On the Complexity of the Multiplication Method for Monotone CNF/DNF Dualization

Given the irredundant CNF representation φ of a monotone Boolean function f : {0, 1} 7→ {0, 1}, the dualization problem calls for finding the corresponding unique irredundant DNF representation ψ of f . The (generalized) multiplication method works by repeatedly dividing the clauses of φ into (not necessarily disjoint) groups, multiplying-out the clauses in each group, and then reducing the result by applying the absorption law. We present the first non-trivial upper-bounds on the complexity of this multiplication method. Precisely, we show that if the grouping of the clauses is done in an output-independent way, then multiplication can be performed in sub-exponential time (n|ψ|) √ |φ|. On the other hand, multiplication can be carried-out in quasi-polynomial time poly(n, |ψ|) · |φ| , provided that the grouping is done depending on the intermediate outputs produced during the multiplication process.