We prove that the Minimum Equivalent DNF problem is Sigma/sub 2//sup p/ 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 Sigma/sub 2//sup p/ complete, and Sigma/sub 2//sup p/ hard to approximate to within an n/sup 1/2 - epsilon/ factor. When the input is a circuit, approximation is Sigma/sub 2//sup p/ hard to within an n/sup 1 - epsilon/ factor. However, when the input is a DNF formula, the Shortest Implicant problem cannot be Sigma/sub 2//sup p/ complete unless Sigma/sub 2//sup p/ = NP[log/sup 2/ n]/sup NP/.