Most probable explanations in Bayesian networks: Complexity and tractability

Abstract

An overview is given of definitions and complexity results of a number of variants of the problem of probabilistic inference of the most probable explanation of a set of hypotheses given observed phenomena.

Tree-Width and the Computational Complexity of MAP Approximations in Bayesian Networks

“…The parameterized complexity class FPT consists of all fixed parameter tractable problems κ−Π. While traditionally κ is defined as a mapping from problem instances to natural numbers (e.g., Flum & Grohe, 2006, p. 4), one can easily enhance the theory for rational parameters (Kwisthout, 2011). In the context of this paper, we will in particular consider rational parameters in the range [0, 1], and we will liberally mix integer and rational parameters.…”

“…This result holds for networks with only binary variables, with at most three incoming arcs per variable, and no evidence. In addition, Kwisthout (2011) showed that it is NP-hard in general to find an explanation approxsol B with Pr(approxsol B , e) > for any constant > 0, and thus that Pr(optsol B , e) − Pr(approxsol B , e) ≤ ρ for ρ > Pr(optsol B , e) − . The latter result holds even for networks with only binary variables, at most two incoming arcs per variable, a single evidence variable, and no intermediate variables (i.e., when we approximate an MPE problem).…”

“…Result 3.2 (Kwisthout, 2011) It is NP-hard to ρ-additive value-approximate MAP for ρ > Pr(optsol B , e) − for any constant > 0.…”

“…Even when all variables in the network are binary and the network has the (very restricted) polytree topology, MAP remains NP-hard (De Campos, 2011). Only when both the optimization and the inference part of the problem can be computed tractably (for example, if both the tree-width of the network is small, the cardinality of the variables is low, and the most probable joint value assignment has a high probability) MAP can be computed tractably (Kwisthout, 2011). It is known that, for arbitrary probability distributions and under the assumption of the Exponential Time Hypothesis, low tree-width of the moralized graph of a Bayesian network is a necessary condition for the Inference problem in Bayesian networks to be tractable (Kwisthout, Bodlaender, & van der Gaag, 2010); this result can easily be extended to MAP, as we will show in Section 4.…”

“…MAP is also intractable to approximate (Abdelbar & Hedetniemi, 1998;Kwisthout, 2011Kwisthout, , 2013Park & Darwiche, 2004). While it is obviously the case that a particular instance to the MAP problem can be approximated efficiently when it can be computed exactly efficiently, it is as yet unclear whether approximate MAP computations can be rendered tractable under different conditions than exact MAP computations.…”