Almost Perfect Matrices and Graphs

We introduce the notions of ω-projection and κ-projection that map almost integral polytopes associated with almost perfect graphs G with n nodes from ℝⁿ into ℝ^n−ω, where ω is the maximum clique size in G. We show that C. Berge's strong perfect graph conjecture is correct if and only if the projection (of either kind) of such polytopes is again almost integral in ℝ^n−ω. Several important properties of ω-projections and κ-projections are established. We prove, among other results, that the strong perfect graph conjecture is wrong if an ω-projection and a related κ-projection of an almost integral polytope with 2 ≤ ω ≤ (n − 1)/2 produce different polytopes in ℝ^n−ω.