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Backdoors to Acyclic SAT

  • Serge Gaspers
  • Stefan Szeider
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

Backdoor sets contain certain key variables of a CNF formula F that make it easy to solve the formula. More specifically, a weak backdoor set of F is a set X of variables such that there exits a truth assignment τ to X that reduces F to a satisfiable formula F[τ] that belongs to a polynomial-time decidable base class \(\mathcal C\). A strong backdoor set is a set X of variables such that for all assignments τ to X, the reduced formula F[τ] belongs to \(\mathcal C\).

We study the problem of finding backdoor sets of size at most k with respect to the base class of CNF formulas with acyclic incidence graphs, taking k as the parameter. We show that
  1. 1

    the detection of weak backdoor sets is W[2]-hard in general but fixed-parameter tractable for r-CNF formulas, for any fixed r ≥ 3, and

     
  2. 2

    the detection of strong backdoor sets is fixed-parameter approximable.

     
Result 1 is the the first positive one for a base class that does not have a characterization with obstructions of bounded size. Result 2 is the first positive one for a base class for which strong backdoor sets are more powerful than deletion backdoor sets.

Not only SAT, but also #SAT can be solved in polynomial time for CNF formulas with acyclic incidence graphs. Hence Result 2 establishes a new structural parameter that makes #SAT fixed-parameter tractable and that is incomparable with known parameters such as treewidth and clique-width. We obtain the algorithms by a combination of an algorithmic version of the Erdős-Pósa Theorem, Courcelle’s model checking for monadic second order logic, and new combinatorial results on how disjoint cycles can interact with the backdoor set.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Serge Gaspers
    • 1
    • 2
  • Stefan Szeider
    • 2
  1. 1.School of Computer Science and EngineeringThe University of New South WalesSydneyAustralia
  2. 2.Institute of Information SystemsVienna University of TechnologyViennaAustria

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