Monte Carlo algorithms are very effective in finding the largest independent set in sparse random graphs

The effectiveness of stochastic algorithms based on Monte Carlo dynamics in solving hard optimization problems is mostly unknown. Beyond the basic statement that at a dynamical phase transition the ergodicity breaks and a Monte Carlo dynamics cannot sample correctly the probability distribution in times linear in the system size, there are almost no predictions or intuitions on the behavior of this class of stochastic dynamics. The situation is particularly intricate because, when using a Monte Carlo-based algorithm as an optimization algorithm, one is usually interested in the out-of-equilibrium behavior, which is very hard to analyze. Here we focus on the use of parallel tempering in the search for the largest independent set in a sparse random graph, showing that it can find solutions well beyond the dynamical threshold. Comparison with state-of-the-art message passing algorithms reveals that parallel tempering is definitely the algorithm performing best, although a theory explaining its behavior is still lacking.