Property {A} and duality in linear programming

TL;DR

This paper links property A, a concept related to group and metric space amenability, to linear programming on graphs, revealing new insights into graph connectivity and providing novel proofs for existing theorems.

Contribution

It reduces property A to linear programming problems on graphs and explores their duals, connecting them to graph connectivity and isoperimetric inequalities.

Findings

Property A can be characterized via linear programming on finite graphs.

Dual problems relate to maximum flow and Cheeger constant.

New proofs of theorems regarding graphs without property A.

Abstract

Property A is a form of weak amenability for groups and metric spaces introduced as an approach to the famous Novikov higher signature conjecture, one of the most important unsolved problems in topology. We show that property A can be reduced to a sequence of linear programming optimization problems on finite graphs. We explore the dual problems, which turn out to have interesting interpretations as combinatorial problems concerning the maximum total supply of flows on a network. Using isoperimetric inequalities, we relate the dual problems to the Cheeger constant of the graph and explore the role played by symmetry of a graph to obtain a striking characterization of the difference between an expander and a graph without property A. Property A turns out to be a new measure of connectivity of a graph that is relevant to graph theory. The dual linear problems can be solved using a variety of methods, which we demonstrate on several enlightening examples. As a demonstration of the power of this linear programming approach we give elegant proofs of theorems of Nowak and Willett about graphs without property A.