Satisfiability of acyclic and almost acyclic CNF formulas

  • Authors:

  • Sebastian Ordyniak;Daniel Paulusma;Stefan Szeider

  • Affiliations:

  • Institute of Information Systems, Vienna University of Technology, A-1040 Vienna, Austria;School of Engineering and Computing Sciences, Durham University, Durham DH1 3LE, UK;Institute of Information Systems, Vienna University of Technology, A-1040 Vienna, Austria

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2013

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Abstract

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