When Is Weighted Satisfiability FPT?

  • Iyad A. Kanj
  • Ge Xia
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)

Abstract

The weighted monotone and antimonotone satisfiability problems on normalized circuits, abbreviated wsat + [t] and wsat[t], are canonical problems in the parameterized complexity theory. We study the parameterized complexity of wsat[t] and wsat + [t], where t ≥ 2, with respect to the genus of the circuit. For wsat[t], we give a fixed-parameter tractable (FPT) algorithm when the genus of the circuit is no(1), where n is the number of the variables in the circuit. For wsat + [2] (i.e., weighted monotone cnf-sat) and wsat + [3], which are both W[2]-complete, we also give FPT algorithms when the genus is no(1). For wsat + [t] where t > 3, we give FPT algorithms when the genus is \(O(\sqrt{\log{n}})\). We also show that both wsat[t] and wsat + [t] on circuits of genus nΩ(1) have the same W-hardness as the general wsat + [t] and wsat[t] problem (i.e., with no restriction on the genus), thus drawing a precise map of the parameterized complexity of wsat[t], and of wsat + [t], for t = 2,3, with respect to the genus of the underlying circuit.

As a byproduct of our results, we obtain, via standard parameterized reductions, tight results on the parameterized complexity of several problems with respect to the genus of the underlying graph.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Iyad A. Kanj
    • 1
  • Ge Xia
    • 2
  1. 1.School of ComputingDePaul UniversityChicagoUSA
  2. 2.Dept. of Computer ScienceLafayette CollegeEastonUSA

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