In this paper, we introduce a novel algorithm to solve projected model counting (PMC). PMC asks to count solutions of a Boolean formula with respect to a given set of projection variables, where multiple solutions that are identical when restricted to the projection variables count as only one solution. Inspired by the observation that the so-called “treewidth” is one of the most prominent structural parameters, our algorithm utilizes small treewidth of the primal graph of the input instance. More precisely, it runs in time $\mathcal{O}(2^{2^{k+4}}{n}^2)$ where k is the treewidth and n is the input size of the instance. In other words, we obtain that the problem PMC is fixed-parameter tractable when parameterized by treewidth. Further, we take the exponential time hypothesis (ETH) into consideration and establish lower bounds of bounded treewidth algorithms for PMC, yielding asymptotically tight runtime bounds of our algorithm.

While the algorithm above serves as a first theoretical upper bound and although it might be quite appealing for small values of k, unsurprisingly a naive implementation adhering to this runtime bound suffers already from instances of relatively small width. Therefore, we turn our attention to several measures in order to resolve this issue towards exploiting treewidth in practice: We present a technique called nested dynamic programming, where different levels of abstractions of the primal graph are used to (recursively) compute and refine tree decompositions of a given instance. Further, we integrate the concept of hybrid solving, where subproblems hidden by the abstraction are solved by classical search-based solvers, which leads to an interleaving of parameterized and classical solving. Finally, we provide a nested dynamic programming algorithm and an implementation that relies on database technology for PMC and a prominent special case of PMC, namely model counting (#Sat). Experiments indicate that the advancements are promising, allowing us to solve instances of treewidth upper bounds beyond 200.

在本文中，我们介绍了一种解决投影模型计数( PMC ) 的新算法。PMC要求计算关于给定投影变量集的布尔公式的解，其中在受限于投影变量时相同的多个解仅计为一个解。受观察到所谓的“树宽”是最突出的结构参数之一的启发，我们的算法利用了输入实例的原始图的小树宽。更准确地说，它及时运行$\mathcal{○}(2^{2^{ķ+4}}{n}^2)$其中k是树宽，n是实例的输入大小。换句话说，我们得到问题PMC在通过树宽参数化时是固定参数可处理的。此外，我们将指数时间假设 (ETH) 考虑在内，并为PMC建立有界树宽算法的下界，从而产生算法的渐近紧运行时界限。

虽然上面的算法作为第一个理论上限，虽然它可能对k的小值很有吸引力，不出所料，遵循此运行时界限的幼稚实现已经遭受宽度相对较小的实例的影响。因此，我们将注意力转向几个措施，以解决在实践中利用树宽的问题：我们提出了一种称为嵌套动态规划的技术，其中原始图的不同抽象级别用于（递归）计算和细化树分解给定实例的。此外，我们整合了混合求解的概念，其中被抽象隐藏的子问题由经典的基于搜索的求解器解决，这导致了参数化求解和经典求解的交错。最后，我们提供了一种嵌套动态规划算法和一个依赖于 PMC 数据库技术的实现以及 PMC 的一个突出特例，即模型计数 (#星期六）。实验表明，这些进步是有希望的，使我们能够解决树宽上限超过 200 的实例。