[Paper Review] Distributed Maximal Matching: Greedy is Optimal

Juho Hirvonen, Jukka Suomela|arXiv (Cornell University)|Oct 3, 2011

Complexity and Algorithms in Graphs17 references22 citations

TL;DR

This paper proves that the greedy algorithm for finding a maximal matching in anonymous, edge-coloured graphs is optimal, requiring exactly $k-1$ rounds when edges are coloured with $k$ colours. It establishes a tight $\Theta(\Delta + \log^* k)$ round complexity for the maximal matching problem, providing the first linear-in-$\Delta$ lower bound for a classical distributed 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 $Δ$, it is known that a maximal matching can be found in $O(Δ+ \log^* k)$ rounds, and prior work implies a lower bound of $Ω(\polylog(Δ) + \log^* k)$ rounds. Our work closes the gap between upper and lower bounds: the complexity is $Θ(Δ+ \log^* k)$ rounds. To our knowledge, this is the first linear-in-$Δ$ lower bound for the distributed complexity of a classical graph problem.

Motivation & Objective

• To close the gap between upper and lower bounds for the distributed maximal matching problem in terms of maximum degree $\Delta$ and number of colours $k$.
• To establish a tight complexity bound of $\Theta(\Delta + \log^* k)$ rounds for deterministic distributed maximal matching in anonymous, $k$-edge-coloured graphs.
• To prove that the greedy algorithm, which runs in $k-1$ rounds, is optimal in the general case.
• To provide the first linear-in-$\Delta$ lower bound for a classical distributed graph problem, resolving a long-standing open question.

Proposed method

• Uses inductive construction of $h$-templates and $h$-critical pairs to model indistinguishability in distributed networks.
• Defines pruning operations on templates to simulate local network views and propagate information over distance.
• Applies a parity argument on matched and unmatched nodes in constructed subgraphs to identify nodes with inconsistent outputs.
• Constructs a pair of $d$-templates $(S_d, \sigma_d)$ and $(T_d, \tau_d)$ that are $d$-compatible but differ in output at a root node, proving a lower bound.
• Leverages the structure of edge-coloured graphs and the greedy matching process to simulate worst-case scenarios requiring $k-1$ rounds.
• Combines known upper bounds ($O(\Delta + \log^* k)$) with new lower bounds ($\Omega(\Delta)$ and $\Omega(\log^* k)$) to achieve tight complexity.

Experimental results

Research questions

• RQ1 Is the greedy algorithm for maximal matching in $k$-edge-coloured graphs optimal in terms of round complexity?
• RQ2 What is the exact distributed complexity of maximal matching as a function of $\Delta$ and $k$?
• RQ3 Can a linear-in-$\Delta$ lower bound be established for a classical distributed graph problem?
• RQ4 Does the $\Omega(\log^* k)$ lower bound combine with a $\Omega(\Delta)$ lower bound to yield a tight $\Theta(\Delta + \log^* k)$ complexity?

Key findings

• The greedy algorithm for maximal matching in $k$-edge-coloured anonymous graphs requires exactly $k-1$ rounds, and this is optimal.
• Any deterministic distributed algorithm for maximal matching in anonymous, $k$-edge-coloured graphs requires $\Omega(\Delta)$ rounds, establishing a linear-in-$\Delta$ lower bound.
• The distributed complexity of maximal matching is tightly $\Theta(\Delta + \log^* k)$ rounds, closing the gap between known upper and lower bounds.
• This is the first known linear-in-$\Delta$ lower bound for a classical distributed graph problem, resolving a major open question in the field.
• The lower bound holds even in anonymous networks without unique identifiers, and applies to deterministic algorithms.
• The proof technique using $h$-templates and $h$-critical pairs provides a general framework for deriving lower bounds in distributed computing.