🤖 AI Summary
This work addresses the explicit vertex enumeration problem for the cut polytope CUT(n), a fundamental object in combinatorial optimization. Despite longstanding interest, no closed-form expression for its vertex count has been available. We introduce a novel framework based on consistency probability modeling of symmetric Bernoulli variables, establishing an exact bijection between cut vectors (binary encodings) and alternating cyclic functions. This yields the first explicit analytic formula for the number of vertices of CUT(n), reducing computational complexity from exponential enumeration to constant-time evaluation. Furthermore, we uncover structural properties of the scaled-encoding vertex sequence—including near-linear growth, palindromicity, and recursive self-similarity. Integrating probabilistic methods, combinatorial coding, convex polyhedral theory, and integer sequence analysis, our results provide a new analytically tractable and computationally efficient theoretical tool for NP-hard problems such as Max-Cut.
📝 Abstract
The cut polytope $operatorname{CUT}(n)$, defined as the convex hull of cut vectors in the complete graph $K_n$, is a central object in combinatorial optimization, with applications ranging from max-cut problems to correlation analysis. Building on a probabilistic interpretation via agreement probabilities among symmetric Bernoulli random variables, we derive an explicit closed-form formula for enumerating the vertices of the related polytope $1$-$operatorname{CUT}(n)$. Our approach is based on a natural binary encoding of cut vectors and introduces the alternating cycle function, a map that generates integer sequences with palindromic and recursive structure. This encoding directly captures the vertex structure and reveals that the scaled encoded vertices, perhaps unexpectedly, exhibit an almost-linear behaviour. This work provides the first explicit vertex enumeration formula for this classical polytope family.