Time Lower Bounds for the Metropolis Process and Simulated Annealing

Abstract

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 $Ω\left(\frac{1}{n^{1-\varepsilon}}\right)$ 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 $Ω\left(\frac{\log (d)}{d}\right)$ 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.