Alternatives for Testing Total Dual Integrality

TL;DR

This paper explores characterizations of TDI systems, linking algebraic and geometric properties, and examines their computational aspects, especially in relation to perfect graphs and set packing polyhedra.

Contribution

It introduces new characterizations of TDI systems using Hilbert bases and test-sets, connecting algebraic, geometric, and combinatorial perspectives.

Findings

TDI systems are characterized by Hilbert bases and specific test-sets.

The paper relates TDI properties to algebraic structures like toric ideals.

Results provide geometric insights into perfect graphs and set packing polyhedra.

Abstract

In this paper we provide characterizing properties of 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.