Continuous-time dynamical systems approach to hard constraint satisfaction problems

by Melinda Varga There are many problems, which cannot be solved with today’s digital computers. One of the most studied such problem is Boolean satisfiability (k-SAT), which asks to find the truth-values for a set of Boolean variables in a way to satisfy a given number of constraints. This problem appears in many real-world applications, and it has a key role in the theory of computational complexity and in particular NP-completeness: if one would find an efficient (polynomial-time) algorithm to solve k-SAT (for k ≥ 3), then we would be able to generate solutions efficiently to all problems from the NP class (Cook-Levin theorem), i.e., to a very large number of hard problems. The Thesis focuses on the k-SAT problem and presents a novel approach to it, using a deterministic continuous-time dynamical system. This dynamical system solves the problem efficiently (in polynomial continuous-time) at the expense of exponential fluctuations in its energy function, while it also shows that problem hardness is translated into a transiently chaotic behavior of the analog trajectories by this system. We use the escape rate, an invariant measure of transient chaos, to show that hardness appears through a second-order phase transition in the random 3-SAT ensemble and a similar behavior is found in 4-SAT as well, however, such transition does not occur for 2-SAT (which is in class P, hence easily solvable by a polynomial-time algorithm). Since the solver (i.e., the dynamical system expressed as ordinary differential