Monte Carlo dynamics of optimization problems: A scaling description

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Abstract
We show that some hard optimization problems studied by Monte Carlo methods, such as simulated annealing, have features that can be estimated by a statistical analysis of the data, well before being actually observed. This applies, for instance, to the estimation of the ground-state energy of the problem. We start by showing that the density of states and the distribution of extremes of energy seen in a given time interval in the Monte Carlo dynamics of combinatorial optimization problems are strongly related to each other through the first-passage-time distribution of the stochastic dynamics of the system. We then introduce a scaling ansatz for this last quantity, which allows an estimate of the ground-state energy. Finally, we demonstrate the method on a ‘‘traveling-salesman’’ problem with known ground-state energy and apply it to the simulated annealing of a graph-bipartitioning problem.