Lifting Integer Variables in Minimal Inequalities Corresponding to Lattice-Free Triangles

Recently, Andersen et al. [1] and Borozan and Cornu´ejols [3] characterized the minimal inequalities of a system of two rows with two free integer variables and nonnegative continuous variables. These in- equalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these minimal inequalities to ob- tain cuts from two rows of a general simplex tableau, it is necessary to ex- tend the system to include integer variables (giving the two-dimensional mixed integer infinite group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding in- equalities. In this paper, we analyze the lifting of minimal inequalities derived from lattice-free triangles. Maximal lattice-free triangles in R2 can be classified into three cate- gories: those with multiple integral points in the relative interior of one of its sides, those with integral vertices and one integral point in the relative interior of each side, and those with non integral vertices and one integral point in the relative interior of each side. We prove that the lifting functions are unique for each of the first two categories such that the resultant inequality is minimal for the mixed integer infinite group problem, and characterize them. We show that the lifting function is not necessarily unique in the third category. For this category we show that a fill-in inequality (Johnson [11]) yields minimal inequalities for mixed in- teger infinite group problem under certain sufficiency conditions. Finally, we present conditions for the fill-in inequality to be extreme.