We study the parameterized complexity of the propositional satisfiability (SAT) and the more general model counting (#SAT) problems and obtain novel fixed-parameter algorithms that exploit the structural properties of input formulas. In the first part of the paper, we parameterize by the treewidth of the following two graphs associated with CNF formulas: the consensus graph and the conflict graph. Both graphs have as vertices the clauses of the formula; in the consensus graph two clauses are adjacent if they do not contain a complementary pair of literals, while in the conflict graph two clauses are adjacent if they do contain a complementary pair of literals. We show that #SAT is fixed-parameter tractable when parameterized by the treewidth of the former graph, but SAT is W[1]-hard when parameterized by the treewidth of the latter graph.

In the second part of the paper, we turn our attention to a novel structural parameter we call h-modularity which is loosely inspired by the well-established notion of community structure. The new parameter is defined in terms of a partition of clauses of the given CNF formula into strongly interconnected communities which are sparsely interconnected with each other. Each community forms a hitting formula, whereas the interconnections between communities form a graph of small treewidth. Our algorithms first identify the community structure and then use them for an efficient solution of SAT and #SAT, respectively.

我们研究了命题可满足性（SAT）和更一般的模型计数（#SAT）问题的参数化复杂性，并获得了利用输入公式的结构特性的新颖固定参数算法。在本文的第一部分中，我们通过与CNF公式关联的以下两个图的树宽进行参数化：共识图和冲突图。两个图都具有公式的子句作为顶点；在共识图中，如果两个子句不包含一对互补的文字，则它们是相邻的；而在冲突图中，如果两个子句确实包含一对互补的文字，则它们是相邻的。我们显示，当通过前一个图的树宽进行参数化时，＃SAT是固定参数可处理的，但是当通过后一个图的树宽进行参数化时，SAT是W [1] -hard。

在本文的第二部分中，我们将注意力转移到一个称为h模数的新结构参数上，该参数是受到公认的社区结构概念的宽松启发。根据给定CNF公式的子句划分为相互之间稀疏互连的强互连社区来定义新参数。每个社区构成一个重要的公式，而社区之间的相互联系形成一个小树宽的图。我们的算法首先确定社区结构，然后将其分别用于SAT和#SAT的有效解决方案。