A (k+1)-Slope Theorem for the k-Dimensional Infinite Group Relaxation

TL;DR

This paper generalizes a theorem about the extremality of certain minimal valid functions in the k-dimensional infinite group relaxation, showing that piecewise linear functions with at most k+1 slopes are extreme under specific conditions.

Contribution

It extends the (k+1)-Slope Theorem to higher dimensions, providing a broader understanding of extremal functions in infinite group relaxations.

Findings

Proves extremality of piecewise linear functions with up to k+1 slopes in k-dimensional relaxations.

Generalizes previous results from 1D and 2D to arbitrary dimensions.

Identifies conditions under which such functions do not factor through non-trivial linear maps.

Abstract

We prove that any minimal valid function for the k-dimensional infinite group relaxation that is piecewise linear with at most k+1 slopes and does not factor through a linear map with non-trivial kernel is extreme. This generalizes a theorem of Gomory and Johnson for k=1, and Cornuejols and Molinaro for k=2.