The volume of relaxed Boolean-quadric and cut polytopes

Abstract

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