A Constructive Proof Of The Lovasz Local Lemma

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 variable with at most 2(k-2) other clauses is always satisfiable. All hiterto 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 call 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 aysmptotic optimum of 2(k-5) - 1 other clauses and thus runs is expected time polynomial in the size of the formula, irrespective of k. If k is considered a constant, we call also give a deterministic variant. In contrast to all previous approaches, our analysis does tool, anymore invoke the, standard non-constructive versions of the Local Lemma and can therefore be considered all alternative,. constructive proof of it.