W-hierarchies defined by symmetric gates

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

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


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 C of connectives we construct the corresponding W(C)-hierarchy. We derive some general conditions which guarantee that the W-hierarchy and the W(C)-hierarchy coincide levelwise. If 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.

Original languageEnglish
Pages (from-to)311-339
Number of pages29
JournalTheory of Computing Systems
Issue number2
StatePublished - 1 Feb 2010
Externally publishedYes


  • Bounded weft circuits
  • Parameterized complexity
  • Symmetric gates
  • W-hierarchy

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics


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