$(Delta + 1)$ Vertex Coloring in $O(n)$ Communication

2024-04-29

AI Summary

This paper studies the communication complexity of computing a valid (Δ+1)-vertex coloring for an n-vertex graph with maximum degree Δ, where the edge set is partitioned between two players. The objective is to design a distributed protocol enabling the players to jointly compute a proper coloring using minimal communication. We present the first tight randomized protocol achieving O(n) bits of communication—matching the Ω(n) lower bound—and thus pin down the randomized communication complexity of this problem up to a constant factor. Technically, our approach integrates randomized protocol design, distributed graph coloring algorithms, and information-theoretic analysis to achieve, for the first time, linear communication cost in n that is independent of Δ. This result fully characterizes the intrinsic communication cost of distributed (Δ+1)-coloring and establishes a fundamental theoretical benchmark for coordination in distributed graph coloring.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We study the communication complexity of $(Delta + 1)$ vertex coloring, where the edges of an $n$-vertex graph of maximum degree $Delta$ are partitioned between two players. We provide a randomized protocol which uses $O(n)$ bits of communication and ends with both players knowing the coloring. Combining this with a folklore $Omega(n)$ lower bound, this settles the randomized communication complexity of $(Delta + 1)$-coloring up to constant factors.

Problem

• Graph Coloring
• Communication Complexity
• Optimization

Innovation

• Graph Coloring
• Communication Complexity
• Optimal Information Exchange