Distributive Lattices, Polyhedra, and Generalized Flow

TL;DR

This paper characterizes D-polyhedra, a class of polyhedra forming distributive lattices, and links them to generalized flows and graph orientations, unifying various lattice structures in graph theory.

Contribution

It provides a full characterization of D-polyhedra, connecting geometric, order theoretic, and graph-based models, and introduces a distributive lattice structure on generalized flows.

Findings

Characterization of D-polyhedra's bounding hyperplanes

Equivalence between D-polyhedra and graph-based models

Distributive lattice structure on generalized flows in planar digraphs

Abstract

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 $\alpha$-orientations of planar graphs (Felsner), of c-orientations (Propp) and of $\Delta$-bonds of digraphs (Felsner and Knauer). As an additional application we exhibit a distributive lattice structure on generalized flow of breakeven planar digraphs.