A probabilistic comparison of the strength of split, triangle, and quadrilateral cuts (extended version)

TL;DR

This paper analyzes the relative strength of various cutting planes in mixed integer linear sets using a probabilistic model, revealing that certain inequalities become less beneficial as lattice width decreases.

Contribution

It introduces a probabilistic framework to compare the effectiveness of split, type 2, type 3, and quadrilateral inequalities in strengthening mixed integer linear sets.

Findings

Type 2 inequalities are less beneficial as lattice width decreases.

Results suggest similar trends for type 3 and quadrilateral inequalities.

Adding non-split inequalities provides diminishing returns with smaller lattice width.

Abstract

We consider mixed integer linear sets defined by two equations involving two integer variables and any number of non-negative continuous variables. The non-trivial valid inequalities of such sets can be classified into split, type 1, type 2, type 3, and quadrilateral inequalities. We use a strength measure of Goemans to analyze the benefit from adding a non-split inequality on top of the split closure. Applying a probabilistic model, we show that the importance of a type 2 inequality decreases with decreasing lattice width, on average. Our results suggest that this is also true for type 3 and quadrilateral inequalities.