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

TL;DR

This paper explores different generalizations of facets in the infinite-dimensional Gomory-Johnson model, proving their equivalences in some cases and distinguishing them with examples, advancing the theoretical understanding of the model.

Contribution

The paper proves the equivalence of extreme functions and facets for continuous piecewise linear functions without rational breakpoints and establishes an if-and-only-if version of the Gomory-Johnson Facet Theorem.

Findings

Extreme functions and facets coincide for continuous piecewise linear functions.

An if-and-only-if version of the Gomory-Johnson Facet Theorem is established.

The three notions are separated using discontinuous examples.

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 prove an if-and-only-if version of the Gomory-Johnson Facet Theorem. Finally, we separate the three notions using discontinuous examples.