Coloring Graphs with Few Colors in the Streaming Model

TL;DR

This paper investigates the space complexity of graph coloring in streaming models, establishing bounds for distinguishing between graphs with few colors under adversarial, random, and dynamic stream conditions, and introduces new technical tools.

Contribution

It provides the first bounds on space complexity for coloring graphs with few colors in streaming models, exploring different stream types and introducing novel techniques.

Findings

Lower bounds of near-quadratic space for certain colorability distinctions in adversarial streams.

Efficient algorithms for distinguishing between q- and q^t-colorable graphs in random order streams.

Nearly optimal bounds for distinguishing q- vs q·t-colorable graphs in dynamic streams.

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 $2^{\Omega(q)}$-colorable graphs, lower bounds of $n^{2-o(1)}$ space for $q$ up to $(\log{n})^{1/2-o(1)}$, and $n^{1+\Omega(1/\log\log{n})}$ 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 $\tilde{O}(n^{1+1/t})$ 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 $\Omega(q^2)$-colorable graphs requires $\Omega(n^2)$ 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 $\tilde{\Theta}(n^2/t^2)$ space. We develop several new technical tools along the way: cluster packing graphs, a generalization of Ruzsa-Szemer\'edi graphs; a player elimination framework based on cluster packing graphs; and new edge and vertex sampling lemmas tailored to graph coloring.