Schoening in 1999 presented a simple randomized algorithm for k-SAT with running time an * poly(n) for a = 2(k-1)/k. We give a deterministic version of this algorithm running in time an+o(n).

The Lovász Local Lemma discovered by Erdős and Lovász in 1975 is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In 1991, József Beck was the first to demonstrate that a constructive variant can be given under certain more restrictive conditions, starting a whole line of research aimed at improving his algorithm's performance and relaxing its restrictions. In the present article, we improve upon recent findings so as to provide a method for making almost all known applications of the general Local Lemma algorithmic.

The Lovasz Local Lemma [2] is a powerful tool to prove the existence of combinatorial objects meeting a prescribed collection of criteria. The technique can directly be applied to the satisfiability problem, yielding that a k-CNF formula in which each clause has common variables with at most 2(k-2) other clauses is always satisfiable. All hitherto known proofs of the Local Lemma are non-constructive and do thus not provide a recipe as to how a satisfying assignment to such a formula can be efficiently found. In his breakthrough paper [3], Beck demonstrated that if the neighbourhood of each clause be restricted to O(2(k/48)), a polynomial time algorithm for the search problem exists. Alon simplified and randomized his procedure and improved the bound to O(2(k/8)) [4]. Srinivasan presented in [9] a variant that achieves a bound of essentially O(2(k/4)). In [11], we improved this to O(2(k/2)). In the present paper, we give a randomized algorithm that finds a satisfying assignment to every k-CNF formula in which each clause has a neighbourhood of at most the asymptotic optimum of 2(k-5)-1 other clauses and that runs in expected time polynomial in the size of the formula, irrespective of k. If k is considered a constant, we can also give a deterministic variant. In contrast to all previous approaches, our analysis does not anymore invoke the standard non-constructive versions of the Local Lemma and can therefore be considered an alternative, constructive proof of it.