Parameterized complexity of weighted satisfiability problems

Authors:
Nadia Creignou;Heribert Vollmer
Affiliations:
Laboratoire d'Informatique Fondamentale de Marseille, CNRS UMR 7279, Aix-Marseille Université, Marseille Cedex 9, France;Institut für Theoretische Informatik, Leibniz Universität Hannover, Hannover, Germany
Venue:
SAT'12 Proceedings of the 15th international conference on Theory and Applications of Satisfiability Testing
Year:
2012

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Abstract

We consider the weighted satisfiability problem for Boolean circuits and propositional formulæ, where the weight of an assignment is the number of variables set to true. We study the parameterized complexity of these problems and initiate a systematic study of the complexity of its fragments. Only the monotone fragment has been considered so far and proven to be of same complexity as the unrestricted problems. Here, we consider all fragments obtained by semantically restricting circuits or formulæ to contain only gates (connectives) from a fixed set B of Boolean functions. We obtain a dichotomy result by showing that for each such B, the weighted satisfiability problems are either W[P]-complete (for circuits) or W[SAT]-complete (for formulæ) or efficiently solvable. We also consider the related counting problems.