Coloring Graphs with Few Colors in the Streaming Model

TL;DR

This work initiates a systematic study of graph coloring in the streaming model when the number of colors is small. It develops a suite of techniques—cluster packing graphs, a player-elimination framework, and edge/vertex sampling lemmas—to establish tight (or near-tight) space bounds across adversarial, random-order, and dynamic streams for distinguishing q-colorable from more color-rich instances. The authors prove strong lower bounds in adversarial streams via multi-player communication complexity and cluster-packing constructions, while giving matching or near-matching upper bounds in random-order and dynamic streams, including a near-optimal $\tilde{O}(n^{1+1/t})$-space algorithm and a $\tilde{Θ}(n^2/t^2)$-space lower bound for dynamic streams. The results reveal a sharp separation between models and highlight fundamental limits of semi-streaming approaches for small-color-ability questions, with potential implications for NP-hard problem streaming and new combinatorial constructions. Overall, the paper lays a foundational framework and tools that can be leveraged in broader streaming-graph problems beyond coloring.”

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 $n^{1+Ω(1/\log\log{n})}$ 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 $Ω(q^2)$-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 $\tildeΘ(n^2/t^2)$ 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.

Figures (10)

• Figure 1: An illustration of our two-player communication lower bound. Each circle denotes the vertices of a $3$-clique in $G_{cluster}$ and each small box (gray) denotes an induced $(3,3)$-cluster. The filled in circles mean the edges of those cliques are not dropped from Alice's graph and the empty circles shows the one that were dropped. The solid lines inside the cluster denote the edges of Bob between vertices of all cliques of the cluster. We shall note that this figure is not accurate in the sense of the clusters in $G_{cluster}$ are not necessarily vertex-disjoint (unlike what is depicted) as we have $\Omega(n^2)$ many of them.
• Figure 2: An illustration of forming a $k^p$-clique by combining $p$$k$-colorable graphs according to a discrete $p$-dimensional grid; here, $k=p=3$. To avoid cluttering the figures, not all edges of the clique are shown.
• Figure 3: A $(2,3,3)$-cluster packing graph ($r=2$, $t=3$, $k=3$). Each one of the three $(2,3)$-clusters are shown separately; notice that there are no other edges between vertices of each cluster (they are induced).
• Figure 4: An illustration of input of the first player in our multi-player communication lower bound. Here, the bigger boxes show two different induced clusters in the graph $G_1$. Each circle is a $k$-clique in the cluster and empty circles mean the edges of the clique are dropped from $G_1$ while filled in one means the clique is present. The graph $G_1$ can only have remaining cliques inside sets $S_1,\ldots,S_t$ for each induced cluster. The set $T$ that defines vertices of the subsequent players is also depicted. After player one, all other players only have edges inside the set $T$.
• Figure 5: An illustration of our simultaneous communication lower bound for $k=4$ and $\binom{4}{2} = 6$ players.

Theorems & Definitions (128)

• Definition 4.1
• Proposition 4.2
• proof
• Proposition 4.3: KNR95
• Theorem 1
• Lemma 4.4
• proof
• Lemma 4.5
• proof
• proof : Proof of \ref{['thm:two-player']}