We consider the Max-Cut problem. Let $G = (V,E)$ be a graph with adjacency matrix $(a_{ij})_{i,j=1}^{n}$. Burer, Monteiro & Zhang proposed to find, for $n$ angles $\left\{\theta_1, \theta_2, \dots, \theta_n\right\} \subset [0, 2\pi]$, minima of the energy $$ f(\theta_1, \dots, \theta_n) = \sum_{i,j=1}^{n} a_{ij} \cos{(\theta_i - \theta_j)}$$ because configurations achieving a global minimum leads to a partition of size 0.878 Max-Cut(G). This approach is known to be computationally viable and leads to very good results in practice. We prove that by replacing $\cos{(\theta_i - \theta_j)}$ with an explicit function $g_{\varepsilon}(\theta_i - \theta_j)$ global minima of this new functional lead to a $(1-\varepsilon)$Max-Cut(G). This suggests some interesting algorithms that perform well. It also shows that the problem of finding approximate global minima of energy functionals of this type is NP-hard in general.
翻译：我们认为 Max-Cut 问题。 让 $G = (V, E) 是一个配对矩阵 $(a ⁇ ij}) ⁇ i,j=1 ⁇ n}$的图表。 Burer, Monteiro & Zhang 提议寻找 $n 角度 $left\\\ theta_ 1,\theta_ 2,\theta_ n\\ subset [0, 2\ pi]$, 能量的迷你值 f(theta_ 1,\dots,\ dots,\dotes,\theta_n) ================ = = = = a\ j} a\ Burer, Mteiroi & Zhang 和 Zhang $, 因为实现全球最小值的配置导致0. 878 Max-Cut (G) 的分布。 众所周知, 这种方法是可行的, 并且在实践中可以很好的结果。 我们证明, 通过 $\\\\\\ meval_ mexal_ gal_ leal_ lection_ gal_ pal_ gal_ pal_ gal_ pal_ pal_ pal_ pal_ pal_ pal_ pal_ pal_ gal_ gal_ gal_ gal_ gal_ gal_ gal_ sal_ sal_ sal_ sal_ gal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ g_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ g_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ sal_ ex
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