Existence of new inequalities for representable polymatroids

TL;DR

This paper introduces new inequalities for representable polymatroids and demonstrates that existing inequalities are insufficient for their complete characterization, impacting network coding theory.

Contribution

It presents a novel method to combine polymatroids, constructing a counterexample polymatroid that satisfies known inequalities but is not representable.

Findings

Constructed a polymatroid outside the convex cone of representable polymatroids.

Showed this polymatroid satisfies Ingleton and recent inequalities.

Proved these inequalities are not sufficient for representability.

Abstract

An Ingletonian polymatroid satisfies, in addition to the polymatroid axioms, the inequalities of Ingleton (Combin. Math. Appln., 1971). These inequalities are required for a polymatroid to be representable. It is has been an open question as to whether these inequalities are also sufficient. Representable polymatroids are of interest in their own right. They also have a strong connection to network coding. In particular, the problem of finding the linear network coding capacity region is equivalent to the characterization of all representable, entropic polymatroids. In this paper, we describe a new approach to adhere two polymatroids together to produce a new polymatroid. Using this approach, we can construct a polymatroid that is not inside the minimal closed and convex cone containing all representable polymatroids. This polymatroid is proved to satisfy not only the Ingleton inequalities, but also the recently reported inequalities of Dougherty, Freiling and Zeger. A direct consequence is that these inequalities are not sufficient to characterize representable polymatroids.