Theory of Computing Systems

, Volume 46, Issue 2, pp 311–339 | Cite as

W-Hierarchies Defined by Symmetric Gates

  • Michael Fellows
  • Jörg Flum
  • Danny Hermelin
  • Moritz Müller
  • Frances Rosamond


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.


Parameterized complexity W-hierarchy Symmetric gates Bounded weft circuits 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Michael Fellows
    • 1
  • Jörg Flum
    • 2
  • Danny Hermelin
    • 3
  • Moritz Müller
    • 2
  • Frances Rosamond
    • 4
  1. 1.Research UnitUniversity of NewcastleNewcastleAustralia
  2. 2.Mathematics DepartmentUniversity of FreiburgFreiburgGermany
  3. 3.Caesaria Rothschild InstituteUniversity of HaifaHaifaIsrael
  4. 4.Parameterized Complexity Research UnitUniversity of NewcastleNewcastleAustralia

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