Parameterized Complexity Results for Bayesian Inference
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We present completeness results for inference in Bayesian networks with respect to two different parameterizations, namely the number of variables and the topological vertex separation number. For this we introduce the parameterized complexity classes 𝖶[1]𝖯𝖯 and 𝖷𝖫𝖯𝖯, which relate to 𝖶[1] and 𝖷𝖭𝖫𝖯 respectively as 𝖯𝖯 does to 𝖭𝖯. The second parameter is intended as a natural translation of the notion of pathwidth to the case of directed acyclic graphs, and as such it is a stronger parameter than the more commonly considered treewidth. Based on a recent conjecture, the completeness results for this parameter suggest that deterministic algorithms for inference require exponential space in terms of pathwidth and by extension treewidth. These results are intended to contribute towards a more precise understanding of the parameterized complexity of Bayesian inference and thus of its required computational resources in terms of both time and space.
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