三角形闭包是一个多面体
The Triangle Closure is a Polyhedron
摘要：
近年来，由极大无格凸集导出的割平面已成为整数规划界的研究热点。在这个研究领域中的一个重要问题是决定与某些无格集族相关的闭包是否是多面体。长期以来，唯一已知的结果是著名的库克定理，Kannan和Schrijver谁表明分裂封闭是一个多面体。虽然Andersen，Louveaux和Weismantel [ An analysis of mixed integer linear sets based on lattice point free convex sets，Math. Oper. Res. 35（2010），233- 256]和Averkov [On Jumper generated closures in the theory of cutting planes，离散最优化9（2012），no. 4，209- 215]，一些基本问题仍然没有解决。例如，极大无格三角形是除分裂族之外的自然研究族，并且决定三角形闭包是否是多面体一直是一个悬而未决的问题。在本文中，我们表明，当整数变量的数量$m=2$的三角形闭包确实是一个多面体和它的面的数量可以由一个多项式的输入数据的大小有界。该证明的技术还用于对Cornuéjols和Margot [关于具有两个整数变量和两个约束的混合整数规划的方面，数学规划120（2009），429--456]定义的有效不等式的必要条件进行细化，并获得有关混合整数壳的多项式复杂度结果。
英文摘要：
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 Cornuéjols 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.
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