An algorithm for the SAT problem for formulae of linear length

  • Authors:

  • Magnus Wahlström

  • Affiliations:

  • Department of Computer and Information Science, Linköping University, Linköping, Sweden

  • Venue:
  • ESA'05 Proceedings of the 13th annual European conference on Algorithms
  • Year:
  • 2005

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Abstract

We present an algorithm that decides the satisfiability of a CNF formula where every variable occurs at most k times in time $O(2^{N(1-c/(k+1)+O(1/k^{2}))})$ for some c (that is, O(αN) with αk). For k ≤ 4, the results coincide with an earlier paper where we achieved running times of O(20.1736 N) for k = 3 and O(20.3472N) for k = 4 (for k ≤ 2, the problem is solvable in polynomial time). For k4, these results are the best yet, with running times of O(20.4629N) for k = 5 and O(20.5408N) for k = 6. As a consequence of this, the same algorithm is shown to run in time O(20.0926L) for a formula of length (i.e.total number of literals) L. The previously best bound in terms of L is O(20.1030L). Our bound is also the best upper bound for an exact algorithm for a 3sat formula with up to six occurrences per variable, and a 4sat formula with up to eight occurrences per variable.