Facet Generating Techniques

Given a polyhedron P which is of interest, a major goal of polyhedral combinatorics is to find classes of essential, i.e. facet inducing inequalities which describe P . In general this is a difficult task. We consider the case in which we have knowledge of facets for a face F of P , and present some general theory and methods for exploiting the close relationship between such polyhedra in order to obtain facets for P from facets of F . We demonstrate the usefulness of this theory by applying it in three instances where this relationship holds, namely the linear ordering polytope and the acyclic subgraph polytope, the asymmetric travelling salesman polytope and the monotone asymmetric travelling salesman polytope, and the symmetric travelling salesman polytope and the two-connected spanning subgraph polytope. In the last case we obtain a new class of facet inducing inequalities for the two-connected spanning subgraph polytope by our procedure. This theory has also been applied by Boyd and Hao (1993) to show a new class of inequalities is facet inducing for the two edge connected spanning subgraph polytope, and by Leung and Lee (1992) to show a new class of inequalities is facet inducing for the acyclic subgraph