可表示多拟阵新不等式的存在性
Existence of new inequalities for representable polymatroids
摘要：
一个英格尔顿多拟阵满足，除了多拟阵公理，不等式英格尔顿（组合。数学应用，1971年）。这些不等式是多拟阵可表示所必需的。这是一个悬而未决的问题，这些不平等是否也足够。可表示的多元拟阵本身就是令人感兴趣的。它们与网络编码也有很强的联系。特别是，寻找线性网络编码容量区域的问题等价于所有可表示的熵多拟阵的表征。在本文中，我们描述了一种新的方法来连接两个多拟阵在一起，以产生一个新的多拟阵。利用这种方法，我们可以构造一个多拟阵，它不在包含所有可表示的多拟阵的最小闭凸锥内。证明了该多拟阵不仅满足Ingleton不等式，而且还满足最近报道的Doughterson，Freiling和Zeger不等式.一个直接的结果是，这些不等式不足以表征可表示的多拟阵。
英文摘要：
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.
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