# On the Relative Strength of Split, Triangle and Quadrilateral Cuts

## TL;DR

This paper compares the effectiveness of split, triangle, and quadrilateral inequalities in approximating the integer hull of certain integer programs, finding that triangles and quadrilaterals provide good approximations while splits can be arbitrarily weak.

## Contribution

It provides a theoretical comparison of the strength of these three inequality families in approximating the integer hull for specific integer programs.

## Key findings

- Triangle inequalities closely approximate the integer hull.
- Quadrilateral inequalities also provide a good approximation.
- Split inequalities can be arbitrarily weak in approximation.

## Abstract

Integer programs defined by two equations with two free integer variables and nonnegative continuous variables have three types of nontrivial facets: split, triangle or quadrilateral inequalities. In this paper, we compare the strength of these three families of inequalities. In particular we study how well each family approximates the integer hull. We show that, in a well defined sense, triangle inequalities provide a good approximation of the integer hull. The same statement holds for quadrilateral inequalities. On the other hand, the approximation produced by split inequalities may be arbitrarily bad.