Distributed Lov\'{a}sz Local Lemma under Bandwidth Limitations

TL;DR

This paper develops bandwidth- and time-efficient distributed algorithms for Lovász Local Lemma problems in the CONGEST model, enabling faster graph coloring and subgraph sampling under bandwidth constraints.

Contribution

It introduces the first bandwidth-efficient algorithms for LLL in the CONGEST model, including applications to graph coloring and subgraph sampling.

Findings

Algorithms are exponentially faster than previous LOCAL model methods.

Effective solutions for subgraph sampling problems formulated as LLLs.

New coloring algorithms for sparse and triangle-free graphs with fewer colors.

Abstract

The constructive Lov\'{a}sz Local Lemma has become a central tool for designing efficient distributed algorithms. While it has been extensively studied in the classic LOCAL model that uses unlimited bandwidth, much less is known in the bandwidth-restricted CONGEST model. In this paper, we present bandwidth- and time-efficient algorithms for various subclasses of LLL problems, including a large class of subgraph sampling problems that are naturally formulated as LLLs. Lastly, we use our LLLs to design efficient CONGEST algorithms for coloring sparse and triangle-free graphs with few colors. These coloring algorithms are exponentially faster than previous LOCAL model algorithms.