Technical Report 025-2003

Title

On Compact Formulations for Integer Programs Solved by Column Generation

Authors

Daniel Villeneuve, Jacques Desrosiers, Marco E. Lübbecke, and Francois Soumis

Publication

Submitted to: Annals of Operations Research

Source

Download as [PDF] [ps.gz]

Classification

MSC:
primary:	90C10	Integer programming
secondary:	49M27	Decomposition methods
	65K05	Mathematical programming algorithms
	90C06	Large-scale problems
	90C57	Polyhedral combinatorics, branch-and-bound, branch-and-cut

Keywords

Integer programming, branch and bound, column generation

Abstract

Column generation has become a powerful tool in solving large scale integer programs. It is well known that most of the often reported compatibility issues between pricing subproblem and branching rule disappear when branching decisions are based on imposing constraints on the subproblem's variables. This can be generalized to branching on variables of a so-called compact formulation. We constructively show that such a formulation always exists under mild assumptions. It has a block diagonal structure with identical subproblems, each of which contributes only one column in an integer solution. This construction has an interpretation as reversing a Dantzig-Wolfe decomposition. Our proposal opens the way for the development of branching rules adapted to the subproblem's structure and to the linking constraints.