Submodular setfunctions on sigma-algebras, version 2

TL;DR

This paper explores the connections between analytic and combinatorial theories of submodular setfunctions, aiming to extend the rich finite set theory to infinite and analytic contexts, with implications for potential theory and matroid limits.

Contribution

It describes the parallels between analytic and combinatorial submodular setfunctions and proposes problems for extending finite set theories to the analytic setting.

Findings

Identifies connections between analytic and combinatorial submodular theories.

Proposes problems for generalizing submodular setfunctions to infinite sets.

Develops the analytic theory with references to combinatorial optimization.

Abstract

Submodular setfunctions play an important role in potential theory, and a perhaps even more important role in combinatorial optimization. The analytic line of research goes back to the work of Choquet; the combinatorial, to the work of Rado and Edmonds. The two research lines have not had much interaction though. Recently, with the development of graph limit theory, the question of a limit theory for matroids has been considered by several people; such a theory will, most likely, involve submodular setfunctions both on finite and infinite sets. The goal of this paper is to describe several connections between the analytic and combinatorial theory, to show parallels between them, and to propose problems arising by trying the generalize the rich theory of submodular setfunctions on finite sets to the analytic setting. It is aimed more at combinatorialists, and it spends more time on developing the analytic theory, often referring to results in combinatorial optimization by name or sketch only.