Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Faster Distributed $Δ$-Coloring via Ruling Subgraphs

Yann Bourreau, Sebastian Brandt, Alexandre Nolin

TL;DR

This work studies the Δ-coloring problem in the LOCAL distributed model, introducing ruling subgraph families as a generic symmetry-breaking tool to overcome Bottlenecks of previous ruling-set approaches. The authors develop a formal framework around orchids, distance-d conflict graphs, and contention digraphs, and show how to compute ruling subgraph families in roughly Õ(log n) rounds. They then apply this tool to obtain near-optimal deterministic Δ-coloring in LOCAL, achieving O(log n log^* n) rounds on constant-degree graphs (and approaching tight bounds) and improved O(Δ)-dependent bounds for larger Δ. In the randomized setting, the method yields O(√(Δ log Δ)(log_Δ log n log^* n + log log n)) rounds or Õ(log^{8/3} log n) depending on the coloring strategy, significantly narrowing the gap to the Ω(log log n) lower bound and advancing toward the Chang–Pettie conjecture. Overall, ruling subgraphs offer a generic, broadly applicable technique with practical implications for Δ-coloring and related problems in distributed graph algorithms.

Abstract

Brooks' theorem states that all connected graphs but odd cycles and cliques can be colored with $Δ$ colors, where $Δ$ is the maximum degree of the graph. Such colorings have been shown to admit non-trivial distributed algorithms [Panconesi and Srinivasan, Combinatorica 1995] and have been studied intensively in the distributed literature. In particular, it is known that any deterministic algorithm computing a $Δ$-coloring requires $Ω(\log n)$ rounds in the LOCAL model [Chang, Kopelowitz, and Pettie, FOCS 2016], and that this lower bound holds already on constant-degree graphs. In contrast, the best upper bound in this setting is given by an $O(\log^2 n)$-round deterministic algorithm that can be inferred already from the works of [Awerbuch, Goldberg, Luby, and Plotkin, FOCS 1989] and [Panconesi and Srinivasan, Combinatorica 1995] roughly three decades ago, raising the fundamental question about the true complexity of $Δ$-coloring in the constant-degree setting. We answer this long-standing question almost completely by providing an almost-optimal deterministic $O(\log n \log^* n)$-round algorithm for $Δ$-coloring, matching the lower bound up to a $\log^* n$-factor. Similarly, in the randomized LOCAL model, we provide an $O(\log \log n \log^* n)$-round algorithm, improving over the state-of-the-art upper bound of $O(\log^2 \log n)$ [Ghaffari, Hirvonen, Kuhn, and Maus, Distributed Computing 2021] and almost matching the $Ω(\log \log n)$-round lower bound by [BFHKLRSU, STOC 2016]. Our results make progress on several important open problems and conjectures. One key ingredient for obtaining our results is the introduction of ruling subgraph families as a novel tool for breaking symmetry between substructures of a graph, which we expect to be of independent interest.

Faster Distributed $Δ$-Coloring via Ruling Subgraphs

TL;DR

This work studies the Δ-coloring problem in the LOCAL distributed model, introducing ruling subgraph families as a generic symmetry-breaking tool to overcome Bottlenecks of previous ruling-set approaches. The authors develop a formal framework around orchids, distance-d conflict graphs, and contention digraphs, and show how to compute ruling subgraph families in roughly Õ(log n) rounds. They then apply this tool to obtain near-optimal deterministic Δ-coloring in LOCAL, achieving O(log n log^* n) rounds on constant-degree graphs (and approaching tight bounds) and improved O(Δ)-dependent bounds for larger Δ. In the randomized setting, the method yields O(√(Δ log Δ)(log_Δ log n log^* n + log log n)) rounds or Õ(log^{8/3} log n) depending on the coloring strategy, significantly narrowing the gap to the Ω(log log n) lower bound and advancing toward the Chang–Pettie conjecture. Overall, ruling subgraphs offer a generic, broadly applicable technique with practical implications for Δ-coloring and related problems in distributed graph algorithms.

Abstract

Brooks' theorem states that all connected graphs but odd cycles and cliques can be colored with colors, where is the maximum degree of the graph. Such colorings have been shown to admit non-trivial distributed algorithms [Panconesi and Srinivasan, Combinatorica 1995] and have been studied intensively in the distributed literature. In particular, it is known that any deterministic algorithm computing a -coloring requires rounds in the LOCAL model [Chang, Kopelowitz, and Pettie, FOCS 2016], and that this lower bound holds already on constant-degree graphs. In contrast, the best upper bound in this setting is given by an -round deterministic algorithm that can be inferred already from the works of [Awerbuch, Goldberg, Luby, and Plotkin, FOCS 1989] and [Panconesi and Srinivasan, Combinatorica 1995] roughly three decades ago, raising the fundamental question about the true complexity of -coloring in the constant-degree setting. We answer this long-standing question almost completely by providing an almost-optimal deterministic -round algorithm for -coloring, matching the lower bound up to a -factor. Similarly, in the randomized LOCAL model, we provide an -round algorithm, improving over the state-of-the-art upper bound of [Ghaffari, Hirvonen, Kuhn, and Maus, Distributed Computing 2021] and almost matching the -round lower bound by [BFHKLRSU, STOC 2016]. Our results make progress on several important open problems and conjectures. One key ingredient for obtaining our results is the introduction of ruling subgraph families as a novel tool for breaking symmetry between substructures of a graph, which we expect to be of independent interest.

Paper Structure

This paper contains 44 sections, 43 theorems, 8 equations, 3 figures, 9 algorithms.

Table of Contents

  1. Introduction
  2. Model of computation.
  3. A fundamental dichotomy.
  4. Logarithmic dependencies on $n$?
  5. $\Delta$-coloring.
  6. The importance of $\Delta$-coloring.
  7. Our Contributions
  8. A generic approach.
  9. Ruling subgraph families.
  10. Deterministic $\Delta$-coloring in LOCAL.
  11. Randomized $\Delta$-coloring in LOCAL.
  12. Organization of the Paper
  13. Preliminaries and Notation
  14. Structural graph theoretical results.
  15. Algorithmic subroutines relevant for our algorithms.
  16. ...and 29 more sections

Key Result

Theorem 1.1

Let $\mathcal{R}$ be a family of connected subgraphs of the input graph $G$, and let $k$ be an integer such that each member of $\mathcal{R}$ has at most $k$ vertices. Let $t \leq k$ be an integer, and, for each member $H\in\mathcal{R}$, let $X(H)$ be an induced connected subgraph of $H$ containing

Figures (3)

  • Figure 1: Three representations of $H$ and its orchid. Left: the graph with its stem highlighted and its root node represented as the bigger node. Middle: the orchid is a zone surrounding and including the stem. Right: the stem with highlighted branch nodes, which we introduce later (\ref{['def:branch-nodes']}).
  • Figure 2: Three possibilities for a nice LDCC's flexible node: (1) its root node is flexible; (2) another node in the stem is flexible; (3) a neighbor of the root in the stem is made flexible by coloring the root with a color that is used for this neighbor. In all three cases, nodes are colored in order of decreasing distance to the flexible node.
  • Figure 3: Examples of nice LDCCs. Each of them is 2-connected, has two nodes of degree $3$, while all other nodes have degree $2$.

Theorems & Definitions (76)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Degree-choosable component
  • Definition 2.2: $\Delta$-extendable
  • Definition 2.3: Blocks and block graph
  • Proposition 2.4: See e.g. Diestel_GT_5th_ed
  • Definition 2.5
  • Lemma 2.6: KSW_dm96Thomassen_jct97a
  • Lemma 2.7: GHKM_dc21
  • ...and 66 more