Distributed Maximal Matching: Greedy is Optimal

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.

💡 Research Summary

The paper investigates deterministic distributed algorithms for finding a maximal matching in anonymous networks where edges are properly coloured with k distinct colours. The authors first revisit the obvious greedy algorithm: in round i all edges of colour i are examined in parallel, and any edge whose endpoints are still unmatched is added to the matching. Because the first colour can be processed without any communication, the algorithm finishes in exactly k − 1 synchronous rounds.

The main contribution is a matching lower bound that shows this greedy approach is optimal. To prove it, the authors construct a family of hard instances using the free Coxeter group Gₖ (the product of k copies of ℤ₂) and its Cayley graph Γₖ, which is a k‑regular, k‑edge‑coloured infinite tree. Subsets of vertices that are prefix‑closed in Γₖ are called colour systems; they capture the local view of any node after a certain number of communication rounds. The paper introduces the notions of templates (partial colour systems together with a “forbidden colour” function τ) and colour pickers that assign a set of free colours to each node. By repeatedly extending a template with a colour picker, the authors grow larger colour systems while controlling the set of colours that remain available at each node.

Using an inductive construction, they produce two d‑regular k‑colour systems U and V (where d = k − 1) that are indistinguishable up to distance d from the root but require different outputs at the root (matched vs. unmatched). Any deterministic algorithm that terminates in fewer than d rounds would have to produce the same output on both instances, contradicting the maximal‑matching specification. Consequently, any algorithm needs at least k − 1 = d rounds, proving the greedy algorithm’s optimality.

When the maximum degree Δ is bounded, the known upper bound of O(Δ + log* k) rounds (obtained by a simple adaptation of the greedy method together with colour‑reduction techniques) matches the new lower bound Ω(Δ). Hence the exact distributed complexity of maximal matching in anonymous, edge‑coloured graphs is Θ(Δ + log* k). This is the first linear‑in‑Δ lower bound for a classic graph problem in the distributed setting.

The work also discusses related results, noting that prior lower bounds were only polylogarithmic in Δ, and emphasizes that breaking the Ω(Δ) barrier would require the use of unique node identifiers. Future directions include extending the lower bound to models where identifiers are available, and exploring whether randomness can help reduce the dependence on Δ.