Online Edge Coloring: Sharp Thresholds

2025-07-29

AI Summary

This paper studies the online edge coloring problem: assigning colors to edges as they arrive sequentially, using nearly optimal $(1+o(1))Delta$ colors for graphs with maximum degree $Delta$. To overcome limitations of classical greedy approaches, we propose the first deterministic online algorithm achieving $(1+o(1))Delta$-coloring when $Delta = omega(log n)$, and advance the regime for randomized algorithms to the weaker condition $Delta = omega(sqrt{log n})$, establishing the broadest known threshold. Our approach integrates probabilistic analysis, structured color assignment schemes, and novel online design principles, breaking long-standing performance barriers. The resulting guarantees match the classical lower bound of $Delta$, thereby achieving the theoretically optimal asymptotic approximation ratio.

Technology Category

• Machine Learning: Online Learning & Bandits
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Vizing's theorem guarantees that every graph with maximum degree $Δ$ admits an edge coloring using $Δ+ 1$ colors. In online settings - where edges arrive one at a time and must be colored immediately - a simple greedy algorithm uses at most $2Δ- 1$ colors. Over thirty years ago, Bar-Noy, Motwani, and Naor [IPL'92] proved that this guarantee is optimal among deterministic algorithms when $Δ= O(log n)$, and among randomized algorithms when $Δ= O(sqrt{log n})$. While deterministic improvements seemed out of reach, they conjectured that for graphs with $Δ= ω(log n)$, randomized algorithms can achieve $(1 + o(1))Δ$ edge coloring. This conjecture was recently resolved in the affirmative: a $(1 + o(1))Δ$-coloring is achievable online using randomization for all graphs with $Δ= ω(log n)$ [BSVW STOC'24]. Our results go further, uncovering two findings not predicted by the original conjecture. First, we give a deterministic online algorithm achieving $(1 + o(1))Δ$-colorings for all $Δ= ω(log n)$. Second, we give a randomized algorithm achieving $(1 + o(1))Δ$-colorings already when $Δ= ω(sqrt{log n})$. Our results establish sharp thresholds for when greedy can be surpassed, and near-optimal guarantees can be achieved - matching the impossibility results of [BNMN IPL'92], both deterministically and randomly.

Problem

• Deterministic online edge coloring for large degree graphs
• Randomized online edge coloring for moderate degree graphs
• Sharp thresholds surpassing greedy algorithm limitations

Innovation

• Deterministic online algorithm for near-optimal edge coloring
• Randomized algorithm improves threshold to ω(√log n)
• Achieves (1 + o(1))Δ-coloring for large Δ