Approximate counting and the Lovasz local lemma

We introduce a new approach for approximately counting in bounded degree systems with higher-order constraints. Our main result is an algorithm to approximately count the number of solutions to a CNF formula where the degree is exponential in the number of variables per clause. Moreover our algorithm extends straightforwardly to approximate sampling, which shows that under Lovasz Local Lemma-like conditions, it is possible to generate a satisfying assignment approximately uniformly at random.

Our program is a significant departure from earlier techniques in approximate counting, and is a based on a framework to bootstrap an oracle for computing marginal probabilities on individual variables.