By prior work, there is a distributed algorithm that finds a maximal fractional matching (maximal edge packing) in O(Δ) rounds, where Δ is the maximum degree of the graph. We show that this is optimal: there is no distributed algorithm that finds a maximal fractional matching in o(Δ) rounds. Our work gives the first linear-in-Δ lower bound for a natural graph problem in the standard model of distributed computing---prior lower bounds for a wide range of graph problems have been at best logarithmic in Δ.
@article{arxiv.1304.1007,
title = {Linear-in-$\Delta$ Lower Bounds in the LOCAL Model},
author = {Mika Göös and Juho Hirvonen and Jukka Suomela},
journal= {arXiv preprint arXiv:1304.1007},
year = {2019}
}