On Hashing-Based Approaches to Approximate DNF-Counting

Abstract

Propositional model counting is a fundamental problem in artificial intelligence with a wide variety of applications, such as probabilistic inference, decision making under uncertainty, and probabilistic databases. Consequently, the problem is of theoretical as well as practical interest. When the constraints are expressed as DNF formulas, Monte Carlo-based techniques have been shown to provide a fully polynomial randomized approximation scheme (FPRAS). For CNF constraints, hashing-based approximation techniques have been demonstrated to be highly successful. Furthermore, it was shown that hashing-based techniques also yield an FPRAS for DNF counting without usage of Monte Carlo sampling. Our analysis, however, shows that the proposed hashing-based approach to DNF counting provides poor time complexity compared to the Monte Carlo-based DNF counting techniques. Given the success of hashing-based techniques for CNF constraints, it is natural to ask: Can hashing-based techniques provide an efficient FPRAS for DNF counting? In this paper, we provide a positive answer to this question. To this end, we introduce two novel algorithmic techniques: \emph{Symbolic Hashing} and \emph{Stochastic Cell Counting}, along with a new hash family of \emph{Row-Echelon hash functions}. These innovations allow us to design a hashing-based FPRAS for DNF counting of similar complexity (up to polylog factors) as that of prior works. Furthermore, we expect these techniques to have potential applications beyond DNF counting.