Hyperfinite graphings and combinatorial optimization

TL;DR

This paper explores the analogy between measure-preserving transformations and combinatorial optimization, extending finite techniques like linear programming to infinite hyperfinite graphings, revealing invariance properties.

Contribution

It introduces a novel infinite analog of a combinatorial optimization problem for hyperfinite graphings and adapts finite proof techniques to this infinite setting.

Findings

Invariance of hyperfiniteness under local isomorphism is established.

Linear relaxation and duality techniques are extended to infinite graphings.

An analogy between measure-preserving maps and combinatorial optimization is demonstrated.

Abstract

We exhibit an analogy between the problem of pushing forward measurable sets under measure preserving maps and linear relaxations in combinatorialoptimization. We show how invariance of hyperfiniteness of graphings under local isomorphism can be reformulated as an infinite version of a natural combinatorial optimization problem, and how one can prove it by extending well-known proof techniques (linear relaxation, greedy algorithm, linear programming duality) from the finite case to the infinite.