[Paper Review] Split rank of triangle and quadrilateral inequalities

Santanu S. Dey, Quentin Louveaux|ArXiv.org|Jun 4, 2009

Advanced Optimization Algorithms Research20 references4 citations

TL;DR

This paper establishes that all facet-defining triangle and quadrilateral inequalities for the convex hull of a two-row mixed integer program relaxation have finite split rank, except for one specific class of triangle inequalities previously shown to have infinite split rank. The authors provide an explicit constructive sequence of split inequalities that generate all such finite-split-rank inequalities, resolving a key question in cutting plane theory for mixed integer programming.

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.

Motivation & Objective

• To determine which facet-defining inequalities of the convex hull of a two-row mixed integer program relaxation have finite split rank.
• To resolve the open question of whether all triangle and quadrilateral inequalities (except one known class) can be generated via repeated split cuts.
• To provide an explicit, constructive sequence of split inequalities that generate all facet-defining triangle and quadrilateral inequalities with finite split rank.

Proposed method

• The authors analyze the structure of the convex hull of the two-row relaxation set P(R,f), defined by two free integer variables and non-negative continuous variables.
• They leverage known characterizations of facet-defining inequalities as either split cuts or intersection cuts from lattice-free triangles and quadrilaterals.
• The proof uses geometric and polyhedral techniques to analyze split rank by reducing the problem to sets with at most four continuous variables.
• A constructive algorithm is developed to generate the required sequence of split inequalities for each finite-split-rank facet-defining inequality.
• The analysis distinguishes between different types of inequalities based on their geometric support and uses convex combinations and vertex enumeration to verify validity and tightness.
• The authors apply results from Cook et al. [15] on the infinite split rank of a specific triangle inequality class to establish the boundary case.

Experimental results

Research questions

• RQ1 Which facet-defining inequalities of conv(P(R,f)) have finite split rank?
• RQ2 Can all triangle and quadrilateral inequalities (except the known infinite-split-rank class) be generated through repeated application of split cuts?
• RQ3 Is there a constructive method to generate a finite sequence of split inequalities that yields any given finite-split-rank facet-defining triangle or quadrilateral inequality?
• RQ4 What geometric and algebraic properties distinguish the one class of triangle inequalities with infinite split rank from all others?
• RQ5 How can the split rank of intersection cuts derived from lattice-free triangles and quadrilaterals be systematically analyzed and bounded?

Key findings

• All facet-defining triangle and quadrilateral inequalities for conv(P(R,f)) have finite split rank, except for one specific class of triangle inequalities identified by Cook et al. [15].
• For every facet-defining inequality with finite split rank, the paper constructs an explicit finite sequence of split inequalities that can be used to generate it.
• The proof demonstrates that the infinite split rank is unique to a particular class of triangle inequalities, while all others are finitely generated via split cuts.
• The analysis confirms that the split rank is finite for all inequalities derived from lattice-free triangles and quadrilaterals except for this exceptional class.
• The authors establish that the split rank of any such finite-split-rank inequality is bounded and constructively computable through geometric decomposition and vertex analysis.
• The paper provides a complete classification of the split rank behavior of all facet-defining inequalities in the two-row relaxation, resolving a long-standing open question in cutting plane theory.