SIAM Journal on Computing, Ahead of Print.
Let ${A}\in{\mathbb R}^{n\times n}$ be a symmetric random matrix with independent and identically distributed (i.i.d.) Gaussian entries above the diagonal. We consider the problem of maximizing $\langle{\sigma},{A}{\sigma}\rangle$ over binary vectors ${\sigma}\in\{+1,-1\}^n$. In the language of statistical physics, this amounts to finding the ground state of the Sherrington--Kirkpatrick model of spin glasses. The asymptotic value of this optimization problem was characterized by Parisi via a celebrated variational principle, subsequently proved by Talagrand. We give an algorithm that, for any $\varepsilon>0$, outputs ${\sigma}_*\in\{-1,+1\}^n$ such that $\langle{\sigma}_*,\boldsymbol{A}{\sigma}_*\rangle$ is at least $(1-\varepsilon)$ of the optimum value, with probability converging to one as $n\to\infty$. The algorithm's time complexity is $C(\varepsilon)\, n^2$. We generalize it to matrices with i.i.d., but not necessarily Gaussian, entries, and obtain an algorithm that computes the MAXCUT of a dense Erdös--Renyi random graph to within a factor $(1-\varepsilon\cdot n^{-1/2})$. As a side result, we prove that, at (low) nonzero temperature, the algorithm constructs approximate solutions of the Thouless--Anderson--Palmer equations.

中文翻译：

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Sherrington-Kirkpatrick哈密顿量的优化

《 SIAM计算杂志》，预印本。
令$ {A} \ in {\ mathbb R} ^ {n \ times n} $是一个对称随机矩阵，在对角线上方具有独立且分布均匀的（iid）高斯项。我们考虑在二进制矢量$ {\ sigma} \ in \ {+ 1，-1 \} ^ n $上最大化$ \ langle {\ sigma}，{A} {\ sigma} \ rangle $的问题。用统计物理学的语言来说，这等于找到Sherrington-Kirkpatrick旋转玻璃模型的基态。Parisi通过著名的变分原理来表征此优化问题的渐近值，随后由Talagrand证明。我们给出一种算法，对于任何$ \ varepsilon> 0 $，输出$ {\ sigma} _ * \ in \ {-1，+ 1 \} ^ n $使得$ \ langle {\ sigma} _ *，\ boldsymbol {A} {\ sigma} _ * \ rangle $至少是最佳值的$（1- \ varepsilon）$，概率收敛为$ n \ to \ infty $。算法' s的时间复杂度是$ C（\ varepsilon）\，n ^ 2 $。我们将其推广到具有iid项（但不一定是高斯项）的矩阵，并获得一种算法，该算法可将密集的Erdös-Renyi随机图的MAXCUT计算到因子$（1- \ varepsilon \ cdot n ^ {-1 / 2}）$。附带的结果是，我们证明了在（低）非零温度下，该算法构造了Thouless-Anderson-Palmer方程的近似解。