Some polyhedra related to combinatorial problems

Abstract

This paper first describes a theory and algorithms for asymptotic integer programs. Next, a class of polyhedra is introduced. The vertices of these polyhedra provide solutions to the asymptotic integer programming problem; their faces are cutting planes for the general integer programming problem and, to some extent, the polyhedra coincide with the convex hull of the integer points satisfying a linear programming problem. These polyhedra are next shown to be cross sections of more symmetric higher dimensional polyhedra whose properties are then studied. Some algorithms for integer programming, based on a knowledge of the polyhedra, are outlined.