On the dimension of projected polyhedra

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. (C) 1998 Elsevier Science B.V All rights reserved.