Deterministic Edge Coloring with few Colors in CONGEST

2026-03-03

AI Summary

This work presents the first deterministic edge-coloring algorithm in the CONGEST model that achieves fewer than $2\Delta - 1$ colors in a polynomial number of rounds. By carefully analyzing and extending online edge-coloring techniques, and integrating local foresight, derandomization methods, and efficient distributed communication strategies, the algorithm uses only $(1+\varepsilon)\Delta + O(\sqrt{\log n})$ colors. It runs in $\widetilde{O}(\log^{2.5} n + \log^2 \Delta \cdot \log n)$ rounds, significantly improving upon the previous best bound of $O(\log^8 n)$. This result not only advances the state of the art in distributed edge coloring but also establishes a novel connection between online algorithms and distributed graph algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Machine Learning: Learning on the Edge & Model Compression

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

As the main contribution of this work we present deterministic edge coloring algorithms in the CONGEST model. In particular, we present an algorithm that edge colors any $n$-node graph with maximum degree $Δ$ with with $(1+\varepsilon)Δ+O(\sqrt{\log n})$ colors in $\tilde{O}(\log^{2.5} n+\log^2 Δ\log n)$ rounds. This brings the upper bound polynomially close to the lower bound of $Ω(\log n/\log\log n)$ rounds that also holds in the more powerful LOCAL model [Chang, He, Li, Pettie, Uitto; SODA'18]. As long as $Δ\geq c\sqrt{\log n}$ our algorithm uses fewer than $2Δ-1$ colors and to the best of our knowledge is the first polylogarithmic-round CONGEST algorithm achieving this for any range of $Δ$. As a corollary we also improve the complexity of edge coloring with $2Δ-1$ colors for all ranges of $Δ$ to $\tilde{O}(\log^{2.5} n+\log^2 Δ\log n)$. This improves upon the previous $O(\log^8 n)$-round algorithm from [Fischer, Ghaffari, Kuhn; FOCS'17]. Our approach builds on a refined analysis and extension of the online edge-coloring algorithm of Blikstad, Svensson, Vintan, and Wajc [FOCS'25], and more broadly on new connections between online and distributed graph algorithms. We show that their algorithm exhibits very low locality and, if it can additionally have limited local access to future edges (as distributed algorithms can), it can be derandomized for smaller degrees. Under this additional power, we are able to bypass classical online lower bounds and translate the results to efficient distributed algorithms. This leads to our CONGEST algorithm for $(1+\varepsilon)Δ+O(\sqrt{\log n})$-edge coloring. Since the modified online algorithm can be implemented more efficiently in the LOCAL model, we also obtain (marginally) improved complexity bounds in that model.

Problem

• edge coloring
• deterministic algorithm
• CONGEST model
• distributed graph algorithms
• color complexity

Innovation

• deterministic edge coloring
• CONGEST model
• online-to-distributed reduction
• low-locality algorithm
• derandomization