Combinatorica
A parallel algorithmic version of the Local Lemma
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
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
On Dependency Graphs and the Lattice Gas
Combinatorics, Probability and Computing
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)
New Constructive Aspects of the Lovasz Local Lemma
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
An algorithmic approach to the lovász local lemma. I
Random Structures & Algorithms
Journal of the ACM (JACM)
Constraint satisfaction, packet routing, and the lovasz local lemma
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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Beck's early work [3] gave an efficient version of the Lovász Local Lemma(LLL) with significant compromise in the parameters. Following several improvements [1,7,4,13], Moser [8], and Moser and Tardos [9] obtained asymptotically optimal results in terms of the maximal degree. For a fixed dependency graph G the exact criterion under which LLL applies is given by Shearer in [12]. For a dependency structure G, let LO(G) be the set of those probability assignments to the nodes of G for which the Lovász Local Lemma holds. We show that: Both the sequential and parallel ersions of the Moser-Tardos algorithm are efficient up to the Shearer's bound, by giving a tighter analysis. We also prove that, whenever p ∈ LO(G)/(1+ε), the expected running times of the sequential and parallel versions are at most n/ε and O(1/ε log n/ε), the later when ε general LLL). Our alternative proof for the Shearer's bound not only highlights the connection between the variable and general versions of LLL, but also illustrates that variants of the Moser-Tardos algorithm can be useful in existence proofs. We obtain new formulas for phase transitions in the hardcore lattice gas model, non-trivially equivalent to the ones studied by Scott and Sokal [10]. We prove that if p ∈ LO(G)/(1+ε), the running time of the Moser-Tardos algorithm is polynomial not only in the number of events, but also in the number of variables. This extends one of the results from the more recent work of Haeupler, Saha, and Srinivasan [6]. Our new formulas immediately give a majorizing lemma that connects LLL bounds on different graphs. We show that the LLL bound for the (special case of the) variable version is sometimes larger than for the general version. This is the first known separation between the variable and the general versions of LLL.