A purely democratic characterization of W[1]

  • Authors:

  • Michael Fellows;Danny Hermelin;Moritz Müller;Frances Rosamond

  • Affiliations:

  • Parameterized Complexity Research Unit, The University of Newcastle, Callaghan, NSW, Australia;Department of Computer Science, University of Haifa, Haifa, Israel;Mathematisches Institut, Albert Ludwigs Universität Freiburg, Freiburg, Germany;Parameterized Complexity Research Unit, The University of Newcastle, Callaghan, NSW, Australia

  • Venue:
  • IWPEC'08 Proceedings of the 3rd international conference on Parameterized and exact computation
  • Year:
  • 2008

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Abstract

We give a novel characterization of W[1], the most importantfixed-parameter intractability class in the W-hierarchy, using Booleancircuits that consist solely of majority gates. Such gates have a Booleanvalue of 1 if and only if more than half of their inputs have value 1. Usingmajority circuits, we define an analog of the W-hierarchy which wecall the W-hierarchy, and show that W[1] = W[1] and W[t] ⊆ W[t] forall t. This gives the first characterization of W[1] based on the weightedsatisfiability problem for monotone Boolean circuits rather than antimonotone.Our results are part of a wider program aimed at exploringthe robustness of the notion of weft, showing that it remains a key parametergoverning the combinatorial nondeterministic computing strength ofcircuits, no matter what type of gates these circuits have.