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Characterizations of Total Dual Integrality ?

In this paper we provide new characterizing properties of TDI systems, among others the following: A system of linear inequalities is TDI if and only if there exists a test set for its dual integer linear programs where all positive coordinates are equal to 1, and correspond to a set of linearly independent columns. Cook, Lovász and Schrijver’s result on the recognition – in polynomial time – of TDI systems of inequalities in fixed dimension can be simply deduced from this result, while the problem remains open in general. Another corollary is Sturmfels’ theorem relating toric initial ideals generated by square-free monomials to unimodular triangulations. A reformulation of these test-sets to polynomial ideals actually generalizes the existence of square-free monomials to arbitrary TDI systems, providing new relations between integer programming and Gröbner bases of toric ideals. We finally show that stable set polytopes of perfect graphs are characterized by a refined fan that is a triangulation consisting only of unimodular cones, a fact that endows the Weak Perfect Graph Theorem with a computationally advantageous geometric feature. Three ways of implementing the results are described and some experience about one of these is reported.