Cut polytope has vertices on a line

Abstract

The cut polytope $\mathrm{CUT}(n)$ is the convex hull of the cut vectors in a complete graph with vertex set $\{1,\dots ,n\}$. 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 vertices of the polytope $1$-CUT($n$) encoded as integers are approximately on the line $y=x-1/2$.

6 pages