# Lower bounds for maximal matchings and maximal independent sets

FOCS 2019 · 60th Annual IEEE Symposium on Foundations of Computer Science, Baltimore, MD, USA, November 2019 ·

FOCS 2019 Best Paper Award

# 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 maximal matchings and maximal independent sets cannot be found in $o(\Delta + \log \log n / \log \log \log n)$ rounds with any randomized algorithm in the LOCAL model of distributed computing.

As a corollary, it follows that there is no deterministic algorithm for maximal matchings or maximal independent sets that runs in $o(\Delta + \log n / \log \log n)$ rounds; this is an improvement over prior lower bounds also as a function of $n$.

# Publication

2019 IEEE 60th Annual Symposium on Foundations of Computer Science, FOCS 2019, pages 481–497, IEEE, Piscataway, 2019

ISBN 978-1-7281-4952-3

This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author’s copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.