The volume of relaxed Boolean-quadric and cut polytopes

For n 2, the boolean quadric polytope P n is the convex hull in d := ? n+1 2 dimensions of the binary solutions of x i x j = y ij , for all i < j in N := f1; 2; :::; ng. The polytope is naturally modeled by a somewhat larger polytope; namely, Q n the solution set of y ij x i , y ij x j , x i + x j 1 + y ij , y ij 0, for all i; j in N. In a rst step toward seeing how well Q n approximates P n , we establish that the d-dimensional volume of Q n is 2 2n?d n!=(2n)!. Using a well-known connection between P n and the \cut polytope" of a complete graph on n + 1 vertices, we also establish the volume of a relaxation of this cut polytope.