NDLI: Parallel algorithms and concentration bounds for the lovász local lemma via witness-DAGs

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Parallel algorithms and concentration bounds for the lovász local lemma via witness-DAGs

Content Provider	ACM Digital Library
Author	Harris, David G. Haeupler, Bernhard
Abstract	The Lovász Local Lemma (LLL) is a cornerstone principle in the probabilistic method of combinatorics, and a seminal algorithm of Moser & Tardos (2010) provides an efficient randomized algorithm to implement it. This algorithm can be parallelized to give an algorithm that uses polynomially many processors and runs in $O(log^{3}$ 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 $O(log^{2}$ 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 $O(log^{2}$ 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 $O(log^{2}$ n) as well. We also provide improved bounds on the run-times of the sequential and parallel resampling-based algorithms originally developed by Moser & Tardos. Our 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.
Starting Page	1170
Ending Page	1187
Page Count	18
File Format	PDF
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2017-01-16
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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