This paper presents a method for representing the convex hull of 0-1 valued solutions of a system of linear inequalities as the projection of a higher-dimensional polyhedron. This approach allows for efficient optimization of linear objectives over the convex hull in polynomial time. The method is applied to the vertex packing polytope, where it is shown that a sequence of systems of inequalities can approximate the polytope, with the first system including constraints for cliques, odd holes, odd antiholes, wheels, and orthogonality. For perfect graphs, this first system gives the vertex packing polytope. The method is also extended to submodular functions and the Möbius function of a lattice. The paper discusses the use of matrix cuts to construct higher-dimensional polyhedra that approximate the convex hull of 0-1 solutions. These cuts are derived from linear inequalities valid for the convex hull and are used to project back to the original space, preserving all 0-1 solutions. The method is related to the work of Sherali and Adams, who introduced new variables for products of original ones and characterized the convex hull of 0-1 solutions in a high-dimensional space. The method is also related to the work of Pemantle, Propp, and Ullman on tensor powers of linear programs. The paper also discusses the algorithmic aspects of these constructions, including the use of separation oracles and the polynomial time separability of certain polytopes. The method is applied to the stable set polyhedra, where it is shown that the stable set polytope can be represented as the projection of a polytope with a polynomial number of facets. The paper also introduces stronger cut operators and discusses their properties, as well as the use of positive semidefiniteness in the construction of these operators. The paper concludes with a discussion of the implications of these results for combinatorial optimization, including the polynomial time solvability of the weighted stable set problem for certain classes of graphs. The results are shown to have applications in the study of polyhedra and their projections, as well as in the development of efficient algorithms for optimization problems.This paper presents a method for representing the convex hull of 0-1 valued solutions of a system of linear inequalities as the projection of a higher-dimensional polyhedron. This approach allows for efficient optimization of linear objectives over the convex hull in polynomial time. The method is applied to the vertex packing polytope, where it is shown that a sequence of systems of inequalities can approximate the polytope, with the first system including constraints for cliques, odd holes, odd antiholes, wheels, and orthogonality. For perfect graphs, this first system gives the vertex packing polytope. The method is also extended to submodular functions and the Möbius function of a lattice. The paper discusses the use of matrix cuts to construct higher-dimensional polyhedra that approximate the convex hull of 0-1 solutions. These cuts are derived from linear inequalities valid for the convex hull and are used to project back to the original space, preserving all 0-1 solutions. The method is related to the work of Sherali and Adams, who introduced new variables for products of original ones and characterized the convex hull of 0-1 solutions in a high-dimensional space. The method is also related to the work of Pemantle, Propp, and Ullman on tensor powers of linear programs. The paper also discusses the algorithmic aspects of these constructions, including the use of separation oracles and the polynomial time separability of certain polytopes. The method is applied to the stable set polyhedra, where it is shown that the stable set polytope can be represented as the projection of a polytope with a polynomial number of facets. The paper also introduces stronger cut operators and discusses their properties, as well as the use of positive semidefiniteness in the construction of these operators. The paper concludes with a discussion of the implications of these results for combinatorial optimization, including the polynomial time solvability of the weighted stable set problem for certain classes of graphs. The results are shown to have applications in the study of polyhedra and their projections, as well as in the development of efficient algorithms for optimization problems.