The minimum equivalent DNF problem and shortest implicants

We prove that the Minimum Equivalent DNF problem is Σ₂ ^p-complete, resolving a conjecture due to L.J. Stockmeyer (1976). 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 Σ₂^p-complete, and Σ₂^p-hard to approximate to within an n^1/2-ε factor. When the input is a circuit, approximation is Σ₂^p-hard to within an n^1-ε factor. However, when the input is a DNF formula, the shortest implicant problem cannot be Σ₂^p-complete unless Σ₂^p=NP[log²n]^NP