# Lower bounds for maximal matchings and maximal independent sets

## TL;DR

This paper establishes tight lower bounds for the round complexity of distributed algorithms solving maximal matchings and independent sets, matching the known upper bounds and resolving a long-standing open problem regarding their dependence on maximum degree.

## Contribution

It proves that existing upper bounds are optimal by providing matching lower bounds, especially clarifying the dependency on maximum degree in distributed algorithms.

## Key findings

- Lower bounds match upper bounds for maximal matchings and independent sets.
- Dependency on maximum degree $\Delta$ is proven to be tight.
- Improves prior lower bounds as a function of $n$.

## Abstract

There are distributed graph algorithms for finding maximal matchings and maximal independent sets in $O(\Delta + \log^* n)$ communication rounds; here $n$ is the number of nodes and $\Delta$ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on $n$ is optimal: these problems cannot be solved in $o(\log^* n)$ rounds even if $\Delta = 2$. However, the dependency on $\Delta$ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that any algorithm that finds a maximal matching or maximal independent set with probability at least $1-1/n$ requires $\Omega(\min\{\Delta,\log \log n / \log \log \log n\})$ rounds in the LOCAL model of distributed computing. As a corollary, it follows that any deterministic algorithm that finds a maximal matching or maximal independent set requires $\Omega(\min\{\Delta, \log n / \log \log n\})$ rounds; this is an improvement over prior lower bounds also as a function of $n$.