Theory of Nonequilibrium Local Search on Random Satisfaction Problems

Abstract

We study local search algorithms to solve instances of the random k-satisfiability problem, equivalent to finding (if they exist) zero-energy ground states of statistical models with disorder on random hypergraphs. It is well known that the best such algorithms are akin to non-equilibrium processes in a high-dimensional space. In particular, algorithms known as focused, and which do not obey detailed balance, outperform simulated annealing and related methods in the task of finding the solution to a complex satisfiability problem, that is to find (exactly or approximately) the minimum in a complex energy landscape. A physical question of interest is if the dynamics of these processes can be well predicted by the well-developed theory of equilibrium Gibbs states. While it has been known empirically for some time that this is not the case, an alternative systematic theory that does so has been lacking. In this paper we introduce such a theory based on the recently developed technique of cavity master equations and test it on the paradigmatic random 3-satisfiability problem. Our theory predicts the solution process very accurately away from the algorithm phase boundary and also predicts the qualitative form of this boundary.

From random point processes to hierarchical Cavity Master Equations for the stochastic dynamics of disordered systems in Random Graphs: Ising models and epidemics

“…Now we must keep only the order ∆t term in (7). Let's expand first Q t+∆t (X) and leave the rest untouched.…”

“…As before, we will write P t+∆t ( σ) as the marginalization of Q t+∆t ( X) and then expand to first order in ∆t. Explicitly writing that marginalization as in (7):…”

“…The core of this method is a differential equation for the cavity conditional probability densities: the Cavity Master Equation (CME). This approach has proven to be very general as it has been applied to several models in graphs with finite connectivity like the Ising ferromagnet, the random field Ising model and the Viana-Bray spin-glass model [1], the ferromagnetic p-spin under Glauber dynamics [6] and more recently also the dynamics of a focused search algorithm to solve the three-SAT problem in a random graph [7]. In this work we generalize the CME derived in [1], providing master equations for the probability densities of any group of connected variables.…”

“…A brief description of the theory of RPP is given in subsection II. Section III presents our hierarchical system of corrections to the original equations derived in [1] and used in [6,7,13]. They are numerically tested at section IV.…”

“…Unlike the two approaches previously described, our method gives information about local probability distributions, and can be straightforwardly used to study problems which are intrinsically out of equilibrium [7], i.e., where detailed balance is not conserved. As an example, the last subsection of IV is devoted to the exploration of susceptible-infectioussusceptible model for epidemics.…”