Alternatives for testing total dual integrality

• 成果类型: Article
• 署名作者: O'Shea, Edwin; Seboe, Andras
• 署名单位: University College Cork; Communaute Universite Grenoble Alpes; Institut National Polytechnique de Grenoble; Universite Grenoble Alpes (UGA); Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-010-0383-5
• 发表日期: 2012
• 页码: 57-78
• 关键词: integer analog; complexity; graph

摘要

In this paper we provide characterizing properties of totally dual integral (TDI) systems, among others the following: a system of linear inequalities is TDI if and only if its coefficient vectors form a Hilbert basis, and there exists a test-set for the system's dual integer programs where all test vectors have positive entries equal to 1. Reformulations of this provide relations between computational algebra and integer programming and they contain Applegate, Cook and McCormick's sufficient condition for the TDI property and Sturmfels' theorem relating toric initial ideals generated by square-free monomials to unimodular triangulations. We also study the theoretical and practical efficiency and limits of the characterizations of the TDI property presented here. In the particular case of set packing polyhedra our results correspond to endowing the weak perfect graph theorem with an additional, computationally interesting, geometric feature: the normal fan of the stable set polytope of a perfect graph can be refined into a regular triangulation consisting only of unimodular cones.