Lattice Based Extended Formulations for Integer Linear Equality Systems

CORE Discussion Paper No. 2007/17

19 Pages Posted: 25 Jun 2007

See all articles by Karen Aardal

Karen Aardal

Eindhoven University of Technology (TUE)

Laurence A. Wolsey

Center for Operations Research and Econometrics (CORE)

Date Written: February 2007

Abstract

We study different extended formulations for the set X = {x E Zn | Ax = Ax0} in order to tackle the feasibility problem for the set X+ = X N Zn+. Here the goal is not to find an improved polyhedral relaxation of conv(X+), but rather to reformulate in such a way that the new variables introduced provide good branching directions, and in certain circumstances permit one to deduce rapidly that the instance is infeasible. For the case that A has one row a we analyze the reformulations in more detail. In particular, we determine the integer width of the extended formulations in the direction of the last coordinate, and derive a lower bound on the Frobenius number of a. We also suggest how a decomposition of the vector a can be obtained that will provide a useful extended formulation. Our theoretical results are accompanied by a small computational study.

Keywords: integer programming feasibility, integer width, branching directions, reduced lattice bases, Frobenius number

Suggested Citation

Aardal, Karen and Wolsey, Laurence A., Lattice Based Extended Formulations for Integer Linear Equality Systems (February 2007). CORE Discussion Paper No. 2007/17. Available at SSRN: https://ssrn.com/abstract=994808 or http://dx.doi.org/10.2139/ssrn.994808

Karen Aardal (Contact Author)

Eindhoven University of Technology (TUE) ( email )

PO Box 513
Eindhoven, 5600 MB
Netherlands

Laurence A. Wolsey

Center for Operations Research and Econometrics (CORE) ( email )

34 Voie du Roman Pays
1348 Louvain-la-Neuve, 1348
Belgium

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