On the dimension of projected polyhedra Original Research Article

We address several basic questions that arise in the use of projection in combinatorial optimization. Central to these is the connection between the dimension of a polyhedron Q and the dimension of its projection on a subspace. We give the exact relationship between the two dimensions. As a byproduct we characterize the relationship between the equality subsystem of a polyhedron and that of its projection. We also derive a necessary and sufficient condition for a face (in particular, a facet) of a polyhedron Q to project into a face (a facet) of the projection of Q, and give a necessary and sufficient condition for the existence of a 1-1 correspondence between the faces of Q and those of its projection. More generally, we characterize the dimensional relationship between the projection of Q and that of an arbitrary proper face of Q. We also show that the projection of a monotonized polyhedron on a subspace is the monotonization of the projection of the polyhedron on the same subspace.