Quotient-convergence of Submodular Setfunctions

TL;DR

This paper introduces quotient-convergence for submodular set functions, providing a new framework for analyzing the convergence of matroids and extending graph limit theories to more abstract combinatorial structures.

Contribution

It defines quotient-convergence for submodular functions, constructs converging sequences for various classes, and proves the completeness of the space under this convergence.

Findings

Any bounded set function can be approximated by quotient-convergent sequences.

Constructs explicit sequences for increasing, submodular, upper continuous functions.

Establishes the completeness of the space under quotient-convergence.

Abstract

We introduce the concept of quotient-convergence for sequences of submodular set functions, providing, among others, a new framework for the study of convergence of matroids through their rank functions. Extending the limit theory of bounded degree graphs, which analyzes graph sequences via neighborhood sampling, we address the challenge posed by the absence of a neighborhood concept in matroids. We show that any bounded set function can be approximated by a sequence of finite set functions that quotient-converges to it. In addition, we explicitly construct such sequences for increasing, submodular, and upper continuous set functions, and prove the completeness of the space under quotient-convergence.