Light on the Infinite Group Relaxation

TL;DR

This survey reviews the infinite group problem in integer linear optimization, highlighting recent advances in extremality testing algorithms, multi-row problem solutions, and new extreme functions, supported by computational tools.

Contribution

It consolidates recent developments in the infinite group problem, introduces new extremal functions, and provides computational tools for researchers.

Findings

Development of algorithms for extremality testing

Breakthroughs in multi-row problem solutions for all k ≥ 1

Discovery of new piecewise linear extreme functions with multiple slopes

Abstract

This is a survey on the infinite group problem, an infinite-dimensional relaxation of integer linear optimization problems introduced by Ralph Gomory and Ellis Johnson in their groundbreaking papers titled "Some continuous functions related to corner polyhedra I, II" [Math. Programming 3 (1972), 23-85, 359-389]. The survey presents the infinite group problem in the modern context of cut generating functions. It focuses on the recent developments, such as algorithms for testing extremality and breakthroughs for the k-row problem for general k >= 1 that extend previous work on the single-row and two-row problems. The survey also includes some previously unpublished results; among other things, it unveils piecewise linear extreme functions with more than four different slopes. An interactive companion program, implemented in the open-source computer algebra package Sage, provides an updated compendium of known extreme functions.