# Sublogarithmic Distributed Algorithms for Lov\'asz Local lemma, and the Complexity Hierarchy

## TL;DR

This paper advances the understanding of distributed algorithms by significantly improving the upper bound on the complexity of solving the Lovász Local Lemma in bounded degree graphs, and applies these results to enhance algorithms for various graph coloring problems.

## Contribution

It proves a sublogarithmic upper bound of 2^{O(√log log n)} for the LLL complexity, improving previous bounds and enabling faster algorithms for multiple graph coloring problems.

## Key findings

- Established T_{LLL}(n)= 2^{O(√log log n)}
- Improved coloring algorithms from O(log n) to 2^{O(√log log n)}
- Connected LLL complexity to a broad class of graph problems

## Abstract

Locally Checkable Labeling (LCL) problems include essentially all the classic problems of $\mathsf{LOCAL}$ distributed algorithms. In a recent enlightening revelation, Chang and Pettie [arXiv 1704.06297] showed that any LCL (on bounded degree graphs) that has an $o(\log n)$-round randomized algorithm can be solved in $T_{LLL}(n)$ rounds, which is the randomized complexity of solving (a relaxed variant of) the Lov\'asz Local Lemma (LLL) on bounded degree $n$-node graphs. Currently, the best known upper bound on $T_{LLL}(n)$ is $O(\log n)$, by Chung, Pettie, and Su [PODC'14], while the best known lower bound is $\Omega(\log\log n)$, by Brandt et al. [STOC'16]. Chang and Pettie conjectured that there should be an $O(\log\log n)$-round algorithm. Making the first step of progress towards this conjecture, and providing a significant improvement on the algorithm of Chung et al. [PODC'14], we prove that $T_{LLL}(n)= 2^{O(\sqrt{\log\log n})}$. Thus, any $o(\log n)$-round randomized distributed algorithm for any LCL problem on bounded degree graphs can be automatically sped up to run in $2^{O(\sqrt{\log\log n})}$ rounds. Using this improvement and a number of other ideas, we also improve the complexity of a number of graph coloring problems (in arbitrary degree graphs) from the $O(\log n)$-round results of Chung, Pettie and Su [PODC'14] to $2^{O(\sqrt{\log\log n})}$. These problems include defective coloring, frugal coloring, and list vertex-coloring.