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Boolean satisfiability [1] (k-SAT) is one of the most studied optimizationproblems, as an efficient (that is, polynomial-time) solution to k-SAT (for$k\geq 3$) implies efficient solutions to a large number of hard optimizationproblems [2,3]. Here we propose a mapping of k-SAT into a deterministiccontinuous-time dynamical system with a unique correspondence between itsattractors and the k-SAT solution clusters. We show that beyond a constraintdensity threshold, the analog trajectories become transiently chaotic [4-7],and the boundaries between the basins of attraction [8] of the solutionclusters become fractal [7-9], signaling the appearance of optimizationhardness [10]. Analytical arguments and simulations indicate that the systemalways finds solutions for satisfiable formulae even in the frozen regimes ofrandom 3-SAT [11] and of locked occupation problems [12] (considered among thehardest algorithmic benchmarks); a property partly due to the system'shyperbolic [4,13] character. The system finds solutions in polynomialcontinuous-time, however, at the expense of exponential fluctuations in itsenergy function.

Optimization hardness as transient chaos in an analog approach to constraint satisfaction