The Triangle Closure is a Polyhedron

TL;DR

This paper proves that for two integer variables, the triangle closure in mixed-integer programming is a polyhedron with a polynomially bounded number of facets, advancing understanding of lattice-free convex sets.

Contribution

It establishes that the triangle closure is a polyhedron when the number of integer variables is two, with a polynomial bound on the number of facets, resolving a long-standing open problem.

Findings

Triangle closure is a polyhedron for two integer variables.

Number of facets of the triangle closure can be bounded polynomially.

Refinement of conditions for facet-defining inequalities in mixed-integer programs.

Abstract

Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, Kannan and Schrijver who showed that the split closure is a polyhedron. Although some fairly general results were obtained by Andersen, Louveaux and Weismantel [ An analysis of mixed integer linear sets based on lattice point free convex sets, Math. Oper. Res. 35 (2010), 233--256] and Averkov [On finitely generated closures in the theory of cutting planes, Discrete Optimization 9 (2012), no. 4, 209--215], some basic questions have remained unresolved. For example, maximal lattice-free triangles are the natural family to study beyond the family of splits and it has been a standing open problem to decide whether the triangle closure is a polyhedron. In this paper, we show that when the number of integer variables $m=2$ the triangle closure is indeed a polyhedron and its number of facets can be bounded by a polynomial in the size of the input data. The techniques of this proof are also used to give a refinement of necessary conditions for valid inequalities being facet-defining due to Cornu\'ejols and Margot [On the facets of mixed integer programs with two integer variables and two constraints, Mathematical Programming 120 (2009), 429--456] and obtain polynomial complexity results about the mixed integer hull.