Decentralized Distributed Graph Coloring: Cluster Graphs

2024-05-13

Proceedings of the ACM Symposium on Principles of Distributed Computing

AI Summary

This paper addresses the (Δ+1)-coloring problem for cluster graphs in distributed systems, presenting the first CONGEST-model algorithm achieving sublogarithmic round complexity—specifically, O(log* n)—thereby breaking the prior poly(log n) barrier. Methodologically, the algorithm integrates localized information exchange, cluster-structure decomposition, iterative graph contraction, and coordinated coloring, operating efficiently on cluster graphs with minimum degree Ω(polylog n). Key contributions include: (i) the first sublogarithmic-round distributed (Δ+1)-coloring algorithm for cluster graphs, establishing the first sublogarithmic upper bound for this setting; (ii) an extension of classical graph coloring theory to the cluster graph model; and (iii) a substantial improvement over the state-of-the-art SODA’24 result, offering a novel paradigm for decentralized graph processing. The work advances foundational understanding of distributed symmetry breaking in structured graph families and opens avenues for efficient large-scale graph algorithms in constrained communication models.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Graph coloring is fundamental to distributed computing. We give the first sub-logarithmic distributed algorithm for coloring cluster graphs. These graphs are obtained from the underlying communication network by contracting nodes and edges, and they appear frequently as components in the study of distributed algorithms. In particular, we give a O (log* n)-round algorithm to (Δ + 1)-color cluster graphs of at least polylogarithmic degree. The previous best bound known was poly(log n) [Flin et al., SODA'24]. This properly generalizes results in the CONGEST model and shows that distributed graph problems can be solved quickly even when the node itself is decentralized.

Problem

• Develop sub-logarithmic algorithm for cluster graph coloring
• Improve coloring complexity from poly(log n) to O(log* n)
• Generalize distributed coloring results for decentralized nodes

Innovation

• Sub-logarithmic algorithm for cluster graphs
• O(log^* n)-round (Δ+1)-coloring method
• Decentralized node problem-solving approach