The volume of relaxed Boolean-quadric and cut polytopes

For n ⩾ 2, the boolean quadric polytope Pn is the convex hull in d:=(n+12) dimensions of the binary solutions xixj = yij, for all i < j in N ≔ 1,2,. …,n. The polytope is naturally modeled by a somewhat larger polytope; namely, Ln the solution set of uij ⩽ xij, yij ⩽ xj, xi + xj ⩽ 1 + yij, yij ⩾ 0, for all i, j in N. In a first step toward seeing how well Ln approximates Pn we establish that the d-dimensional volume of Ln is 22n−dn!/(2n)!. Using a well-known connection between Pn and the ‘cut polytope’ of a complete graph n + 1 vertices, we also establish the volume of a relaxation of this cut polytope.