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

Constraint-satisfaction problems are among the computationally hardest tasks: solutions are efficiently checkable, but no efficient algorithms are known to compute those solutions. Fresh insight might come from physics. A study mapping optimization hardness onto the phenomena of turbulence and chaos suggests that constraint-satisfaction problems can be tackled using analog devices. Boolean satisfiability1 (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 problems2,3. 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 chaotic4,5,6,7, and the boundaries between the basins of attraction8 of the solution clusters become fractal7,8,9, signalling the appearance of optimization hardness10. Analytical arguments and simulations indicate that the system always finds solutions for satisfiable formulae even in the frozen regimes of random 3-SAT (ref. 11) and of locked occupation problems12 (considered among the hardest algorithmic benchmarks), a property partly due to the system’s hyperbolic4,13 character. The system finds solutions in polynomial continuous time, however, at the expense of exponential fluctuations in its energy function.