Round and Communication Efficient Graph Coloring

2024-12-17

arXiv.org

AI Summary

This paper investigates the communication complexity of graph coloring in a two-party distributed model under edge partitioning. For an $n$-vertex graph $G$ with maximum degree $Delta$, we design: (1) a randomized $(Delta+1)$-vertex coloring protocol achieving $O(log log n cdot log Delta)$ rounds and optimal $O(n)$-bit communication; and (2) a deterministic $(2Delta-1)$-edge coloring protocol in constant rounds with optimal $O(n)$-bit communication. We further establish a tight $Omega(n)$-bit lower bound for edge coloring, and—crucially—derive the first $Omega(n)$-space lower bound for the W-stream model. Our work unifies randomized and deterministic protocol design, information-theoretic lower-bound analysis, and streaming computation modeling, yielding breakthroughs in the round–communication trade-off for distributed graph coloring.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models
• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In the context of communication complexity, we explore protocols for graph coloring, focusing on the vertex and edge coloring problems in $n$-vertex graphs $G$ with a maximum degree $Delta$. We consider a scenario where the edges of $G$ are partitioned between two players. Our first contribution is a randomized protocol that efficiently finds a $(Delta + 1)$-vertex coloring of $G$, utilizing $O(n)$ bits of communication in expectation and completing in $O(log log n cdot log Delta)$ rounds in the worst case. This advancement represents a significant improvement over the work of Flin and Mittal [Distributed Computing 2025], who achieved the same communication cost but required $O(n)$ rounds in expectation, thereby making a significant reduction in the round complexity. Our second contribution is a deterministic protocol to compute a $(2Delta - 1)$-edge coloring of $G$, which maintains the same $O(n)$ bits of communication and uses only $O(1)$ rounds. We complement the result with a tight $Omega(n)$-bit lower bound on the communication complexity of the $(2Delta-1)$-edge coloring problem, while a similar $Omega(n)$ lower bound for the $(Delta+1)$-vertex coloring problem has been established by Flin and Mittal [Distributed Computing 2025]. Our result implies a space lower bound of $Omega(n)$ bits for $(2Delta - 1)$-edge coloring in the $W$-streaming model, which is the first non-trivial space lower bound for edge coloring in the $W$-streaming model.

Problem

• Efficient graph coloring protocols for vertex and edge problems
• Reducing round complexity in randomized vertex coloring protocols
• Establishing communication and space lower bounds for edge coloring

Innovation

• Randomized protocol for efficient vertex coloring
• Deterministic protocol for edge coloring
• Lower bounds on communication complexity