Affiliations: Aix-Marseille Université, CNRS, LIF UMR 7279, 13288, Marseille, France. [email protected] | Institut für Theoretische Informatik, Leibniz Universität Hannover, Appelstr. 4, 30167 Hannover, Germany. [email protected]
Note: [] Supported by Campus France Projet No 28292TE. Address for correspondence: Aix-Marseille Univ., CNRS, LIF UMR 7279, 163 avenue de Luminy, 13288, Marseille, France
Note: [] Supported by DAAD Projekt-ID 55892324. Part of this work done while on leave at the University of Oxford, supported by EPSRC EP/G055114/1.
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 enumeration and counting problems.
