<p>We consider the multilinear polytope, defined as the convex hull of the feasible region of a lifted binary polynomial optimization problem. We define a relaxation in an extended space for this polytope, which we call the complete edge relaxation. The complete edge relaxation is stronger than several well-known relaxations of the multilinear polytope, including the standard linearization, the flower relaxation, and the intersection of all possible recursive McCormick relaxations. In addition, for fixed-degree binary polynomial optimization problems, the case of primary practical interest, the complete edge relaxation is of polynomial size and is computationally efficient in practice. We prove that the complete edge relaxation is an extension of the multilinear polytope if and only if the corresponding hypergraph is <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-acyclic, the most general type of hypergraph acyclicity. This is in stark contrast with the widely-used standard linearization, which describes the multilinear polytope if and only if the hypergraph is Berge-acyclic, the most restrictive type of hypergraph acyclicity. Finally, we introduce a new class of facet-defining inequalities for the multilinear polytope of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-cycles of length three, which serve as the generalization of the well-known triangle inequalities for the Boolean quadric polytope.</p>