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

Abstract

Abstract. Recently, Andersen et al. [1] and Borozan and Cornuéjols [3] characterized the minimal inequalities of a system of two rows with two free integer variables and nonnegative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these minimal inequalities to obtain cuts from two rows of a general simplex tableau, it is necessary to extend 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 inequalities. In this paper, we analyze the lifting of minimal inequalities derived from lattice-free triangles. Maximal lattice-free triangles in R 2 can be classified into three categories: 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 integer infinite group problem under certain sufficiency conditions. Finally, we present conditions for the fill-in inequality to be extreme.

A Geometric Perspective on Lifting

“…Dey and Wolsey [10] studied the case n = 2. In this case maximal lattice free polytopes are either triangles or quadrilaterals [15].…”

On the Practical Strength of Two-Row Tableau Cuts

“…Proposition 2.1 ( [20]) Let M be a full-dimensional maximal lattice-free convex set in R 2 . Then M is one of the following:…”

“…For an infinite version of the relaxation (1), it has been shown [20] that this strengthening yields extreme inequalities if π was obtained using (3) with M being a type 1 or type 2 triangle. Unfortunately however the valid inequalities for (2) strengthened in the way described above do not provide the complete list of facet-defining inequalities of the convex hull of (1).…”

“…For type 2 triangles, the function φ(r j ) leads to extreme inequalities for the infinite relaxation of (2) ( [20]). …”

“…Several variants of (2) have been studied; see for example [2,3,10,11,14,15,17,20,21,29] and [19] for a recent survey on the topic. This relaxation (2) has at least two appealing features : i) it is possible to obtain a complete characterization of all facet-defining inequalities using intersections cuts derived from the so-called maximal lattice-free convex sets and ii) the cutting planes derived from these facet-defining inequalities have the strongest possible coefficients for the continuous variables.…”