# How much does randomness help with locally checkable problems?

## TL;DR

This paper investigates the impact of randomness on locally checkable labeling problems (LCLs), revealing that some LCLs benefit from subexponential speedups due to randomness, which was previously unknown.

## Contribution

The paper demonstrates the existence of LCLs that benefit from randomness with subexponential improvements, specifically showing an LCL with a gap between deterministic and randomized complexities.

## Key findings

- Existence of LCLs with subexponential randomized speedup
- An LCL with deterministic complexity Θ(log^2 n)
- Randomized complexity Θ(log n log log n)

## Abstract

Locally checkable labeling problems (LCLs) are distributed graph problems in which a solution is globally feasible if it is locally feasible in all constant-radius neighborhoods. Vertex colorings, maximal independent sets, and maximal matchings are examples of LCLs. On the one hand, it is known that some LCLs benefit exponentially from randomness---for example, any deterministic distributed algorithm that finds a sinkless orientation requires $\Theta(\log n)$ rounds in the LOCAL model, while the randomized complexity of the problem is $\Theta(\log \log n)$ rounds. On the other hand, there are also many LCLs in which randomness is useless. Previously, it was not known if there are any LCLs that benefit from randomness, but only subexponentially. We show that such problems exist: for example, there is an LCL with deterministic complexity $\Theta(\log^2 n)$ rounds and randomized complexity $\Theta(\log n \log \log n)$ rounds.