An Algorithm for the SAT Problem for Formulae of Linear Length

  • Magnus Wahlström
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3669)


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(α N ) with α< 2 for every fixed k). 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 k>4, 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 ( 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.


Local Search Piecewise Linear Function Conjunctive Normal Form Deterministic Algorithm Probabilistic Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Magnus Wahlström
    • 1
  1. 1.Department of Computer and Information ScienceLinköping UniversityLinköpingSweden

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