MAP is the problem of finding a most probable instantiation of a set of
variables given evidence. MAP has always been perceived to be significantly
harder than the related problems of computing the probability of a variable
instantiation Pr, or the problem of computing the most probable explanation
(MPE). This paper investigates the complexity of MAP in Bayesian networks.
Specifically, we show that MAP is complete for NP^PP and provide further
negative complexity results for algorithms based on variable elimination. We
also show that MAP remains hard even when MPE and Pr become easy.