Cut polytope has vertices on a line

Abstract

The cut polytope CUT(n) is the convex hull of the cut vectors in a complete graph with vertex set {1,…, n}. It is well known in the area of combinatorial optimization and recently has also been studied in a direct relation with admissible correlations of symmetric Bernoulli random variables. That probabilistic interpretation is a starting point of this work in conjunction with a natural binary encoding of the CUT(n). We show that for any n, with appropriate scaling, all encoded vertices of the polytope 1-CUT(n) are approximately on the line y=x−1/2.