Facets of the Asymmetric Traveling Salesman Polytope

Abstract

In this paper we consider the Asymmetric Traveling Salesman polytope, Pn, defined as the convex hull of the incidence vectors of tours in a complete digraph with n vertices, and its monotonization, P̃n. Several classes of valid inequalities for both there polytopes have been introduced in the literature which are not known to define facets of Pn or P̃n. We describe a general technique which can often be used to prove that a given inequality defines a facet of Pn. The method is applied to prove that the so-called Dk+, Dk−, C3, comb and C2 inequalities define facets of Pn, thus solving open problems. As for the monotone polytope, a necessary and sufficient condition for a facet-defining inequality of Pn to define a facet of P̃n as well, is introduced which applies to the Dk+, Dk−, C3, comb and C2 inequalities. In addition, a simple procedure for transforming any facet-defining inequality of Pn into an equivalent one which also defines a facet of P̃n, is given.