On the Approximability of Weighted Model Integration on DNF Structures
This work addresses a computational challenge in probabilistic reasoning and AI, providing an efficient approximation method for a generalization of weighted model counting, though it is incremental as it builds on classical results.
The paper tackles the problem of approximating weighted model integration on DNF structures, showing that it admits a fully polynomial randomized approximation scheme for a class of weight functions, with experimental validation scaling to instances with up to 1K variables.
Weighted model counting (WMC) consists of computing the weighted sum of all satisfying assignments of a propositional formula. WMC is well-known to be #P-hard for exact solving, but admits a fully polynomial randomized approximation scheme (FPRAS) when restricted to DNF structures. In this work, we study weighted model integration, a generalization of weighted model counting which involves real variables in addition to propositional variables, and pose the following question: Does weighted model integration on DNF structures admit an FPRAS? Building on classical results from approximate volume computation and approximate weighted model counting, we show that weighted model integration on DNF structures can indeed be approximated for a class of weight functions. Our approximation algorithm is based on three subroutines, each of which can be a weak (i.e., approximate), or a strong (i.e., exact) oracle, and in all cases, comes along with accuracy guarantees. We experimentally verify our approach over randomly generated DNF instances of varying sizes, and show that our algorithm scales to large problem instances, involving up to 1K variables, which are currently out of reach for existing, general-purpose weighted model integration solvers.