2}) are shown to be Matúš‑type and Chen‑Yuang‑type respectively, with χ_{U_{2,4}} = { v ≥ 3, v ≠ 6 } highlighting “missing pieces” at v = 2 and v = 6.

Overall, the paper extends earlier results on low‑dimensional entropy faces (Matúš 2006, Chen‑Yuang 2012, Liu‑Chen 2020) to arbitrary n, delivering a complete taxonomy of all two‑dimensional faces that involve a rank‑one matroid on one extreme ray. The introduction of the p‑characteristic set and the systematic use of circuit restrictions provide a powerful framework for analyzing non‑Shannon information inequalities. The findings have potential implications for network coding, distributed storage, and any setting where understanding the fine structure of the entropy region is crucial.