Lifted Cycle Inequalities for the Asymmetric Traveling Salesman Problem

We investigate the family of facet defining inequalities for the asymmetric traveling salesman (ATS) polytope obtainable by lifting the cycle inequalities. We establish several properties of this family that earmark it as the most important among the asymmetric inequalities for the ATS polytope known to date: (i) The family is shown to contain members of unbounded Chvatal rank, whereas most known asymmetric inequalities are of Chvatal rank 1. (ii) For large classes within the family a coefficient pattern is identified that makes it easy to develop efficient separation routines, (iii) Each member of the family is shown to have a counterpart for the symmetric TS (STS) polytope that is often new, and is obtainable by mapping the inequality for the ATS polytope into a certain face of the STS polytope and then lifting the resulting inequality into one for the STS polytope itself.