Community Structure Inspired Algorithms for SAT and #SAT

We introduce h-modularity, a structural parameter of CNF formulas, and present algorithms that render the decision problem SAT and the model counting problem #SAT fixed-parameter tractable when parameterized by h-modularity. 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 it for an efficient solution of SAT and #SAT, respectively. We further show that h-modularity is incomparable with known parameters under which SAT or #SAT is fixed-parameter tractable.