A simple parallel algorithm for the maximal independent set problem
SIAM Journal on Computing
Small-bias probability spaces: efficient constructions and applications
SIAM Journal on Computing
Further algorithmic aspects of the local lemma
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Efficient approximation of product distributions
Random Structures & Algorithms
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Improved algorithms via approximations of probability distributions
Journal of Computer and System Sciences
On allocations that maximize fairness
Proceedings of the nineteenth 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
A constructive proof of the Lovász local lemma
Proceedings of the forty-first annual ACM symposium on Theory of computing
A constructive proof of the general lovász local lemma
Journal of the ACM (JACM)
Derandomizing HSSW algorithm for 3-SAT
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
New Constructive Aspects of the Lovász Local Lemma
Journal of the ACM (JACM)
Strengthening hash families and compressive sensing
Journal of Discrete Algorithms
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The Lovász Local Lemma [5] (LLL) is a powerful result in probability theory that states that the probability that none of a set of bad events happens is nonzero if the probability of each event is small compared to the number of events that depend on it. It is often used in combination with the probabilistic method for non-constructive existence proofs. A prominent application is to k-CNF formulas, where LLL implies that, if every clause in the formula shares variables with at most d ≤ 2k / e other clauses then such a formula has a satisfying assignment. Recently, a randomized algorithm to efficiently construct a satisfying assignment was given by Moser [13]. Subsequently Moser and Tardos [14] gave a randomized algorithm to construct the structures guaranteed by the LLL in a very general algorithmic framework. We address the main problem left open by Moser and Tardos of derandomizing these algorithms efficiently. Specifically, for a k-CNF formula with m clauses and d ≤ 2k/(1+ε) / e for some ε ε (0, 1), we give an algorithm that finds a satisfying assignment in time Õ(m2(1+1ε)). This improves upon the deterministic algorithms of Moser and of Moser-Tardos with running times mΩ(k2) and mΩ(k·1/ε) which are superpolynomial for k = ϖ(1) and upon other previous algorithms which work only for d ≤ 2k/16 / e. Our algorithm works efficiently for the asymmetric version of LLL under the algorithmic framework of Moser and Tardos [14] and is also parallelizable, i.e., has polylogarithmic running time using polynomially many processors.