Optimization hardness as transient chaos in an analog approach to constraint satisfaction
Boolean satisfiability (k-SAT) is one of the most studied optimization problems, as an efficient (that is, polynomial-time) solution to k-SAT (for k≥3) implies efficient solutions to a large number of hard optimization problems. Here we propose a mapping of k-SAT into a deterministic continuous-time dynamical system with a unique correspondence between its attractors and the k-SAT solution clusters. We show that beyond a constraint density threshold, the analog trajectories become transiently chaotic and the boundaries between the basins of attraction8 of the solution clusters become fractal signalling the appearance of optimization hardness. Analytical arguments and simulations indicate that the system always finds solutions for satisfiable formulae even in the frozen regimes of random 3-SAT and of locked occupation problems (considered among the hardest algorithmic benchmarks), a property partly due to the system’s hyperbolic character. The system finds solutions in polynomial continuous time, however, at the expense of exponential fluctuations in its energy function.