Weighted model counting (WMC) is the task of computing the weighted sum of all satisfying assignments (i.e., models) of a propositional formula. Similarly, weighted model sampling (WMS) aims to randomly generate models with probability proportional to their respective weights. Both WMC and WMS are hard to solve exactly, falling under the #-hard complexity class. However, it is known that the counting problem may sometimes be tractable, if the propositional formula can be compactly represented and expressed in first-order logic. In such cases, model counting problems can be solved in time polynomial in the domain size, and are known as . The following question then arises: Is it also the case for WMS? This paper addresses this question and answers it affirmatively. Specifically, we prove the for the two-variables fragment of first-order logic with counting quantifiers in this paper, by devising an efficient sampling algorithm for this fragment that runs in time polynomial in the domain size. We then further show that this result continues to hold even in the presence of cardinality constraints. To empirically validate our approach, we conduct experiments over various first-order formulas designed for the uniform generation of combinatorial structures and sampling in statistical-relational models. The results demonstrate that our algorithm outperforms a state-of-the-art WMS sampler by a substantial margin, confirming the theoretical results.

中文翻译：

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对称加权一阶模型采样的提升算法

加权模型计数（WMC）是计算命题公式的所有满足分配（即模型）的加权和的任务。类似地，加权模型抽样（WMS）旨在随机生成模型，其概率与其各自的权重成正比。 WMC 和 WMS 都很难精确求解，属于 #-hard 复杂度类别。然而，众所周知，如果命题公式可以用一阶逻辑紧凑地表示和表达，则计数问题有时可能是容易处理的。在这种情况下，模型计数问题可以用域大小的时间多项式来求解，称为 。那么问题来了：WMS也是这样吗？本文针对这个问题做出了肯定的回答。具体来说，我们在本文中通过为该片段设计一种在域大小中以时间多项式运行的有效采样算法，证明了具有计数量词的一阶逻辑的二变量片段。然后我们进一步表明，即使存在基数约束，这个结果仍然成立。为了凭经验验证我们的方法，我们对各种一阶公式进行了实验，这些一阶公式是为组合结构的统一生成和统计关系模型中的采样而设计的。结果表明，我们的算法大幅优于最先进的 WMS 采样器，证实了理论结果。