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Truncation Technique for Characterizing Linear Polymatroids

机译：截断技术表征线性多类拟物

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摘要

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.
机译：线性多类拟阵与网络编码有很强的联系。找到线性网络编码能力区域的问题等同于所有线性多拟阵的表征。众所周知，线性多拟阵必须满足Ingleton的不等式（Combin。Math。Appln。，1971）。然而，多年来，关于这些不平等是否足够，一直是一个悬而未决的问题。直到最近，才发现了新的子空间秩不等式（由Kinser和Dougherty独立地发现）。在本文中，我们提出了一种研究线性多类拟物的特性的新方法。具体来说，我们演示了如何构造不仅满足Ingleton和DFZ不等式而且还满足包含所有线性多类拟似体的最小封闭和凸锥之外的新多类拟似体。使用该多类拟态，我们证明了所有保留截断的不等式（包括Ingleton不等式和DFZ不等式）不足以表征线性多类拟态。

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