Satisfiability of Acyclic and Almost Acyclic CNF Formulas

Sebastian Ordyniak, Daniel Paulusma, and Stefan Szeider.

Proceedings of FSTTCS 2010, 30th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, December 15-18, 2010, IMSc, Chennai, India. Kamal Lodaya and Meena Mahajan (eds.), Leibniz International Proceedings in Informatics (LIPIcs), pp. 84-95, 2010.


We study the propositional satisfiability problem (SAT) on classes of CNF formulas (formulas in Conjunctive Normal Form) that obey certain structural restrictions in terms of their hypergraph structure, by associating to a CNF formula the hypergraph obtained by ignoring negations and considering clauses as hyperedges on variables. We show that satisfiability of CNF formulas with so-called "β-acyclic hypergraphs" can be decided in polynomial time. We also 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 smallest strong backdoor sets 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 (Discrete Applied Mathematics 156, 2008).

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