All Cyclic Group Facets Inject

TL;DR

This paper extends approximation theorems for minimal valid functions in Gomory-Johnson's infinite group problem, demonstrating that extreme functions can be approximated by continuous piecewise linear functions with rational breakpoints, preserving key properties.

Contribution

The paper introduces a new approximation theorem for minimal valid functions with rational breakpoints, preserving function values on these points, and shows that finite group problem extreme functions extend to infinite group problem extreme functions.

Findings

Every finite group problem extreme function is a restriction of an infinite group problem extreme function.

The approximation preserves function values on rational breakpoints.

Supports that the infinite group problem is the master problem for facets of 1-row group relaxations.

Abstract

We give a variant of Basu-Hildebrand-Molinaro's approximation theorem for continuous minimal valid functions for Gomory-Johnson's infinite group problem by piecewise linear two-slope extreme functions [Minimal cut-generating functions are nearly extreme, IPCO 2016]. Our theorem is for piecewise linear minimal valid functions that have only rational breakpoints (in $1/q\,\mathbb{Z}$ for some $q\in \mathbb{N}$) and that take rational values at the breakpoints. In contrast to Basu et al.'s construction, our construction preserves all function values on $1/q\,\mathbb{Z}$. As a corollary, we obtain that every extreme function for the finite group problem on $1/q\,\mathbb{Z}$ is the restriction of a continuous piecewise linear two-slope extreme function for the infinite group problem with breakpoints on a refinement $1/(Mq)\,\mathbb{Z}$ for some $M\in \mathbb{N}$. In combination with Gomory's master theorem [Some Polyhedra related to Combinatorial Problems, Lin. Alg. Appl. 2 (1969), 451-558], this shows that the infinite group problem is the correct master problem for facets (extreme functions) of 1-row group relaxations.