Split rank of triangle and quadrilateral inequalities

TL;DR

This paper proves that all facet-defining inequalities for certain mixed integer sets, derived from triangles and quadrilaterals, have finite split-rank, providing explicit sequences to generate them.

Contribution

It establishes the finiteness of split-rank for all but one known class of facet-defining inequalities, with a constructive proof and explicit sequences.

Findings

All facet-defining triangle and quadrilateral inequalities have finite split-rank.

A constructive method to generate these inequalities via explicit split sequences.

One class of triangle inequalities is known to have infinite split-rank.

Abstract

A simple relaxation of two rows of a simplex tableau is a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. Recently Andersen, Louveaux, Weismantel and Wolsey (2007) and Cornuejols and Margot (2008) showed that the facet-defining inequalities of this set are either split cuts or intersection cuts obtained from lattice-free triangles and quadrilaterals. Through a result by Cook, Kannan and Schrijver (1990), it is known that one particular class of facet-defining triangle inequality does not have a finite split rank. In this paper, we show that all other facet-defining triangle and quadrilateral inequalities have a finite split-rank. The proof is constructive and given a facet-defining triangle or quadrilateral inequality we present an explicit sequence of split inequalities that can be used to generate it.