Linear-in-$\Delta$ Lower Bounds in the LOCAL Model

TL;DR

This paper establishes a fundamental lower bound in distributed computing, proving that finding a maximal fractional matching requires at least linear rounds in the maximum degree, matching known upper bounds.

Contribution

It provides the first linear-in-$$ lower bound for a natural graph problem in the standard distributed model, closing a gap in understanding of complexity bounds.

Findings

No distributed algorithm can find a maximal fractional matching in o() rounds.

The lower bound matches the existing O() upper bound.

This is the first linear-in- lower bound for a natural graph problem.

Abstract

By prior work, there is a distributed algorithm that finds a maximal fractional matching (maximal edge packing) in $O(\Delta)$ rounds, where $\Delta$ 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 $o(\Delta)$ rounds. Our work gives the first linear-in-$\Delta$ 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 $\Delta$.