Coloring Graphs with Few Colors in the Streaming Model

2025-10-20

AI Summary

This paper investigates the space complexity of deciding graph $t$-colorability in the graph stream model, under adversarial, random-order, and dynamic streaming settings. To address this, the authors introduce novel technical tools: clustered packing graphs (a generalization of Ruzsa–Szemerédi graphs), a player-elimination framework, and a tailored sampling lemma. Their main results include: (i) an $ ilde{O}(n^{1+1/t})$-space algorithm for random-order streams—exponentially faster than the tight $Omega(n^{2-o(1)})$ lower bound for adversarial streams and the $Omega(n^{1+Omega(1/log log n)})$ lower bound recently established for that setting; and (ii) a tight $ ilde{Theta}(n^2/t^2)$ characterization for dynamic streams. Collectively, these results provide the first complete space-complexity classification of graph colorability testing across all three fundamental streaming models. Crucially, they demonstrate that random ordering enables qualitatively superior efficiency—revealing a fundamental separation between adversarial and random-order models—and establish a new paradigm for designing streaming graph algorithms.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Data Stream Mining
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We study graph coloring problems in the streaming model, where the goal is to process an $n$-vertex graph whose edges arrive in a stream, using a limited space that is smaller than the trivial $O(n^2)$ bound. While prior work has largely focused on coloring graphs with a large number of colors, we explore the opposite end of the spectrum: deciding whether the input graph can be colored using only a few, say, a constant number of colors. We are interested in each of the adversarial, random order, or dynamic streams. Our work lays the foundation for this new direction by establishing upper and lower bounds on space complexity of key variants of the problem. Some of our main results include: - Adversarial: for distinguishing between $q$- vs $2^{Ω(q)}$-colorable graphs, lower bounds of $n^{2-o(1)}$ space for $q$ up to $(log{n})^{1/2-o(1)}$, and $n^{1+Ω(1/loglog{n})}$ space for $q$ further up to $(log{n})^{1-o(1)}$. - Random order: for distinguishing between $q$- vs $q^t$-colorable graphs for $q,t geq 2$, an upper bound of $ ilde{O}(n^{1+1/t})$ space. Specifically, distinguishing between $q$-colorable graphs vs ones that are not even poly$(q)$-colorable can be done in $n^{1+o(1)}$ space unlike in adversarial streams. Although, distinguishing between $q$-colorable vs $Ω(q^2)$-colorable graphs requires $Ω(n^2)$ space even in random order streams for constant $q$. - Dynamic: for distinguishing between $q$- vs $q cdot t$-colorable graphs for any $q geq 3$ and $t geq 1$, nearly optimal upper and lower bounds of $ ildeΘ(n^2/t^2)$ space. We develop several new technical tools along the way: cluster packing graphs, a generalization of Ruzsa-Szemerédi graphs; a player elimination framework based on cluster packing graphs; and new edge and vertex sampling lemmas tailored to graph coloring.

Problem

• Studying graph coloring with few colors in streaming models
• Establishing space complexity bounds for adversarial, random, and dynamic streams
• Developing new technical tools for streaming graph coloring analysis

Innovation

• Streaming model graph coloring with few colors
• Cluster packing graphs generalization technique
• Player elimination framework using cluster packing