Weighted model counting has emerged as a prevalent approach for probabilistic inference. In this paper, we are interested in weighted DNF counting, or briefly, weighted #DNF, which admits a fully polynomial randomized approximation scheme, as shown by Karp and Luby. To this date, the best algorithm for approximating #DNF is due to Karp, Luby and Madras. The drawback of this algorithm is that it runs in quadratic time and hence is not suitable for fast online reasoning. To overcome this, we propose a novel approach that combines approximate model counting with deep learning. We conduct detailed experiments to validate our approach, and show that our model learns and generalizes from #DNF instances with a very high accuracy.
Two complexity classes, PP and (+)P, are compared with PH (the polynomial-time hierarchy). The main results are as follows: (1) every set in PH is reducible in a certain sense to a set in PP, an (2) every set in PH is reducible to a set in (+)P under randomized polynomial-time reducibility with two-sided bounded error probability. It follows from these results that neither PP nor (+)P is a subset of or equivalent to PH unless PH collapses to a finite level. This is strong evidence that both classes are strictly harder than PH. > Several enumeration and reliability problems are shown to be # P-complete, and hence, at least as hard as NP-complete problems. Included are important problems in network reliability analysis, namely, computing the probability that a graph is connected and counting the number of minimum cardinality @math -cuts or directed network cuts. Also shown to be # P-complete are counting vertex covers in a bipartite graph, counting antichains in a partial order, and approximating the probability that a graph is connected and the probability that a pair of vertices is connected. 