We consider the positive semidefinite (psd) matrices with binary entries, along with the corresponding integer polytopes. We begin by establishing some basic properties of these matrices and polytopes. Then, we show that several families of integer polytopes in the literature -- the cut, boolean quadric, multicut and clique partitioning polytopes -- are faces of binary psd polytopes. Finally, we present some implications of these polyhedral relationships. In particular, we answer an open question in the literature on the max-cut problem, by showing that the rounded psd inequalities define a polytope.
A.N. Letchford & M.M. Sørensen (2012) Binary positive semidefinite matrices and associated integer polytopes. Math. Program., 131(1), 253-271.