Truncation Technique for Characterizing Linear Polymatroids

Linear polymatroids have a strong connection to network coding. The problem of finding the linear network coding capacity region is equivalent to the characterization of all linear polymatroids. It is well known that linear polymatroids must satisfy the inequalities of Ingleton (Combin. Math. Appln., 1971). However, it has been an open question for years as to whether these inequalities are sufficient. It was until recently that new subspace rank inequalities have been discovered (independently by Kinser and Dougherty, ). In this paper, we propose a new approach to investigate properties of linear polymatroids. Specifically, we demonstrate how to construct a new polymatroid that satisfies not only the Ingleton and DFZ inequalities, but also lies outside the minimal closed and convex cone containing all linear polymatroids. Using this polymatroid, we prove that all truncation-preserving inequalities (including Ingleton inequalities and DFZ inequalities) are insufficient to characterize linear polymatroids.