Lattice Based Extended Formulations for Integer Linear Equality Systems; Probability rept

摘要

We study different extended formulations for the set X = (x epsilon Zeta(sup n) (vertical bar) Ax = Ax(sup 0)) in order to tackle the feasibility problem for the set X(sub +) = X Omega Zeta(sup n)(sub +). Here the goal is not to find an improved polyhedral relaxation of conv(X(sub +)), 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.