Weighted model counting consists of computing the weighted sum of all satisfying assignments of a propositional formula. Weighted model counting is well-known to be #P-hard for exact solving, but admits a fully polynomial randomized approximation scheme 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 a fully polynomial randomized approximation scheme? Building on classical results from approximate weighted model counting and approximate volume computation, 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.

加权模型计数包括计算命题公式的所有满足分配的加权和。众所周知，加权模型计数对于精确求解来说是#P困难的，但在仅限于 DNF 结构时允许完全多项式随机逼近方案。在这项工作中，我们研究加权模型集成，加权模型计数的推广，除了命题变量之外还涉及实变量，并提出以下问题：DNF结构上的加权模型集成是否允许完全多项式随机逼近方案？基于近似加权模型计数和近似体积计算的经典结果，我们展示了加权模型集成DNF结构确实可以近似为一类权重函数。我们的近似算法基于三个子程序，每个子程序都可以是弱（即近似）或强（即精确）预言机，并且在所有情况下都具有准确性保证。我们在随机生成的不同大小的DNF实例上通过实验验证了我们的方法，并表明我们的算法可以扩展到涉及多达 1K 个变量的大型问题实例，这些变量目前对于现有的通用加权模型集成求解器来说是遥不可及的。