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

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.