8 research outputs found
Distributive Lattices, Polyhedra, and Generalized Flow
A D-polyhedron is a polyhedron such that if are in 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 -orientations of
planar graphs (Felsner), of c-orientations (Propp) and of -bonds of
digraphs (Felsner and Knauer). As an additional application we exhibit a
distributive lattice structure on generalized flow of breakeven planar
digraphs.Comment: 17 pages, 3 figure