分配格、多面体和广义流
Distributive Lattices, Polyhedra, and Generalized Flow
摘要：
一个D-多面体是一个多面体$P$，如果$x,y$在$P$中，那么它们的分量最大值和最小值也在其中。换句话说，一个D-多面体的点集形成了一个具有优势阶的分配格。我们提供了一个完整的特征的D-多面体的边界超平面。除了是几何和序理论概念的一个很好的结合之外，D-多面体是从图中产生的几个分配格的统一推广。事实上，每个D-多面体对应于一个带有弧参数的有向图，使得多面体中的每个点对应于图上的一个点势。或者，可以给出点集的基于边的描述。该模型中的对象是广义流的对偶，即，二元流的收益和损失。这些模型可以被专门化以产生先前已经研究过的分配格的某些情况。具体专业有：平面有向图的流格（Khuller，Naor和Klein），可平面图的$α$ -定向格（Felsner），c-定向格（Propp）和有向图的$Δ$ -键格（Felsner和Knauer）。作为一个额外的应用，我们展示了一个分配格结构上的广义流的盈亏平面有向图。
英文摘要：
A D-polyhedron is a polyhedron $P$ such that if $x,y$ are in $P$ then so are their componentwise max and min. In other words, the point set of a D-polyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of D-polyhedra. Aside from being a nice combination of geometric and order theoretic concepts, D-polyhedra are a unifying generalization of several distributive lattices which arise from graphs. In fact every D-polyhedron corresponds to a directed graph with arc-parameters, such that every point in the polyhedron corresponds to a vertex potential on the graph. Alternatively, an edge-based description of the point set can be given. The objects in this model are dual to generalized flows, i.e., dual to flows with gains and losses. These models can be specialized to yield some cases of distributive lattices that have been studied previously. Particular specializations are: lattices of flows of planar digraphs (Khuller, Naor and Klein), of $α$-orientations of planar graphs (Felsner), of c-orientations (Propp) and of $Δ$-bonds of digraphs (Felsner and Knauer). As an additional application we exhibit a distributive lattice structure on generalized flow of breakeven planar digraphs.
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