Monte Carlo and mean field slow cooling simulations for spin glasses: relation to NP-completeness

We discuss the dependence of the ground state energy as the Monte Carlo (MC) cooling rate in a variety of spin glass and other optimization problems. Our work is motivated by recent interest in the concept of simulated annealing and the need to find efficient numerical optimization schemes. We find that the ground state energy for six spin-glass models and the optimal tour length for the N-city traveling salesman problem are a sensitive function of the cooling rate. For the one dimensional nearest neighbor Gaussian and two two dimensional spin-glass models, the nearest neighbor ± J and Gaussian models, we find E(r) = E0 + C1 rx, while for the three dimensional t J model, a two layer t J model, the infinite range model and the N-city traveling salesman problem, the dependence on r is slower, E(r) = Eo - C2 (Inr)-1.Here C1 and C2 are constants.We believe that this difference is related to the fact that finding the ground state energy for the latter four problems is an intractable problem, i.e., it is Np-complete. We assert that a logarithmic cooling rate dependence for finding the ground state is a necessary consequence of Np-completeness. Non Np-complete problems may depend on the cooling rate either as a power-law or logarithmically. However, we have found no example for the latter.