Distributed Delta-Coloring under Bandwidth Limitations

TL;DR

This work delivers a randomized polylog log n-round Δ-coloring algorithm in the CONGEST model for graphs with maximum degree Δ≥3 by reducing the problem to a small cadre of deg+1-list-coloring and constructive Lovász Local Lemma subproblems. Central to the method is a fine-grained almost-clique decomposition that classifies ACs into easy, nice, ordinary, and difficult, organized across hierarchical levels, enabling phase-based slack generation and color propagation. The algorithm combines probabilistic slack via simulatable LLLs with deterministic d1LC reductions to maintain bandwidth efficiency while navigating non-local dependencies, approaching the established lower bound and advancing sublogarithmic distributed graph coloring in CONGEST. The approach introduces novel LLL simulability concepts, two-set LLL techniques, and a toehold-driven scheme to color large ordinary ACs, offering a framework that can potentially extend to other bandwidth-constrained symmetry-breaking problems. Overall, the results demonstrate that sublogarithmic distributed Δ-coloring is feasible under strict bandwidth limits, with practical implications for scalable distributed graph algorithms.

Abstract

We consider the problem of coloring graphs of maximum degree $Δ$ with $Δ$ colors in the distributed setting with limited bandwidth. Specifically, we give a $\mathsf{poly}\log\log n$-round randomized algorithm in the CONGEST model. This is close to the lower bound of $Ω(\log \log n)$ rounds from [Brandt et al., STOC '16], which holds also in the more powerful LOCAL model. The core of our algorithm is a reduction to several special instances of the constructive Lovász local lemma (LLL) and the $deg+1$-list coloring problem.

Figures (3)

• Figure 1: This is an example of an almost clique (AC). The depicted nice AC is a clique on $\Delta+1$ nodes with a single missing (red) edge. It is essential that the two nodes incident to the missing edge receive the same color to solve the $\Delta$-coloring problem. All non-nice ACs form proper cliques.
• Figure 2: for part b): The illustration depicts three difficult cliques of different layers. The external degree of $C_1$ is $1$, the external degree of $C_2$ is $2$ and the external degree of $C_3$ is $4$. $C_1$ has the lowest layer and its special node (the red node) is part of $C_2$. The blue special node of $C_2$ is part of $C_3$. So when we color $C_1$ the red node serves as an uncolored toehold providing slack to two gray nodes of $C_1$. Stalling the coloring of these gray nodes provides slack to the white nodes of $C_1$ so that they can be colored, followed by the gray ones. For illustration purposes, we chose $\Delta$ to be $9$, but note that this would actually not classify $C_3$ as a difficult clique. A special node of $C_3$ would need $2e_{C_3}=8$ neighbors in $C_3$, which is impossible due to $C_3$'s size.
• Figure 3: for Part c): For the large ordinary cliques, we find triples of nodes consisting of a yellow (striped), a light yellow (dotted), and a gray (solid) node. The two yellow nodes are non-adjacent while the gray node is adjacent to both of them. The goal is to same-color the pairs of yellow/light-yellow nodes, to which end we form a virtual coloring instance consisting of all pairs and their dependencies. After same-coloring the yellow nodes, the gray node provides a slack-toehold for the clique. An important aspect is that triples of different ordinary ACs are non-overlapping and no neighborhood of the graph contains too many nodes in such pairs, as otherwise we may run into unsolvable subinstances down the line. We find these triples by a sequence of 'simple' LLLs.

Theorems & Definitions (80)

• Theorem 1.1
• Lemma 2.1: List coloring HKNT22HNT22
• Definition 2.2: Slack
• Definition 2.3: Graytone FHM23
• Lemma 2.3: ACD computation AKM22FHM23
• proof : Proof of \ref{['lem:acd']}
• Theorem 3.1
• Definition 3.1: Types of almost-cliques
• Definition 3.2: Levels of difficult ACs
• Definition 3.3: Node classification