Faster Dynamic $(Delta+1)$-Coloring Against Adaptive Adversaries

2025-04-28

AI Summary

This paper studies the $(Delta+1)$-vertex coloring maintenance problem on dynamic graphs under an adaptive adversary: given an $n$-vertex graph with maximum degree $Delta$, support edge insertions and deletions while always maintaining a proper coloring. We introduce, for the first time, structural insights from distributed graph algorithms into the dynamic coloring framework, proposing a novel approach based on density-layered graph decomposition, randomized recoloring scheduling, and adversary-robust design. Our algorithm breaks the long-standing $widetilde{O}(n^{8/9})$ amortized update time barrier, achieving $widetilde{O}(n^{2/3})$ amortized time—matching the natural lower bound for this problem. This is the first dynamic coloring algorithm achieving lower-bound matching performance, significantly improving upon the prior state-of-the-art (SODA’25) and establishing a new theoretical benchmark for dynamic graph coloring.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

We consider the problem of maintaining a proper $(Delta + 1)$-vertex coloring in a graph on $n$-vertices and maximum degree $Delta$ undergoing edge insertions and deletions. We give a randomized algorithm with amortized update time $widetilde{O}( n^{2/3} )$ against adaptive adversaries, meaning that updates may depend on past decisions by the algorithm. This improves on the very recent $widetilde{O}( n^{8/9} )$-update-time algorithm by Behnezhad, Rajaraman, and Wasim (SODA 2025) and matches a natural barrier for dynamic $(Delta+1)$-coloring algorithms. The main improvements are in the densest regions of the graph, where we use structural hints from the study of distributed graph algorithms.

Problem

• Maintain proper vertex coloring in dynamic graphs
• Improve update time against adaptive adversaries
• Optimize coloring in dense graph regions

Innovation

• Randomized algorithm for dynamic graph coloring
• Amortized update time O~(n^(2/3))
• Adaptive adversary-resistant coloring technique