Satisfiability of acyclic and almost acyclic CNF formulas

• DOI: 10.1016/j.tcs.2012.12.039
• Publication Date: 2013-01-08T16:23:42Z
• Authors: Sebastian Ordyniak; Daniel Paulusma; Stefan Szeider
• Keywords: Satisfiability

We show that the Satisfiability (SAT) problem for CNF formulas with ��-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 ��-acyclic formulas and this superclass with a number of known polynomial formula classes. We then study the parameterized complexity of SAT for "almost" ��-acyclic instances, using as parameter the formula's distance from being ��-acyclic. As distance we use the size of a smallest strong backdoor set and the ��-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.<br/>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)<br/>