Improved theoretical guarantees regarding a class of two-row cutting planes

TL;DR

This paper improves theoretical bounds on the approximation of the Relaxed Corner Polyhedron using specific lattice-free convex sets, enhancing understanding of cutting plane effectiveness in mixed-integer programming.

Contribution

It provides tighter upper bounds for quadrilaterals and extends lower bounds for certain quadrilaterals, advancing the theoretical analysis of cutting planes.

Findings

Upper bound for quadrilaterals improved from 2 to 1.71.

Lower bound for triangles extended to an infinite subclass of quadrilaterals.

Enhanced understanding of approximation quality of lattice-free cuts.

Abstract

The corner polyhedron is described by minimal valid inequalities from maximal lattice-free convex sets. For the Relaxed Corner Polyhedron (RCP) with two free integer variables and any number of non-negative continuous variables, it is known that such facet defining inequalities arise from maximal lattice-free splits, triangles and quadrilaterals. We improve on the tightest known upper bound for the approximation of the RCP, purely by minimal valid inequalities from maximal lattice-free quadrilaterals, from 2 to 1.71. We generalize the tightest known lower bound of 1.125 for the approximation of the RCP, purely by minimal valid inequalities from maximal lattice-free triangles, to an infinite subclass of quadrilaterals.