Complexity of Inference in Graphical Models

TL;DR

This paper investigates whether treewidth is the only structural property that guarantees tractable inference in graphical models, concluding that no other such property exists under certain hypotheses.

Contribution

It proves that, assuming a combinatorial hypothesis, low treewidth is the only structural restriction ensuring polynomial-time inference in graphical models.

Findings

Low treewidth is necessary for tractable inference.

No other structural property guarantees polynomial complexity.

Inference complexity is polynomial only for bounded treewidth models.

Abstract

It is well-known that inference in graphical models is hard in the worst case, but tractable for models with bounded treewidth. We ask whether treewidth is the only structural criterion of the underlying graph that enables tractable inference. In other words, is there some class of structures with unbounded treewidth in which inference is tractable? Subject to a combinatorial hypothesis due to Robertson et al. (1994), we show that low treewidth is indeed the only structural restriction that can ensure tractability. Thus, even for the "best case" graph structure, there is no inference algorithm with complexity polynomial in the treewidth.