Satisfiability of Acyclic and Almost Acyclic CNF Formulas
Abstract
We show that the Satisfiability (SAT) problem for CNF formulas with {\beta}-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 {\beta}-acyclic formulas and this superclass with a number of known polynomial formula classes. We then study the parameterized complexity of SAT for "almost" {\beta}-acyclic instances, using as parameter the formula's distance from being {\beta}-acyclic. As distance we use the size of a smallest strong backdoor set and the {\beta}-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.
Cite
@article{arxiv.1104.4279,
title = {Satisfiability of Acyclic and Almost Acyclic CNF Formulas},
author = {Sebastian Ordyniak and Daniel Paulusma and Stefan Szeider},
journal= {arXiv preprint arXiv:1104.4279},
year = {2013}
}
Comments
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)