Satisfiability of acyclic and almost acyclic CNF formulas

We show that the Satisfiability (SAT) problem for CNF formulas with ββ-acyclic hypergraphs can be solved in polynomial time by using a special type of Davis–Putnam resolution in which each resolvent is a subset of a parent clause. We extend this class to CNF formulas for which this type of Davis–Putnam resolution still applies and show that testing membership in this class is NP-complete. We compare the class of ββ-acyclic formulas and this superclass with a number of known polynomial formula classes. We then study the parameterized complexity of SAT for “almost” ββ-acyclic instances, using as parameter the formula’s distance from being ββ-acyclic. As distance we use the size of a smallest strong backdoor set and the ββ-hypertree width. As a by-product we obtain the W[1]W[1]-hardness of SAT parameterized by the (undirected) clique-width of the incidence graph, which disproves a conjecture by Fischer, Makowsky, and Ravve