Given any family ℱ of valid inequalities for the asymmetric traveling salesman polytope P(G) define on the complete digraph G, the authors show that all members of ℱ are facet defining if the primitive members of ℱ (usually a small subclass) are. Based on this result they then introduce a general procedure for identifying new classes of facet inducing inequalities for P(G) by lifting inequalities that are facet inducing for P(G'), where G' is some induced subgraph of G. Unlike traditional lifting, where the lifted coefficients are calculated one by one and their value depends on the lifting sequence, the present lifting procedure replaces nodes of G' with cliques of G and uses closed form expressions for calculating the coefficients of the new arcs, which are sequence-independent. The authors also introduce a new class of facet inducing inequalities, the class of SD (source-destination) inequalities, which subsumes as special cases most known families of facet defining inequalities.
