We prove that Simulated Annealing with an appropriate cooling schedule computes arbitrarily tight constant-factor approximations to the minimum spanning tree problem in polynomial time. This result was conjectured by Wegener (2005). More precisely, denoting by $n, m, w_{\max}$, and $w_{\min}$ the number of vertices and edges as well as the maximum and minimum edge weight of the MST instance, we prove that simulated annealing with initial temperature $T_0 \ge w_{\max}$ and multiplicative cooling schedule with factor $1-1/\ell$, where $\ell = \omega (mn\ln(m))$, with probability at least $1-1/m$ computes in time $O(\ell (\ln\ln (\ell) + \ln(T_0/w_{\min}) ))$ a spanning tree with weight at most $1+\kappa$ times the optimum weight, where $1+\kappa = \frac{(1+o(1))\ln(\ell m)}{\ln(\ell) -\ln (mn\ln (m))}$. Consequently, for any $\epsilon>0$, we can choose $\ell$ in such a way that a $(1+\epsilon)$-approximation is found in time $O((mn\ln(n))^{1+1/\epsilon+o(1)}(\ln\ln n + \ln(T_0/w_{\min})))$ with probability at least $1-1/m$. In the special case of so-called $(1+\epsilon)$-separated weights, this algorithm computes an optimal solution (again in time $O( (mn\ln(n))^{1+1/\epsilon+o(1)}(\ln\ln n + \ln(T_0/w_{\min})))$), which is a significant speed-up over Wegener's runtime guarantee of $O(m^{8 + 8/\epsilon})$.

翻译：我们证明，采用适当冷却计划的模拟退火算法，能在多项式时间内计算最小生成树问题的任意紧密常数因子近似解。该结果验证了Wegener（2005）的猜想。具体而言，设$n, m, w_{\max}$和$w_{\min}$分别表示最小生成树实例的顶点数、边数以及最大与最小边权重，我们证明：初始温度$T_0 \ge w_{\max}$且冷却因子为$1-1/\ell$（其中$\ell = \omega (mn\ln(m))$）的乘性冷却模拟退火算法，以至少$1-1/m$的概率在$O(\ell (\ln\ln (\ell) + \ln(T_0/w_{\min}) ))$时间内生成一棵权重不超过最优值$1+\kappa$倍的生成树，其中$1+\kappa = \frac{(1+o(1))\ln(\ell m)}{\ln(\ell) -\ln (mn\ln (m))}$。因此，对于任意$\epsilon>0$，可通过选择$\ell$使得算法以至少$1-1/m$的概率在$O((mn\ln(n))^{1+1/\epsilon+o(1)}(\ln\ln n + \ln(T_0/w_{\min})))$时间内找到$(1+\epsilon)$-近似解。在所谓$(1+\epsilon)$-分离权重的特殊情形下，该算法能计算最优解（同样在$O( (mn\ln(n))^{1+1/\epsilon+o(1)}(\ln\ln n + \ln(T_0/w_{\min})))$时间内完成），这显著优于Wegener所保证的$O(m^{8 + 8/\epsilon})$运行时间。