Tractable Weighted First-Order Model Counting with Bounded Treewidth Binary Evidence
This work addresses a computational barrier in probabilistic reasoning and combinatorial problems, offering a tractable solution for specific evidence structures.
The paper tackled the intractability of Weighted First-Order Model Counting (WFOMC) when conditioning on evidence by restricting binary evidence to cases with bounded treewidth, resulting in a polynomial-time algorithm for two-variable fragments and solving an open combinatorial problem.
The Weighted First-Order Model Counting Problem (WFOMC) asks to compute the weighted sum of models of a given first-order logic sentence over a given domain. Conditioning WFOMC on evidence -- fixing the truth values of a set of ground literals -- has been shown impossible in time polynomial in the domain size (unless $\mathsf{\#P \subseteq FP}$) even for fragments of logic that are otherwise tractable for WFOMC without evidence. In this work, we address the barrier by restricting the binary evidence to the case where the underlying Gaifman graph has bounded treewidth. We present a polynomial-time algorithm in the domain size for computing WFOMC for the two-variable fragments $\text{FO}^2$ and $\text{C}^2$ conditioned on such binary evidence. Furthermore, we show the applicability of our algorithm in combinatorial problems by solving the stable seating arrangement problem on bounded-treewidth graphs of bounded degree, which was an open problem. We also conducted experiments to show the scalability of our algorithm compared to the existing model counting solvers.