Ehrhart杂波：正则性与最大流最小割
Ehrhart clutters: Regularity and Max-Flow Min-Cut
摘要：
如果C是具有n个顶点和q条边的杂波，其杂波矩阵具有列向量V={v1，.，vq}，我们称C为Ehrhart杂波，如果{（v1，1），.，(vq，1）}是希尔伯特基。设A（P）是P=conv（V）的Ehrhart环，我们能够证明：如果A是均匀、非混合MFMC杂波C的杂波矩阵，则C是Ehrhart杂波，并且在这种情况下，我们给出了A（P）的Castelnuovo-Mumford正则性的精确界。受Conforti-Cornuejols关于填充问题的猜想的启发，我们猜想：如果C既是理想图又是完美图的团簇，则C具有MFMC性质.通过证明Meyniel图的团杂波是Ehrhart杂波，证明了Meyniel图的团杂波是Ehrhart杂波.本着同样的精神，我们提供了一个简单的证明我们的猜想时，C是一个完美的图的均匀团杂波。我们关闭一个概括的Ehrhart杂波，因为它涉及到总的对偶完整性。
英文摘要：
If C is a clutter with n vertices and q edges whose clutter matrix has column vectors V={v1,...,vq}, we call C an Ehrhart clutter if {(v1,1),...,(vq,1)} is a Hilbert basis. Letting A(P) be the Ehrhart ring of P=conv(V), we are able to show that if A is the clutter matrix of a uniform, unmixed MFMC clutter C, then C is an Ehrhart clutter and in this case we provide sharp bounds on the Castelnuovo-Mumford regularity of A(P). Motivated by the Conforti-Cornuejols conjecture on packing problems, we conjecture that if C is both ideal and the clique clutter of a perfect graph, then C has the MFMC property. We prove this conjecture for Meyniel graphs, by showing that the clique clutters of Meyniel graphs are Ehrhart clutters. In much the same spirit, we provide a simple proof of our conjecture when C is a uniform clique clutter of a perfect graph. We close with a generalization of Ehrhart clutters as it relates to total dual integrality.
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