One more occurrence of variables makes satisfiability jump from trivial to NP-complete
SIAM Journal on Computing
Further algorithmic aspects of the local lemma
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Coloring non-uniform hypergraphs: a new algorithmic approach to the general Lovász local lemma
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Improved algorithmic versions of the Lovász Local Lemma
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Conflict-free colourings of graphs and hypergraphs
Combinatorics, Probability and Computing
A constructive proof of the general lovász local lemma
Journal of the ACM (JACM)
Approximate Pure Nash Equilibria via Lovász Local Lemma
WINE '09 Proceedings of the 5th International Workshop on Internet and Network Economics
Asymptotically optimal frugal colouring
Journal of Combinatorial Theory Series B
Proceedings of the forty-second ACM symposium on Theory of computing
On brambles, grid-like minors, and parameterized intractability of monadic second-order logic
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Deterministic algorithms for the Lovász Local Lemma
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Proceedings of the forty-third annual ACM symposium on Theory of computing
A kolmogorov complexity proof of the lovász local lemma for satisfiability
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
Cover-decomposition and polychromatic numbers
ESA'11 Proceedings of the 19th European conference on Algorithms
New Constructive Aspects of the Lovász Local Lemma
Journal of the ACM (JACM)
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Comparing two stochastic local search algorithms for constraint satisfaction problems
CSR'10 Proceedings of the 5th international conference on Computer Science: theory and Applications
Probabilistic existence of rigid combinatorial structures
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Computer Science Review
Non-crossing matchings of points with geometric objects
Computational Geometry: Theory and Applications
Journal of the ACM (JACM)
A Kolmogorov complexity proof of the Lovász Local Lemma for satisfiability
Theoretical Computer Science
An asymptotically tight bound on the adaptable chromatic number
Journal of Graph Theory
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
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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.