Abstract
The classes of the W-hierarchy are the most important classes of intractable problems in parameterized complexity. These classes were originally defined via the weighted satisfiability problem for Boolean circuits. Here, besides the Boolean connectives we consider connectives such as majority, not-all-equal, and unique. For example, a gate labelled by the majority connective outputs true if more than half of its inputs are true. For any finite set \(\mathcal{C}\) of connectives we construct the corresponding W( \(\mathcal{C}\) )-hierarchy. We derive some general conditions which guarantee that the W-hierarchy and the W( \(\mathcal{C}\) )-hierarchy coincide levelwise. If \(\mathcal{C}\) only contains the majority connective then the first levels of the hierarchies coincide. We use this to show that a variant of the parameterized vertex cover problem, the majority vertex cover problem, is W[1]-complete.
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Research supported by the Australian Research Council, Centre in Bioinformatics, and by Fellowships to the Institute of Advanced Studies, Durham University, and to Grey College, Durham.
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Fellows, M., Flum, J., Hermelin, D. et al. W-Hierarchies Defined by Symmetric Gates. Theory Comput Syst 46, 311–339 (2010). https://doi.org/10.1007/s00224-008-9138-6
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DOI: https://doi.org/10.1007/s00224-008-9138-6