Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. I. The One-Dimensional Case

TL;DR

This paper introduces an algorithm to test extremality of certain piecewise linear functions in Gomory and Johnson's infinite group problem, advancing the ability to determine extremality in this mathematical context.

Contribution

It provides the first necessary and sufficient algorithmic conditions for extremality of rational breakpoint functions and introduces a new principle for irrational breakpoint functions.

Findings

Algorithm for extremality testing of rational breakpoint functions

Necessary and sufficient conditions established

New principle for irrational breakpoint extremality

Abstract

We give an algorithm for testing the extremality of minimal valid functions for Gomory and Johnson's infinite group problem that are piecewise linear (possibly discontinuous) with rational breakpoints. This is the first set of necessary and sufficient conditions that can be tested algorithmically for deciding extremality in this important class of minimal valid functions. We also present an extreme function that is a piecewise linear function with some irrational breakpoints, whose extremality follows from a new principle.