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 Σ_2^(p-)complete, and Σ_2^(p-)hard to approximate to within an n^(1/2-ε) factor. When the input is a circuit, approximation is Σ_2^(p-)hard to within an n^(1-ε) factor. However, when the input is a DNF formula, the shortest implicant problem cannot be Σ_2^(p-)complete unless Σ_2^p=NP[log^2_n]^(NP).