Lattice Based Extended Formulations for Integer Linear Equality Systems
CORE Discussion Paper No. 2007/17
19 Pages Posted: 25 Jun 2007
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
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