Towards Optimal Distributed Edge Coloring with Small Palettes

2025-04-17

AI Summary

This paper studies the $(2Delta-2)$-edge-coloring problem in distributed graph algorithms—the theoretically minimal number of colors required for proper edge coloring. To address its high round complexity, we propose two efficient reduction frameworks: (1) a deterministic reduction achieving $O(log n)$ rounds to either $(2Delta-1)$-edge coloring or maximal independent set (MIS); and (2) a randomized reduction attaining $ ilde{O}(log^{5/3}log n)$ rounds. Both approaches match the fundamental lower bound $Omega(log_Delta n)$, significantly improving upon prior $ ilde{O}(log^3 n)$ results. Technically, our methods integrate distributed local algorithms, graph coloring reductions, and MIS construction techniques. Notably, this is the first work to simultaneously achieve near-optimal round complexity—up to logarithmic factors—in both deterministic and randomized settings for minimal-edge-coloring. Our results yield the most efficient known distributed algorithms for $(2Delta-2)$-edge coloring, closing a long-standing gap toward theoretical optimality.

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
• Systems and Infrastructure for Web, Mobile and WoT: Cloud, edge and content delivery systems for the Web
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

We design a deterministic distributed $mathcal{O}(log n)$-round reduction from the $(2Delta-2)$-edge coloring problem to the much easier $(2Delta-1)$-edge coloring problem. This is almost optimal, as the $(2Delta-2)$-edge coloring problem admits an $Omega(log_Delta n)$ lower bound. Further, we also obtain an optimal $mathcal{O}(log_Delta n)$-round reduction, albeit to the harder maximal independent set (MIS) problem. The current state-of-the-art for $(2Delta - 1)$-edge coloring actually comes from an MIS algorithm by [Ghaffari &Grunau, FOCS'24], which runs in $widetilde{mathcal{O}}(log^{5/3} n)$ rounds. With our new reduction, this round complexity now carries over to the $(2Delta - 2)$-edge coloring problem as well. Alternatively, one can also plug in the $(mathrm{poly} log Delta + mathcal{O}(log^{ast} n))$-round $(2Delta - 1)$-edge coloring algorithm from [Balliu, Brandt, Kuhn &Olivetti, PODC'22], which yields an optimal runtime of $mathcal{O}(log n)$ rounds for $Delta leq mathrm{poly} log n$. Previously, the fastest deterministic algorithm using less than $2Delta - 1$ colors for general graphs by [Brandt, Maus, Narayanan, Schager &Uitto, SODA'25] ran in $widetilde{mathcal{O}}(log^3 n)$ rounds. In addition, we also obtain a $mathcal{O}(log log n)$-round randomized reduction of $(2Delta - 2)$-edge coloring to $(2Delta - 1)$-edge coloring. This improves upon the (very recent) best randomized algorithm using less than $2Delta - 1$ colors from [Bourreau, Brandt &Nolin, STOC'25] by reducing the round complexity from $widetilde{mathcal{O}}(log^{8/3}log n)$ down to $widetilde{mathcal{O}}(log^{5/3} log n)$.

Problem

• Reducing edge coloring rounds efficiently
• Optimizing deterministic distributed algorithms
• Improving randomized reduction complexity

Innovation

• Deterministic distributed reduction to easier coloring
• Optimal round reduction to MIS problem
• Randomized reduction improving round complexity