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
Journal of Algorithms
STOC '00 Proceedings of the thirty-second 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
Efficient proper 2-coloring of almost disjoint hypergraphs
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
New Algorithmic Aspects of the Local Lemma with Applications to Routing and Partitioning
SIAM Journal on Computing
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In his seminal result, Beck gave the first algorithmic version of the Lovász Local Lemma by giving polynomial time algorithms for 2-coloring and partitioning uniform hypergraphs. His work was later generalized by Alon, and Molloy and Reed. Recently, Czumaj and Scheideler gave an efficient algorithm for 2-coloring non-uniform hypergraphs. But the partitioning algorithm obtained based on their second paper only applies to a more limited range of hypergraphs, so much so that their work doesn't imply the result of Beck for the uniform case. Here we give an algorithmic version of the general form of the Local Lemma which captures (almost) all applications of the results of Beck and Czumaj and Scheideler, with an overall simpler proof. In particular, if H is a non-uniform hypergraph in which every edge ei intersects at most |ei|2ak other edges of size at most k, for some small constant α, then we can find a partitioning of H in expected linear time. This result implies the result of Beck for uniform hypergraphs along with a speedup in his running time.