Distributive lattices, polyhedra, and generalized flows

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, Dpolyhedra are a unifying generalization of several distributive lattices which arise from graphs. In fact with a D-polyhedron we associate a directed graph with arc-parameters, such that points in the polyhedron correspond to a vertex potentials on the graph. Alternatively, an edge-based description of the points of a D-polyhedron can be given. In this model the points correspond to the duals of generalized flows, i.e., duals of flows with gains and losses. These models can be specialized to yield distributive lattices that have been previously studied. Particular specializations are: flows of planar digraphs (Khuller, Na...

• Type: Journal
• Year: 2011
• Where: EJC
• Authors: Stefan Felsner, Kolja B. Knauer
• Keywords: Breakeven Planar Digraphs, Distributive Lattice, EJC 2011, Information Technology, Planar Digraphs