Distributed Computing

, Volume 30, Issue 4, pp 261–280

# Distributed algorithms for the Lovász local lemma and graph coloring

• Kai-Min Chung
• Seth Pettie
• Hsin-Hao Su
Article

## Abstract

The Lovász local lemma (LLL), introduced by Erdős and Lovász in 1975, is a powerful tool of the probabilistic method that allows one to prove that a set of n “bad” events do not happen with non-zero probability, provided that the events have limited dependence. However, the LLL itself does not suggest how to find a point avoiding all bad events. Since the works of Alon (Random Struct Algorithms 2(4):367–378, 1991) and Beck (Random Struct Algorithms 2(4):343–365, 1991) there has been a sustained effort to find a constructive proof (i.e. an algorithm) for the LLL or weaker versions of it. In a major breakthrough Moser and Tardos (J ACM 57(2):11, 2010) showed that a point avoiding all bad events can be found efficiently. They also proposed a distributed/parallel version of their algorithm that requires $$O(\log ^2 n)$$ rounds of communication in a distributed network. In this paper we provide two new distributed algorithms for the LLL that improve on both the efficiency and simplicity of the Moser–Tardos algorithm. For clarity we express our results in terms of the symmetric LLL though both algorithms deal with the asymmetric version as well. Let p bound the probability of any bad event and d be the maximum degree in the dependency graph of the bad events. When $$epd^2 < 1$$ we give a truly simple LLL algorithm running in $$O(\log _{1/epd^2} n)$$ rounds. Under the weaker condition $$ep(d+1) < 1$$, we give a slightly slower algorithm running in $$O(\log ^2 d\cdot \log _{1/ep(d+1)} n)$$ rounds. Furthermore, we give an algorithm that runs in sublogarithmic rounds under the condition $$p\cdot f(d) < 1$$, where f(d) is an exponential function of d. Although the conditions of the LLL are locally verifiable, we prove that any distributed LLL algorithm requires $${\varOmega }(\log ^* n)$$ rounds. In many graph coloring problems the existence of a valid coloring is established by one or more applications of the LLL. Using our LLL algorithms, we give logarithmic-time distributed algorithms for frugal coloring, defective coloring, coloring girth-4 (triangle-free) and girth-5 graphs, edge coloring, and list coloring.

## Keywords

Lovász local lemma Distributed algorithms Randomized algorithms Coloring Locality

## Notes

### Acknowledgements

Thanks Mohsen Ghaffari for pointing out that by iteratively applying LLL, the range of f can be improved from $${\varOmega }(\log {\varDelta })$$ to any positive integer for f-defective, $$O({\varDelta }/f)$$-colorings.

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## Authors and Affiliations

• Kai-Min Chung
• 1
• Seth Pettie
• 2
• Hsin-Hao Su
• 3