An Explicit Formula for Vertex Enumeration in the CUT(n) Polytope via Probabilistic Methods

TL;DR

This paper derives an explicit formula for the vertices of the cut polytope using probabilistic methods, providing a clear combinatorial construction that advances understanding of this fundamental polytope.

Contribution

It introduces a novel probabilistic approach and a recursive binary encoding to explicitly characterize the vertices of the cut polytope.

Findings

Vertices are characterized via a probabilistic binary encoding.

The encoding sequence exhibits almost-linear behavior approximating y = x - 1/2.

The alternating cycle function has power-of-two invariance, enabling a closed-form enumeration.

Abstract

We present an explicit closed-form formula for the vertices of the classical cut polytope $\operatorname{CUT}(n)$, defined as the convex hull of cut vectors of the complete graph $K_n$. Our derivation proceeds via a related polytope, denoted $\mathbf{1}$-$\operatorname{CUT}(n)$, whose vertices are obtained by flipping all bits of the $\operatorname{CUT}(n)$ vertices. This polytope arises naturally in a probabilistic context involving agreement probabilities among symmetric Bernoulli random variables which serves as the starting point of this work. Our approach constructs the vertex set recursively via a binary encoding that stems from this probabilistic perspective. We prove that the resulting sequence of encoded integers, when appropriately scaled, exhibits an almost-linear behavior closely approximating the line $y = x - \frac{1}{2}$. This structure motivates the introduction of the alternating cycle function, an integer-valued map whose key property is power-of-two composition invariance. The function serves as the foundation for our closed-form enumeration formula. The result provides a rare instance of explicit vertex characterization for a $0$/$1$-polytope and offers a transparent combinatorial construction independent of enumeration algorithms.