Characterizations of Total Dual Integrality

Edwin O'Shea (1) , András Sebő (2)

1 Department of Mathematics
2 G-SCOP_OC - Optimisation Combinatoire

In this paper we provide new characterizing properties of TDI systems. A 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.

Mots clés: Totally Dual Integral; Gröbner basis; perfect graph; computational algebra

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]; Mathématiques générales [math.GM]

Edwin O'Shea, András Sebő. Characterizations of Total Dual Integrality. 12th International IPCO conference, 2007, Ithaca, United States. pp.382-396. ⟨hal-00189411⟩