On the notions of facets, weak facets, and extreme functions of the Gomory-Johnson infinite group problem

TL;DR

This paper explores and clarifies different concepts of facets and extreme functions in the Gomory-Johnson infinite group problem, showing their equivalence in some cases and differences in others.

Contribution

It proves the equivalence of extreme functions and facets for continuous piecewise linear functions without rational breakpoints and provides examples separating the three notions.

Findings

Extreme functions and facets coincide for continuous piecewise linear functions.

Discontinuous examples differentiate the three notions.

The results extend understanding of the structure of the Gomory-Johnson model.

Abstract

We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing the hypothesis regarding rational breakpoints. We then separate the three notions using discontinuous examples.