# Minimal cut-generating functions are nearly extreme

## TL;DR

This paper demonstrates that in integer-variable models, continuous minimal cut generating functions can be closely approximated by extreme functions, indicating their density within the set of minimal functions.

## Contribution

It proves that extreme cut generating functions are dense among continuous minimal functions in the Gomory-Johnson framework and its extension.

## Key findings

- Extreme functions approximate any continuous minimal function arbitrarily closely.
- Density of extreme functions in the set of continuous minimal functions.
- Extension of results to both original and extended Gomory-Johnson models.

## Abstract

We study continuous (strongly) minimal cut generating functions for the model where all variables are integer. We consider both the original Gomory-Johnson setting as well as a recent extension by Cornu\'ejols and Y{\i}ld{\i}z. We show that for any continuous minimal or strongly minimal cut generating function, there exists an extreme cut generating function that approximates the (strongly) minimal function as closely as desired. In other words, the extreme functions are "dense" in the set of continuous (strongly) minimal functions.