Improved bounds and parallel algorithms for the Lovasz Local Lemma

The Lovász Local Lemma (LLL) is a cornerstone principle in the probabilistic method of combinatorics, and a seminal algorithm of Moser & Tardos (2010) provided an efficient randomized algorithm to implement it. This algorithm could be parallelized to give an algorithm that uses polynomially many processors and O(log n) time, stemming from O(log n) adaptive computations of a maximal independent set (MIS). Chung et. al. (2014) developed faster local and parallel algorithms, potentially running in time O(log n), but these algorithms work under significantly more stringent conditions than the LLL. We give a new parallel algorithm, that works under essentially the same conditions as the original algorithm of Moser & Tardos, but uses only a single MIS computation, thus running in O(log n) time. This conceptually new algorithm also gives a clean combinatorial description of a satisfying assignment which might be of independent interest. Our techniques extend to the deterministic LLL algorithm given by Chandrasekaran et al (2013) leading to an NC-algorithm running in time O(log n) as well. We also provide improved bounds on the run-times of the sequential and parallel resamplingbased algorithms originally developed by Moser & Tardos. These bounds extend to any problem instance in which the tighter Shearer LLL criterion is satisfied. We also improve on the analysis of Kolipaka & Szegedy (2011) to give tighter concentration results. Interestingly, we are able to give bounds which are independent of the (somewhat mysterious) weighting function used in formulations of the asymmetric LLL. School of Computer Science, Carnegie Mellon University, Email: [email protected]. Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742. Research supported in part by NSF Awards CNS-1010789 and CCF-1422569. Email: [email protected]. 0 IMPROVED BOUNDS AND PARALLEL ALGORITHMS FOR THE LOVÁSZ LOCAL LEMMA 1