Fast Counting with Bounded Treewidth

Abstract

Many intractable problems have been shown to become tractable if the treewidth of the underlying structure is bounded by a constant. An important tool for deriving such results is Courcelle’s Theorem, which states that all properties defined by Monadic-Second Order (MSO) sentences are fixed-parameter tractable with respect to the treewidth. Arnborg et al. extended this result to counting problems defined via MSO properties. However, the MSO description of a problem is of course not an algorithm. Consequently, proving the fixed-parameter tractability of some problem via Courcelle’s Theorem can be considered as the starting point rather than the endpoint of the search for an efficient algorithm. Gottlob et al. have recently presented a new approach via monadic datalog to actually devise efficient algorithms for decision problems whose tractability follows from Courcelle’s Theorem. In this paper, we extend this approach and apply it to some fundamental counting problems in logic an artificial intelligence.

This work was supported by the Austrian Science Fund (FWF), project P20704-N18.