[Paper Review] Distributive Lattices, Polyhedra, and Generalized Flow

Stefan Felsner, Kolja Knauer|ArXiv.org|Nov 10, 2008

Advanced Graph Theory Research12 references4 citations

TL;DR

This paper introduces D-polyhedra—polyhedra closed under componentwise max and min—characterizing them via bounding hyperplanes and establishing a duality between vertex potentials in arc-parameterized digraphs and generalized flows. The key contribution is a distributive lattice structure on generalized flows in planar breakeven digraphs, unifying and extending known lattice structures on flows, orientations, and bonds.

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 $α$-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.

Motivation & Objective

• To characterize D-polyhedra geometrically as polyhedra closed under componentwise max and min.
• To unify known distributive lattice structures on combinatorial objects (e.g., flows, orientations, bonds) via a common polyhedral framework.
• To establish a duality between vertex potentials in arc-parameterized digraphs and generalized flows, revealing a distributive lattice structure on the latter.
• To extend the theory to generalized flows in planar breakeven digraphs, proving a new distributive lattice structure on such flows.
• To identify conditions under which integral generalized bonds form a distributive lattice, generalizing prior results on integral $Δ$-bonds.

Proposed method

• Define D-polyhedra as polyhedra closed under componentwise max and min, forming a distributive lattice under dominance order.
• Characterize D-polyhedra via their H-representation (intersection of halfspaces), linking geometry and order theory.
• Represent points in a D-polyhedron as vertex potentials on a directed graph with arc-parameters, using the network matrix to encode constraints.
• Establish a duality between edge-based generalized flows (with gains/losses) and vertex potentials, showing that the lattice structure lifts from potentials to flows.
• Prove that for planar breakeven digraphs, the set of generalized flows bounded by capacity constraints forms a distributive lattice via isomorphism to a $Δ$-bond polyhedron.
• Use the logarithmic transformation of arc-parameterizations to construct breakeven parameterizations from $Ø$-bonds, enabling the lattice structure on generalized flows.

Experimental results

Research questions

• RQ1 What is the complete H-representation (bounding hyperplanes) of D-polyhedra, and how does it unify known polyhedral families?
• RQ2 How can vertex potentials in arc-parameterized digraphs be used to describe the integral points of a D-polyhedron?
• RQ3 What conditions ensure that the set of integral generalized bonds forms a distributive lattice?
• RQ4 Can a distributive lattice structure be established on generalized flows in planar digraphs with gains and losses (i.e., breakeven digraphs)?
• RQ5 What is the relationship between the vertex lattice $L(P)$ of a D-polyhedron $P$ and the geometric structure of $P$?

Key findings

• Every D-polyhedron is characterized by a specific H-representation involving inequalities derived from the dominance order and componentwise operations.
• The set of integral $Δ$-bonds of a directed graph with arc-capacities and cycle balance constraints forms a distributive lattice.
• A bijection exists between integral $Δ$-bonds and integer vertex potentials satisfying $c_{ε} \leq N^\top p \leq c_u$ and $p_i = 0$ for some fixed vertex $i$.
• For planar breakeven digraphs, the set of generalized flows bounded by capacity constraints forms a distributive lattice, as shown via isomorphism to a $Δ$-bond polyhedron.
• The lattice structure on generalized flows is preserved under positive arc-parameterizations, and the construction is universal via logarithmic transformation of $Ø$-bonds.
• The vertex lattice $L(P)$ of a D-polyhedron $P$ captures essential geometric and combinatorial information, and its elements correspond to generalized bonds with specific structural properties.