Characterize all linear rank inequalities

Characterize the complete set of linear rank inequalities satisfied by all linear polymatroids.

Background

The paper relies on Ingleton’s inequality as a constraint satisfied by linear polymatroids. It emphasizes that linear polymatroids satisfy additional inequalities beyond the basic polymatroid axioms, but that the full system of such inequalities is not known.

A complete characterization would provide stronger tools for optimizing linear width and for understanding the separation between linear and entropic width.

References

We note that no general characterization of all linear rank inequalities is known; see for the state of the art on five or more variables.

— Separating Notions of Graph Width: the Adaptive, Normal, Linear, Entropic, and Submodular Width  (2609.40018 - Lanzinger et al., 30 Sep 2026) in Section 4, subsection “Inequalities in $\text{Lin}_n$”