Comparing Intersection Cut Closures using Simple Families of Lattice-Free Convex Sets

Mixed integer programs are a powerful mathematical tool, providing a general model for expressing both theoretically difficult and practically useful problems. One important subroutine of algorithms solving mixed integer programs is a cut generation procedure. The job of a cut generation procedure is to produce a linear inequality that separates a given infeasible point x* (usually a basic feasible solution of the linear programming relaxation) from the set of feasible solutions for the problem at hand. Early and well-known cut generation procedures rely on analyzing a single row of the simplex tableau for x*. Andersen et al. renewed interest in d-row cuts (i.e. cuts derived from d rows of the simplex tableau) by showing that these cuts afford some theoretical benefit. One lens from which to study d-row cuts is in the context of the intersection cuts of Balas and, in particular, intersection cuts obtained from lattice-free convex sets. The strongest d-row intersection cuts are obtained from maximal lattice-free convex sets in $R^d$ - all of which are polyhedra with at most $2^d$ facets. This thesis is concerned with theoretical comparison of the d-row cuts generated by different families of maximal lattice-free convex sets. We use the gauge measure to appraise the quality of the approximation. The main area of focus is 2-row cuts. The problem of generating 2-row cuts can be re-posed as generating valid inequalities for a mixed integer linear set F with two free integer variables and any number of non-negative continuous variables, where there are two defining equations. Every minimal valid inequality for the convex hull of F is an intersection cut generated by a maximal lattice-free split, triangle or quadrilateral. The family of maximal lattice-free triangles can be subdivided into the families of type 1, type 2, and type 3 triangles. Previous results of Basu et al. and Awate et al. compare how well the inequalities from one of these families approximates the convex hull of F (a.k.a. the corner polyhedron). In particular, the closure of all type 2 triangle inequalities is shown to be within a factor of 3/2 of the corner polyhedron. The authors also provide an instance where all type 2 triangles inequalities cannot approximate the corner polyhedron better than a factor of 9/8. The same bounds are shown for type 3 triangles and quadrilaterals. These results are obtained not by directly comparing the given closures to the convex hull of F, but rather to each