Biased thermodynamics can explain the behaviour of smart optimization algorithms that work above the dynamical threshold

Random constraint satisfaction problems can display a very rich structure in the space of solutions, with often an ergodicity breaking -- also known as clustering or dynamical -- transition preceding the satisfiability threshold when the constraint-to-variables ratio $\alpha$ is increased. However, smart algorithms start to fail finding solutions in polynomial time at some threshold $\alpha_{\rm alg}$ which is algorithmic dependent and generally bigger than the dynamical one $\alpha_d$. The reason for this discrepancy is due to the fact that $\alpha_d$ is traditionally computed according to the uniform measure over all the solutions. Thus, while bounding the region where a uniform sampling of the solutions is easy, it cannot predict the performance of off-equilibrium processes, that are still able of finding atypical solutions even beyond $\alpha_d$. Here we show that a reconciliation between algorithmic behaviour and thermodynamic prediction is nonetheless possible at least up to some threshold $\alpha_d^{\rm opt}\geq\alpha_d$, which is defined as the maximum value of the dynamical threshold computed on all possible probability measures over the solutions. We consider a simple Monte Carlo-based optimization algorithm, which is restricted to the solution space, and we demonstrate that sampling the equilibrium distribution of a biased measure improving on $\alpha_d$ is still possible even beyond the ergodicity breaking point for the uniform measure, where other algorithms hopelessly enter the out-of-equilibrium regime. The conjecture we put forward is that many smart algorithms sample the solution space according to a biased measure: once this measure is identified, the algorithmic threshold is given by the corresponding ergodicity-breaking transition.