On a generalization of the Chvatal-Gomory closure

TL;DR

This paper extends the concept of Chvatal-Gomory closures to include general linear constraints, demonstrating that the set of points satisfying these strengthened inequalities forms a rational polyhedron, broadening the scope of integer programming techniques.

Contribution

It generalizes previous results by incorporating all linear constraints into the strengthened CG inequalities, proving the resulting feasible set remains a rational polyhedron.

Findings

The set of points satisfying strengthened CG inequalities with general linear constraints is a rational polyhedron.

The polyhedrality result extends to mixed-integer sets with linear constraints.

The work broadens the applicability of CG closures in integer programming.

Abstract

Many practical integer programming problems involve variables with one or two-sided bounds. Dunkel and Schulz (2012) considered a strengthened version of Chvatal-Gomory (CG) inequalities that use 0-1 bounds on variables, and showed that the set of points in a rational polytope that satisfy all these strengthened inequalities is a polytope. Recently, we generalized this result by considering strengthened CG inequalities that use all variable bounds. In this paper, we generalize further by considering not just variable bounds, but general linear constraints on variables. We show that all points in a rational polyhedron that satisfy such strengthened CG inequalities form a rational polyhedron. We also consider mixed-integer sets defined by linear constraints, to which we extend our polyhedrality result by defining the strengthened CG inequalities on integer variables.