Efficient CONGEST Algorithms for the Lovasz Local Lemma

TL;DR

This paper introduces a highly efficient randomized CONGEST algorithm for solving a class of Lovasz Local Lemma instances on constant degree graphs, significantly improving the understanding of distributed complexity for these problems.

Contribution

It provides the first polylog log n time CONGEST algorithm for certain LLL instances and extends network decomposition techniques with minimal dependency on identifiers.

Findings

Achieves poly log log n time complexity for specific LLL problems

Shows no LCL problems have randomized complexity between log n and poly log log n

Extends network decomposition methods with negligible identifier dependency

Abstract

We present a poly $\log \log n$ time randomized CONGEST algorithm for a natural class of Lovasz Local Lemma (LLL) instances on constant degree graphs. This implies, among other things, that there are no LCL problems with randomized complexity between $\log n$ and poly $\log \log n$. Furthermore, we provide extensions to the network decomposition algorithms given in the recent breakthrough by Rozhon and Ghaffari [STOC2020] and the follow up by Ghaffari, Grunau, and Rozhon [SODA2021]. In particular, we show how to obtain a large distance separated weak network decomposition with a negligible dependency on the range of unique identifiers.