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

Abstract

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 other.

In this thesis, we tighten one of the underlying bounds, showing that the closure of all type 2 triangle inequalities are within a factor of 5/4 of the closure of all quadrilateral inequalities. We also consider the sub-family of quadrilaterals where opposite edges have equal slope. We show that these parallelogram cuts can be approximated by all type 2 triangle inequalities within a factor of 9/8 and there exist instances where no better approximation is possible. In proving both these bounds, we use a subset of the family of type 2 triangles; we call the members of this sub-family ray-sliding triangles.

A secondary area of focus in this thesis is d-row cuts for d >= 3. For d-row cuts in general, the underlying maximal lattice-free convex sets in $R^d$ are not easily classified. Absent a classification, Averkov et al, show that all inequalities generated by lattice-free convex sets with at most $i$ facets approximate the corner polyhedron within a finite factor only when $i > 2^{d-1}$. Here we take a different tact and try to prove analogues of 2-row cut results. We extend the proof techniques to obtain a constant factor approximation between two structured families of maximal lattice-free convex sets in $R^d$ for d >= 3.