Lattice based extended formulations for integer linear equality systems

Abstract

We study different extended formulations for the set ${X = \{{\boldsymbol x}\in \mathbb{Z}^n \mid {\boldsymbol A}{\boldsymbol x} = {\boldsymbol A}{\boldsymbol x}^0\}}$ with ${{\boldsymbol A} \in \mathbb{Z}^{m \times n}}$ in order to tackle the feasibility problem for the set ${X \cap \mathbb{Z}^n_+.}$ Pursuing the work of Aardal, Lenstra et al. using the reformulation ${X=\{{\boldsymbol x} \in \mathbb{Z}^n \mid {\boldsymbol x}-{\boldsymbol x}^0={\boldsymbol Q}{\boldsymbol \lambda},\,{\boldsymbol \lambda} \in \mathbb{Z}^{n-m}\}}$ , our aim is to derive reformulations of the form ${\{{\boldsymbol x} \in \mathbb{Z}^n \mid {\boldsymbol P}({\boldsymbol x}-{\boldsymbol x}^0)={\boldsymbol T} {\boldsymbol \mu}, {\boldsymbol \mu} \in \mathbb{Z}^s\}}$ with 0  ≤  s ≤ n − m where preferably all the coefficients of P are small compared to the coefficients of A and T. In such cases the new variables μ appear to be good branching directions, and in certain circumstances permit one to deduce rapidly that the instance is infeasible. We give a polynomial time algorithm for identifying such P, T if possible, and for the case that A has one row a we analyze the reformulation when s = 1, that is, one μ-variable is introduced. In particular, we determine the integer width of the extended formulations in the direction of the μ-variable, and derive a lower bound on the Frobenius number of a. We conclude with some preliminary tests to see if the reformulations are effective when the number s of additional constraints and variables is limited.