An Algorithm for the SAT Problem for Formulae of Linear Length

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(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(2^{0.1736 N}) for k = 3 and O(2^0.3472N) 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(2^0.4629N) for k = 5 and O(2^0.5408N) for k = 6. As a consequence of this, the same algorithm is shown to run in time O(2^0.0926L) for a formula of length (i.e.total number of literals) L. The previously best bound in terms of L is O(2^0.1030L). 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.