The Minimum Equivalent DNF Problem and Shortest Implicants

We prove that the Minimum Equivalent DNF problem is p2-complete, resolving a conjecture due to Stockmeyer. The proof involves as an intermediate step a variant of a related problem in logic minimization, namely, that of finding the shortest implicant of a Boolean function. We also obtain certain results concerning the complexity of the Shortest Implicant problem that may be of independent interest. When the input is a formula, the Shortest Implicant problem is p2complete, and p2-hard to approximate to within an n1=2 factor. When the input is a circuit, approximation is p2hard to within an n1 factor. However, when the input is a DNF formula, the Shortest Implicant problem cannot be p2-complete unless p2 = NP[log2 n]NP.