Tractable Weighted First-Order Model Counting with Bounded Treewidth Binary Evidence

2025-11-12

AI Summary

This paper investigates weighted first-order model counting (WFOMC) with evidence: summing weights over first-order models consistent with a fixed assignment to a subset of atomic propositions. While WFOMC is known to be #P-hard even for decidable fragments such as FO² and C² under arbitrary evidence, we introduce the treewidth of the Gaifman graph induced by binary evidence as a key parameter and present the first domain-polynomial-time algorithm for WFOMC. Our method leverages bounded treewidth to design a dynamic programming decomposition framework that preserves tractability for FO² and C² under evidence constraints. This yields the first polynomial-time solution to the long-standing open problem of stable seating arrangement on bounded-degree graphs. Experimental evaluation demonstrates that our approach significantly outperforms existing WFOMC solvers in both runtime and scalability.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

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 $ ext{FO}^2$ and $ ext{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.

Problem

• Computing weighted model counts with binary evidence constraints efficiently
• Overcoming intractability via bounded treewidth graph restrictions
• Enabling combinatorial problem solving like stable seating arrangements

Innovation

• Bounded treewidth binary evidence restriction
• Polynomial-time algorithm for FO2 and C2 fragments
• Solving stable seating on bounded-treewidth graphs