Lower Complexity Bounds for Lifted Inference

TL;DR

This paper establishes fundamental lower bounds on the complexity of lifted inference in probabilistic relational models, showing that efficient polynomial algorithms are unlikely for broad classes of such models, even approximately.

Contribution

It extends existing complexity bounds to weighted, quantifier-free formulas, demonstrating the inherent computational limitations of lifted inference.

Findings

No polynomial lifted inference algorithms exist under common complexity assumptions.

Approximate inference also faces similar complexity limitations.

Results hold even for models without the equality predicate.

Abstract

One of the big challenges in the development of probabilistic relational (or probabilistic logical) modeling and learning frameworks is the design of inference techniques that operate on the level of the abstract model representation language, rather than on the level of ground, propositional instances of the model. Numerous approaches for such "lifted inference" techniques have been proposed. While it has been demonstrated that these techniques will lead to significantly more efficient inference on some specific models, there are only very recent and still quite restricted results that show the feasibility of lifted inference on certain syntactically defined classes of models. Lower complexity bounds that imply some limitations for the feasibility of lifted inference on more expressive model classes were established early on in (Jaeger 2000). However, it is not immediate that these results also apply to the type of modeling languages that currently receive the most attention, i.e., weighted, quantifier-free formulas. In this paper we extend these earlier results, and show that under the assumption that NETIME =/= ETIME, there is no polynomial lifted inference algorithm for knowledge bases of weighted, quantifier- and function-free formulas. Further strengthening earlier results, this is also shown to hold for approximate inference, and for knowledge bases not containing the equality predicate.