An extreme function which is nonnegative and discontinuous everywhere

TL;DR

This paper constructs a nonnegative, discontinuous, extreme valid function for Gomory and Johnson's infinite group model, with a graph dense in the plane, highlighting complex behavior of such functions.

Contribution

It introduces a novel extreme valid function that is nonnegative, discontinuous everywhere, and has a dense graph, expanding understanding of extreme functions in integer programming.

Findings

Constructed a dense-graph extreme function

Proved the function is discontinuous everywhere

Demonstrated the complexity of extreme valid functions

Abstract

We consider Gomory and Johnson's infinite group model with a single row. Valid inequalities for this model are expressed by valid functions and it has been recently shown that any valid function is dominated by some nonnegative valid function, modulo the affine hull of the model. Within the set of nonnegative valid functions, extreme functions are the ones that cannot be expressed as convex combinations of two distinct valid functions. In this paper we construct an extreme function $\pi:\mathbb{R} \to [0,1]$ whose graph is dense in $\mathbb{R} \times [0,1]$. Therefore, $\pi$ is discontinuous everywhere.