Approximating MAPs for Belief Networks is NP-Hard and Other Theorems

Finding rna.ximum a posteriori (MAP) assignments, also called Most Probable Explanations, is an important problem on Bayesian belief networks. Shimony has shown that finding MAPS is NPhard. In this paper, we show that approximating MAPS with a constant ratio bound is also NP-hard. In addition. we examine the complexity of two related problems which have been mentioned in the literature. We show that given the MAP for a belief network and evidence set, or the family of MAPS if the optimal is not unique, it remains NP-hard to find, or approximate, alternative nextbest explanations. Furthermore, we show that given the MAP, or MAPS, for a belief network and an initial evidence set, it is also NP-hard to find, or approximate, the MAP assignment for the same belief network with a modified evidence set that differs from the initial set by the addition or removal of even a single node assignment. Finally, we show that our main result applies to networks with constrained in-degree and out-degree, applies to randomized approximation, and even still applies if the ratio bound, instead of being constant, is allowed to be a polynomial function of various aspects of the network topology.