Distributed Maximal Matching: Greedy is Optimal

TL;DR

This paper proves that the greedy algorithm for finding a maximal matching in anonymous, edge-coloured graphs is optimal, establishing tight bounds on the number of rounds needed in distributed settings.

Contribution

It shows the greedy algorithm's optimality and closes the gap between known upper and lower bounds for distributed maximal matching complexity.

Findings

Greedy algorithm is optimal for anonymous, edge-coloured graphs.

Distributed maximal matching requires ( ext{ } ext{ }+ ext{ } ext{log}^* k) rounds.

First linear-in- ext{ } ext{ } lower bound for a classical graph problem.

Abstract

We study distributed algorithms that find a maximal matching in an anonymous, edge-coloured graph. If the edges are properly coloured with $k$ colours, there is a trivial greedy algorithm that finds a maximal matching in $k-1$ synchronous communication rounds. The present work shows that the greedy algorithm is optimal in the general case: any algorithm that finds a maximal matching in anonymous, $k$-edge-coloured graphs requires $k-1$ rounds. If we focus on graphs of maximum degree $\Delta$, it is known that a maximal matching can be found in $O(\Delta + \log^* k)$ rounds, and prior work implies a lower bound of $\Omega(\polylog(\Delta) + \log^* k)$ rounds. Our work closes the gap between upper and lower bounds: the complexity is $\Theta(\Delta + \log^* k)$ rounds. To our knowledge, this is the first linear-in-$\Delta$ lower bound for the distributed complexity of a classical graph problem.