Decentralized Distributed Graph Coloring II: degree+1-Coloring Virtual Graphs

TL;DR

This work develops a general framework for coloring a virtual graph $H$ embedded in a network $G$, capturing settings where the input graph differs from the communication topology. It shows that deg+1-coloring of $H$ can be achieved in $O(\mathsf{c}\mathsf{d}\,\log^4\log n)$ rounds in the CONGEST model, where $\mathsf{c}$ is the embedding congestion and $\mathsf{d}$ the dilation, and provides a matching lower bound that scales with congestion and dilation. The core technical contributions introduce slack-based techniques, an $\varepsilon$-almost-clique decomposition, and an adaptation of the Ghaffari–Kuhn coloring framework to virtual graphs, including handling of inaccurate degrees and limited global knowledge. These results extend fast distributed coloring to a broad class of virtual graph problems, with applications to cluster graphs and power graphs, and establish a foundation for future work on scheduling and direct-coloring in embedded settings.

Abstract

Graph coloring is fundamental to distributed computing. We give the first general treatment of the coloring of virtual graphs, where the graph $H$ to be colored is locally embedded within the communication graph $G$. Besides generalizing classical distributed graph coloring (where $H=G$), this captures other previously studied settings, including cluster graphs and power graphs. We find that the complexity of coloring a virtual graph depends on the edge congestion of its embedding. The main question of interest is how fast we can color virtual graphs of constant congestion. We find that, surprisingly, these graphs can be colored nearly as fast as ordinary graphs. Namely, we give a $O(\log^4\log n)$-round algorithm for the deg+1-coloring problem, where each node is assigned more colors than its degree. This can be viewed as a case where a distributed graph problem can be solved even when the operation of each node is decentralized.

Figures (4)

• Figure 1: A virtual graph $H$ (on the left) embedded on a network $G$ (on the right). On this example, there is a unique choice of support trees; they have congestion $\mathsf{c}=1$ and dilation $\mathsf{d} = 3$.
• Figure 2: Three possible inputs to the communication complexity task.
• Figure 3: Examples of a virtual graph $H$ with a single gadget (left), a communication network $G$ (middle) in which $H$ can be embedded, and the support of the top left virtual node (right).
• Figure 4: Examples of a virtual graph $H_{1,x,y}$ (left), a communication network $G_{1,x,y}$ (middle) in which it can be embedded, and the support of the top left virtual node (right).

Theorems & Definitions (108)

• theorem 1.0
• theorem 1.0
• definition 2.1: Virtual Graph
• definition 2.2: Embedded Virtual Graph
• remark 2.3
• corollary 2.4
• lemma 2.5
• proof
• corollary 2.6
• theorem 3.0