Not All FPRASs Are Equal: Demystifying FPRASs for DNF-Counting (Extended Abstract)

Abstract

The problem of counting the number of solutions of a DNF formula, also called #DNF, is a fundamental problem in AI with wide-ranging applications. Owing to the intractability of the exact variant, efforts have focused on the design of approximate techniques. Consequently, several  Fully Polynomial Randomized Approximation Schemes (FPRASs) based on Monte Carlo techniques have been proposed. Recently, it was discovered that hashing-based techniques too lend themselves to FPRASs for #DNF. Despite significant improvements, the complexity of the hashing-based FPRAS is still worse than that of the best Monte Carlo FPRAS by polylog factors. Two questions were left unanswered in previous works: Can the complexity of the hashing-based techniques be improved? How do these approaches compare empirically? In this paper, we first propose a new search procedure for the hashing-based FPRAS that removes the polylog factors from its time complexity. We then present the first empirical study of runtime behavior of different FPRASs for #DNF, which produces a nuanced picture. We observe that there is no single best algorithm for all formulas and that the algorithm with one of the worst time complexities solves the largest number of benchmarks.