Satisfiability of Acyclic and Almost Acyclic CNF Formulas
arXiv:1104.4279 · doi:10.1016/j.tcs.2012.12.039
Abstract
We show that the Satisfiability (SAT) problem for CNF formulas with β-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 β-acyclic formulas and this superclass with a number of known polynomial formula classes. We then study the parameterized complexity of SAT for "almost" β-acyclic instances, using as parameter the formula's distance from being β-acyclic. As distance we use the size of a smallest strong backdoor set and the β-hypertree width. As a by-product we obtain the W[1]-hardness of SAT parameterized by the (undirected) clique-width of the incidence graph, which disproves a conjecture by Fischer, Makowsky, and Ravve.
Extended abstracts appeared in the Proceedings of FSTTCS 2010 and SAT 2011. The latter corresponds to revision 1 of this arXiv paper (arXiv:1104.4279v1)