Satisfiability with exponential families

  • Authors:

  • Dominik Scheder;Philipp Zumstein

  • Affiliations:

  • Institute of Theoretical Computer Science, ETH Zürich, Zürich, Switzerland;Institute of Theoretical Computer Science, ETH Zürich, Zürich, Switzerland

  • Venue:
  • SAT'07 Proceedings of the 10th international conference on Theory and applications of satisfiability testing
  • Year:
  • 2007

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Abstract

Fix a set S ⊆ {0, 1}* of exponential size, e.g. |S ∩ {0, 1}n| ∈ Ω(αn), α 1. The S-SAT problem asks whether a propositional formula F over variables v1, . . . , vn has a satisfying assignment (v1, . . . , vn) ∈ {0, 1}n ∩ S. Our interest is in determining the complexity of S-SAT. We prove that S-SAT is NP-complete for all context-free sets S. Furthermore, we show that if S-SAT is in P for some exponential S, then SAT and all problems in NP have polynomial circuits. This strongly indicates that satisfiability with exponential families is a hard problem. However, we also give an example of an exponential set S for which the S-SAT problem is not NP-hard, provided P ≠ NP.