Treewidth and the Computational Complexity of MAP Approximations

The problem of finding the most probable explanation to a designated set of variables (the MAP problem) is a notoriously intractable problem in Bayesian networks, both to compute exactly and to approximate. It is known, both from theoretical considerations and from practical experiences, that low treewidth is typically an essential prerequisite to efficient exact computations in Bayesian networks. In this paper we investigate whether the same holds for approximating MAP. We define four notions of approximating MAP (by value, structure, rank, and expectation) and argue that all of them are intractable in general. We prove that efficient value-, structure-, and rank-approximations of MAP instances with high treewidth will violate the Exponential Time Hypothesis. In contrast, we hint that expectation-approximation can be done efficiently, even in MAP instances with high treewidth, if the most probable explanation has a high probability.