Towards Optimal Distributed Edge Coloring with Fewer Colors

TL;DR

The paper tackles the hardness gap between distributed edge coloring with $2\Delta-1$ colors and the harder $2\Delta-2$-edge coloring. It introduces a deterministic LOCAL reduction that uses clustering, a two-edge exclusive assignment per cluster, and hypergraph sinkless orientation to reduce $(2\Delta-2)$-edge coloring to $(2\Delta-1)$-edge coloring in $O(\log n)$ rounds, achieving optimality given the $\Omega(\log n)$ lower bound for the harder problem. A randomized counterpart achieves $O(\log \log n)$ rounds, and a variant using MIS as a subroutine yields runtimes like $O(\log_\Delta n) + T_{MIS}(\Delta^2 n, \mathrm{poly}(\Delta))$, leading to general-graph bounds of $\tilde{O}(\log^{5/3} n)$. When plugged into the existing $O(\log^{12} \Delta + \log^{*} n)$-round $(2\Delta-1)$-edge coloring algorithm, the results attain an overall optimal $O(\log n)$-round complexity for moderately large $\Delta$ (e.g., $\Delta = 2^{O(\log^{1/12} n)}$). The techniques rely on a careful extendability analysis (colorful leaves), MIS-based clustering, and hypergraph- oriented reallocation of inter-cluster edges to enable parallel progress, marking a significant advance in the non-greedy regime of distributed edge coloring.

Abstract

There is a huge difference in techniques and runtimes of distributed algorithms for problems that can be solved by a sequential greedy algorithm and those that cannot. A prime example of this contrast appears in the edge coloring problem: while $(2Δ-1)$-edge coloring can be solved in $\mathcal{O}(\log^{\ast}(n))$ rounds on constant-degree graphs, the seemingly minor reduction to $(2Δ-2)$ colors leads to an $Ω(\log n)$ lower bound [Chang, He, Li, Pettie & Uitto, SODA'18]. Understanding this sharp divide between very local problems and inherently more global ones remains a central open question in distributed computing and it is a core focus of this paper. As our main contribution we design a deterministic distributed $\mathcal{O}(\log n)$-round reduction from the $(2Δ-2)$-edge coloring problem to the much easier $(2Δ-1)$-edge coloring problem. This reduction is optimal, as the $(2Δ-2)$-edge coloring problem admits an $Ω(\log n)$ lower bound, whereas the $2Δ-1$-edge coloring problem can be solved in $\mathcal{O}(\log^{\ast}n)$ rounds. By plugging in the $(2Δ-1)$-edge coloring algorithms from [Balliu, Brandt, Kuhn & Olivetti, PODC'22] running in $\mathcal{O}(\log^{12}Δ+ \log^{\ast} n)$ rounds, we obtain an optimal runtime of $\mathcal{O}(\log n)$ rounds as long as $Δ= 2^{\mathcal{O}(\log^{1/12} n)}$. Furthermore, on general graphs our reduction improves the runtime from $\widetilde{\mathcal{O}}(\log^3 n)$ to $\widetilde{\mathcal{O}}(\log^{5/3} n)$. In addition, we also obtain an optimal $\mathcal{O}(\log \log n)$-round randomized reduction of $(2Δ- 2)$-edge coloring to $(2Δ- 1)$-edge coloring. Lastly, we obtain an $\mathcal{O}(\log_Δn)$-round reduction from the $(2Δ-1)$-edge coloring, albeit to the somewhat harder maximal independent set (MIS) problem.

Figures (6)

• Figure 1: A matching of the intercluster edges that assigns two edges to each clusters. The matching edges are marked red. Arrows point towards the cluster the edge belongs to.
• Figure 2: Extendability of $(2\Delta - 2)$-colorings for $\Delta = 3$.
• Figure 3: In order to complete the coloring on the edges within the clusters, our algorithm ensures that each cluster can independently control the colors of two edges adjacent to the cluster. For illustration purposes this example has many cycles that are dealt with separately in our algorithm.
• Figure 4: Extending the coloring to a tree.
• Figure 5: A simplified example for the color switching procedure, where $\Delta = 3$. The edges in the BFS-tree $T_r$ are represented by solid lines, while edges outside of $T_r$ are dashed. The assigned matching edges $e_1, e_2$ are marked orange.

Theorems & Definitions (45)

• Theorem 1
• Theorem 2
• Corollary 2
• Theorem 3
• Corollary 3
• Lemma 4: Simplified version of \ref{['lem:colorful_leaves']}
• Definition 5
• Definition 6: $(\alpha,\beta)$-ruling set
• Definition 7: $(\alpha,\beta)$-clustering
• Remark 8