The Symmetric Traveling Salesman Polytope Revisited

We propose in this paper a tour of the symmetric traveling salesman polytope, focusing on inequalities that can be deened on sets. The most known inequalities are all of this type. Many papers have appeared which give more and more complex valid inequalities for this polytope, but no intuitive idea on why these inequalities are valid has ever been given. In order to help in understanding these inequalities, we develop an intuition into the validity of such inequalities by giving a unifying way of deening them through a sequential lifting procedure. This procedure is based on lifting the slack variables associated with subtour elimination inequalities deened on sets of nodes (called teeth). We apply this procedure to some known classes of valid inequalities for the TSP, respectively Comb, Brushes, Star and Path, Bipartition inequalities , where the lifting coeecients are sequence independent. We also give an example where a facet deening inequality is derived from the lifting procedure, but where the lifting coeecients are sequence dependent. We nally study the Ladder inequalities and show that they can be generated by an extension of the general procedure, where the lifted variables are diierent from the slack variables of subtour elimination inequalities.