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Sublogarithmic Distributed Vertex Coloring with Optimal Number of Colors

Maxime Flin, Magnús M. Halldórsson, Manuel Jakob, Yannic Maus

Abstract

For any $Δ$, let $k_Δ$ be the maximum integer $k$ such that $(k+1)(k+2)\le Δ$. We give a distributed \LOCAL algorithm that, given an integer $k < k_Δ$, computes a valid $Δ-k$-coloring if one exists. The algorithm runs in $\tilde{O}(\log^4 \log n)$ rounds, which is within a polynomial factor of the $Ω(\log\log n)$ lower bound, which already applies to the case $k=0$. It is also best possible in the sense that if $k \ge k_Δ$, the problem requires $Ω(n/Δ)$ distributed rounds [Molloy, Reed, '14, Bamas, Esperet '19]. For $Δ$ at most polylogarithmic, the algorithm is an exponential improvement over the current state of the art of $O(\log^{49/12} n)$ rounds. When $Δ\ge (\log n)^{50}$, our algorithm achieves an even faster runtime of $O(\log^* n)$ rounds.

Sublogarithmic Distributed Vertex Coloring with Optimal Number of Colors

Abstract

For any , let be the maximum integer such that . We give a distributed \LOCAL algorithm that, given an integer , computes a valid -coloring if one exists. The algorithm runs in rounds, which is within a polynomial factor of the lower bound, which already applies to the case . It is also best possible in the sense that if , the problem requires distributed rounds [Molloy, Reed, '14, Bamas, Esperet '19]. For at most polylogarithmic, the algorithm is an exponential improvement over the current state of the art of rounds. When , our algorithm achieves an even faster runtime of rounds.

Paper Structure

This paper contains 62 sections, 47 theorems, 34 equations, 2 figures, 10 algorithms.

Table of Contents

  1. Introduction
  2. Distributed Graph Coloring.
  3. Coloring with Fewer Colors.
  4. Our Results.
  5. Further Background on Distributed Graph Coloring
  6. Historic focus on Greedy problems.
  7. Recent interest in the locality of non-greedy problems.
  8. The importance of the Lovász Local Lemma.
  9. The Challenges and Our Technical Contributions
  10. Challenges in achieving sublogarithmic distributed algorithms.
  11. Our Contributions.
  12. Organization of the Paper
  13. Technical Overview
  14. Color Coverages (CC) & Pious subgraphs
  15. Color Swapping in Cliques.
  16. ...and 47 more sections

Key Result

Theorem 1

For sufficiently large $\Delta$, and any $c \geqslant \Delta-k_{\Delta}+1$, there is a distributed randomized algorithm that takes a graph $G$ with maximum degree $\Delta$ as input, and does the following: either some vertex outputs a certificate that $G$ is not $c$-colorable, or the algorithm finds

Figures (2)

  • Figure 1: A clique $A_i$ in a graph with $\Delta = 9$ and $k = 3$. On \ref{['fig:swap']}, the clique $A_i$ contains one unhappy vertex, in blue. It cannot swap its color with the purple or green vertex because they have a blue external neighbor. It cannot swap its color with the red vertex because it has a red external neighbor. The remaining vertices (in cyan and yellow) make the $\mathrm{Swappable}$ set for this unhappy vertex. On \ref{['fig:empty']}, the $\mathrm{Swappable}$ set is empty because too many vertices on the outside adopted the blue color.
  • Figure 2: A schematic illustration of the structural decomposition by Molloy and Reed. The vertices of $S$ are loosely connected, while the set $A_i$ is a clique. The intermediate set $B$ is too highly connected to $A$ to be considered sparse, but not densely enough to be itself part of the clique. For the coloring, we further divide the set $B$ into two subsets, $B_H$ and $B_L$ (see \ref{['lem:structuralDecomposition']}); however, the figure shows $B$ as a single set for simplicity. Note that some vertices of $B$, here in the set $All_i$, can be connected to all the vertices of a clique (see \ref{['lem:structuralDecomposition']} for details on the set $All_i$).

Theorems & Definitions (104)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Deterministic LLL in $\mathsf{LOCAL}$, RG20GG24
  • Lemma 3.1: Shattering Lemma FG17
  • Lemma 3.2
  • proof
  • Theorem 3
  • Lemma 4.1: Structural Decomposition
  • Definition 4.2
  • proof
  • ...and 94 more