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Hierarchical threshold structure in Max-Cut with geometric edge weights

Mar 09, 2026

This study investigates the structure of optimal Max-Cut solutions in complete graphs with geometrically weighted edges, focusing on the parameter regime \(1 < r < 2\). For instances where edge weights decay geometrically in lexicographic order, the authors analyze the optimality transitions within the family of “k-isolated cuts.” By integrating combinatorial optimization, root analysis of polynomials, and extensive numerical verification, they establish—for the first time—a strictly decreasing hierarchy of threshold values \(r_k(n)\) that govern the dominance of these cuts. These thresholds decrease strictly with \(k\) and converge to 1 as \(n \to \infty\). The resulting phase diagram precisely delineates the regions of dominance among competing cuts, and computational evidence for \(n \leq 100\) supports the conjecture that k-isolated cuts are globally optimal for all \(n \geq 7\).

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

Jun 26, 2025

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.

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Latest Papers

Hierarchical threshold structure in Max-Cut with geometric edge weights

Mar 09, 2026

This study investigates the structure of optimal Max-Cut solutions in complete graphs with geometrically weighted edges, focusing on the parameter regime \(1 < r < 2\). For instances where edge weights decay geometrically in lexicographic order, the authors analyze the optimality transitions within the family of “k-isolated cuts.” By integrating combinatorial optimization, root analysis of polynomials, and extensive numerical verification, they establish—for the first time—a strictly decreasing hierarchy of threshold values \(r_k(n)\) that govern the dominance of these cuts. These thresholds decrease strictly with \(k\) and converge to 1 as \(n \to \infty\). The resulting phase diagram precisely delineates the regions of dominance among competing cuts, and computational evidence for \(n \leq 100\) supports the conjecture that k-isolated cuts are globally optimal for all \(n \geq 7\).

0 citationsRead paper

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

Jun 26, 2025

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.

0 citationsRead paper