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Optimal (degree+1)-Coloring in Congested Clique

Sam Coy, Artur Czumaj, Peter Davies, Gopinath Mishra

TL;DR

The paper resolves the distributed complexity of degree+1-list coloring (D1LC) in the Congested Clique by presenting a deterministic constant-round algorithm. It introduces a tree-structured bucket decomposition that maps nodes and colors to hierarchical buckets, enabling disjoint reduced instances and parallel coloring. The approach derandomizes core steps (ColorTrial, Subsample) via the method of conditional expectations and bounded-independence hashing, ensuring the leftover bad nodes form an $O(n)$-sized graph that can be handled efficiently. The technique extends to general graphs by recursive partitioning and aggregation, yielding a practical, constant-round solution with rigorous guarantees. This work solidifies the parity between D1LC and simpler Δ+1-style coloring in the Congested Clique and provides a robust framework for deterministic distributed coloring at scale.

Abstract

We consider the distributed complexity of the (degree+1)-list coloring problem, in which each node $u$ of degree $d(u)$ is assigned a palette of $d(u)+1$ colors, and the goal is to find a proper coloring using these color palettes. The (degree+1)-list coloring problem is a natural generalization of the classical $(Δ+1)$-coloring and $(Δ+1)$-list coloring problems, both being benchmark problems extensively studied in distributed and parallel computing. In this paper we settle the complexity of the (degree+1)-list coloring problem in the Congested Clique model by showing that it can be solved deterministically in a constant number of rounds.

Optimal (degree+1)-Coloring in Congested Clique

TL;DR

The paper resolves the distributed complexity of degree+1-list coloring (D1LC) in the Congested Clique by presenting a deterministic constant-round algorithm. It introduces a tree-structured bucket decomposition that maps nodes and colors to hierarchical buckets, enabling disjoint reduced instances and parallel coloring. The approach derandomizes core steps (ColorTrial, Subsample) via the method of conditional expectations and bounded-independence hashing, ensuring the leftover bad nodes form an -sized graph that can be handled efficiently. The technique extends to general graphs by recursive partitioning and aggregation, yielding a practical, constant-round solution with rigorous guarantees. This work solidifies the parity between D1LC and simpler Δ+1-style coloring in the Congested Clique and provides a robust framework for deterministic distributed coloring at scale.

Abstract

We consider the distributed complexity of the (degree+1)-list coloring problem, in which each node of degree is assigned a palette of colors, and the goal is to find a proper coloring using these color palettes. The (degree+1)-list coloring problem is a natural generalization of the classical -coloring and -list coloring problems, both being benchmark problems extensively studied in distributed and parallel computing. In this paper we settle the complexity of the (degree+1)-list coloring problem in the Congested Clique model by showing that it can be solved deterministically in a constant number of rounds.
Paper Structure (35 sections, 18 theorems, 23 equations, 2 figures, 4 algorithms)

This paper contains 35 sections, 18 theorems, 23 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background and Related Works
  3. (degree+1)-list coloring ($\mathsf{D1LC}$).
  4. Recent work in $\mathsf{D1LC}$ on $\mathsf{MPC}$.
  5. Derandomization tools for distributed coloring algorithms.
  6. Technical Overview
  7. Preliminaries
  8. Notation
  9. Derandomization in $\mathsf{CongestedClique}$
  10. Bounded Independence
  11. The Method of Conditional Expectations
  12. The $\mathsf{D1LC}$ Algorithm
  13. ColorTrial and Subsample
  14. ColorTrial
  15. Event 1: $|\mathrm{Nom}_v| > \frac{1}{4} d_{G_0}(v) + p_{G_0}(v)^{0.7}$.
  16. ...and 20 more sections

Key Result

Theorem 1.1

There is a deterministic $\mathsf{CongestedClique}$ algorithm which finds a (degree+1)-list coloring of any graph in a constant number of rounds.

Figures (2)

  • Figure 1: Bucket structure
  • Figure 2: Partitioning nodes and colors

Theorems & Definitions (46)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4: Lemma 2.3 of BR94
  • Corollary 2.5
  • proof
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.3
  • ...and 36 more