On the Complexity of Distributed Edge Coloring and Orientation Problems

2025-10-24

AI Summary

This work investigates whether randomization can break the deterministic lower bound Ω(log n) for locally checkable labeling (LCL) problems, focusing on edge coloring and edge orientation—two natural LCL tasks on bounded-degree graphs. Within the LOCAL model, we combine probabilistic methods with graph-theoretic analysis to establish the first exact randomized complexity characterizations for these problems. For bichromatic edge coloring on graphs with maximum degree Δ = O(1), we prove a randomized complexity of poly(log log n), achieving an exponential speedup over deterministic algorithms. For edge orientation, we uncover a clean hierarchical structure in randomized complexity, revealing sharp thresholds governed by degree constraints. These results delineate the precise boundary at which randomness yields asymptotic advantages in LCLs, and provide the first systematic classification of the randomized–deterministic complexity relationship for distributed graph problems.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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: Research challenges in human and human-AI computation
• Security and Privacy: Applications of cryptography

Abstract

Understanding the role of randomness when solving locally checkable labeling (LCL) problems in the LOCAL model has been one of the top priorities in the research on distributed graph algorithms in recent years. For LCL problems in bounded-degree graphs, it is known that randomness cannot help more than polynomially, except in one case: if the deterministic complexity of an LCL problem is in $Ω(log n)$ and its randomized complexity is in $o(log n)$, then the randomized complexity is guaranteed to be $poly(log log n)$. However, the fundamental question of emph{which} problems with a deterministic complexity of $Ω(log n)$ can be solved exponentially faster using randomization still remains wide open. We make a step towards answering this question by studying a simple, but natural class of LCL problems: so-called degree splitting problems. These problems come in two varieties: coloring problems where the edges of a graph have to be colored with $2$ colors and orientation problems where each edge needs to be oriented. For $Δ$-regular graphs (where $Δ=O(1)$), we obtain the following results. - We gave an exact characterization of the randomized complexity of all problems where the edges need to be colored with two colors, say red and blue, and which have a deterministic complexity of $O(log n)$. - For edge orientation problems, we give a partial characterization of the problems that have a randomized complexity of $Ω(log n)$ and the problems that have a randomized complexity of $polyloglog n$. While our results are cleanest to state for the $Δ$-regular case, all our algorithms naturally generalize to nodes of any degree $d<Δ$ in general graphs of maximum degree $Δ$.

Problem

• Characterizing randomized complexity of distributed edge coloring problems
• Determining which orientation problems require logarithmic randomized complexity
• Identifying problems solvable exponentially faster with randomization

Innovation

• Characterizing randomized complexity of edge coloring problems
• Partially characterizing edge orientation problems complexity
• Generalizing algorithms to maximum degree graphs