The Graphical Asymmetric Traveling Salesman Polyhedron: Symmetric Inequalities

Abstract

A present trend in the study of the symmetric traveling salesman polytope is to use, as a relaxation of the polytope, the graphical traveling salesman polyhedron ($\g$). Following a parallel approach for the asymmetric traveling salesman polytope, we define the graphical asymmetric traveling salesman problem on a general digraph $D$ and its associated polyhedron $\ga(D)$. We give basic polyhedral results and lifting theorems for $\ga(D)$ and we give a general condition for a facet-defining inequality for \g to yield a symmetric facet-defining inequality for \ga. Using this approach we show that all known major families of facet-defining inequalities of \g define facets of \ga. Finally, we discuss possible extension of these results to the asymmetric traveling salesman polytope.