The Metropolis process (MP) and Simulated Annealing (SA) are stochastic local search heuristics that are often used in solving combinatorial optimization problems. Despite significant interest, there are very few theoretical results regarding the quality of approximation obtained by MP and SA (with polynomially many iterations) for NP-hard optimization problems. We provide rigorous lower bounds for MP and SA with respect to the classical maximum independent set problem when the algorithms are initialized from the empty set. We establish the existence of a family of graphs for which both MP and SA fail to find approximate solutions in polynomial time. More specifically, we show that for any $\varepsilon \in (0,1)$ there are $n$-vertex graphs for which the probability SA (when limited to polynomially many iterations) will approximate the optimal solution within ratio $\Omega\left(\frac{1}{n^{1-\varepsilon}}\right)$ is exponentially small. Our lower bounds extend to graphs of constant average degree $d$, illustrating the failure of MP to achieve an approximation ratio of $\Omega\left(\frac{\log (d)}{d}\right)$ in polynomial time. In some cases, our impossibility results also go beyond Simulated Annealing and apply even when the temperature is chosen adaptively. Finally, we prove time lower bounds when the inputs to these algorithms are bipartite graphs, and even trees, which are known to admit polynomial-time algorithms for the independent set problem.

翻译：Metropolis过程（MP）和模拟退火（SA）是常用于求解组合优化问题的随机局部搜索启发式算法。尽管备受关注，但关于MP和SA（多项式迭代次数下）对NP难优化问题的近似质量的理论结果却极为稀少。针对经典的最大独立集问题，当算法从空集初始化时，我们为MP和SA建立了严格的下界。我们证明存在一类图，使得MP和SA均无法在多项式时间内找到近似解。具体而言，对于任意$\varepsilon \in (0,1)$，存在$n$顶点图，使得SA（限制为多项式迭代次数）以指数小的概率获得比率$\Omega\left(\frac{1}{n^{1-\varepsilon}}\right)$内的最优解近似。我们的下界可推广至恒定平均度数$d$的图，表明MP无法在多项式时间内达到$\Omega\left(\frac{\log (d)}{d}\right)$的近似比。在某些情形下，我们的不可能性结果甚至超越模拟退火，适用于温度可自适应选择的情况。最后，我们证明当输入为二分图甚至树（已知存在独立集问题的多项式时间算法）时，这些算法的时间下界依然成立。