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S. Ordyniak, D. Paulusma, St. Szeider:
"Satisfiability of acyclic and almost acyclic CNF formulas";
Theoretical Computer Science, 481 (2013), 85 - 99.

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.

Acyclic hypergraph, Chordal bipartite graph, Davis-Putnam resolution

http://dx.doi.org/10.1016/j.tcs.2012.12.039

Project Head Stefan Szeider:
The Parameterized Complexity of Reasoning Problems