Leveraging belief propagation, backtrack search, and statistics for model counting

We consider the problem of estimating the model count (number of solutions) of Boolean formulas, and present two techniques that compute estimates of these counts, as well as either lower or upper bounds with different trade-offs between efficiency, bound quality, and correctness guarantee. For lower bounds, we use a recent framework for probabilistic correctness guarantees, and exploit message passing techniques for marginal probability estimation, namely, variations of Belief Propagation (BP). Our results suggest that BP provides useful information even on structured loopy formulas. For upper bounds, we perform multiple runs of the MiniSat SAT solver with a minor modification, and obtain statistical bounds on the model count based on the observation that the distribution of a certain quantity of interest is often very close to the normal distribution. Our experiments demonstrate that our model counters based on these two ideas, BPCount and MiniCount, can provide very good bounds in time significantly less than alternative approaches.