Matrix cones, projection representations, and stable set polyhedra
László Lovász, Alexander Schrijver · DIMACS series in discrete mathematics and theoretical computer science · 1991
It has been recognized recently that to represent a polyhedron as the projection of a higher-dimensional, but simpler, polyhedron is a powerful tool in polyhedral combinatorics.We develop a general method to construct higher-dimensional polyhedra (or, in some cases, convex sets) whose projection approximates the convex hull of 0-1 valued solutions of a system of linear inequalities.An important feature of these approximations is that one can optimize any linear objective function over them in polynomial time.In the special case of the vertex packing polytope, we obtain a sequence of systems of inequalities, such that already the first system includes clique, odd hole, odd antihole, wheel, and orthogonality constraints.In particular, for perfect (and many other) graphs, this first system gives the vertex packing polytope.For various classes of graphs, including t-perfect graphs, it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets.