Chvátal Closures for mixed Integer Programming Problems

Chvfital introduced the idea of viewing cutting planes as a system for proving that every integral solution of a given set of linear inequalities satisfies another given linear inequality. This viewpoint has proven to be very useful in many studies of combinatorial and integer programming problems. The basic ingredient in these cutting-plane proofs is that for a polyhedron P and integral vector w, if max(wx ]x ~ P, wx integer} = t, then wx ~ t is valid for all integral vectors in P. We consider the variant of this step where the requirement that wx be integer may be replaced by the requirement that #x be integer for some other integral vector #. The cutting-plane proofs thus obtained may be seen either as an abstraction of Gomory 's mixed integer cutting-plane technique or as a proof version of a simple class of the disjunctive cutting planes studied by Balas and Jeroslow. Our main result is that for a given polyhedron P, the set of vectors that satisfy every cutting plane for P with respect to a specified subset of integer variables is again a polyhedron. This allows us to obtain a finite recursive procedure for generating the mixed integer hull of a polyhedron, analogous to the process of repeatedly taking Chv~ital closures in the integer programming case. These results are illustrated with a number of examples from combinatorial optimization. Our work can be seen as a continuation of that of Nemhauser and Wolsey on mixed integer cutting planes.