Lattice based extended formulations for integer linear equality systems

We study different extended formulations for the set $X = \{x \in Z^n \mid Ax = Ax^0\} in order to tackle the feasibility problem for the set $X^+ = X\cap Z^n_+$. 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.