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Faster Distributed $Δ$-Coloring via a Reduction to MIS

Yann Bourreau, Sebastian Brandt, Alexandre Nolin

TL;DR

The paper presents a deterministic reduction of Δ-coloring to MIS and hypergraph sinkless orientation (HSO) in the LOCAL model, showing that Δ-coloring’s complexity is at most the MIS complexity (up to poly(Δ) factors) and becomes tight if MIS is sublogarithmic. Building a three-tier cluster framework (DCC, flex, link) and introducing gatherers, the authors reduce the global Δ-coloring task to a sequence of MIS and HSO subproblems, enabling a layer-based coloring scheme. Applying state-of-the-art MIS and HSO bounds yields new deterministic and randomized complexities for Δ-coloring, specifically ⊂ tilde O(log^{5/3} n) deterministic rounds and sublogarithmic randomized bounds, with improvements extending to graphs of bounded clique number. The approach also provides tight results on constant-degree graphs and connects to Δ+1 coloring for special graph classes, highlighting the MIS problem as a central proxy for Δ-coloring improvements and offering potential routes toward resolving the Chang–Pettie conjecture in this setting.

Abstract

Recent improvements on the deterministic complexities of fundamental graph problems in the LOCAL model of distributed computing have yielded state-of-the-art upper bounds of $\tilde{O}(\log^{5/3} n)$ rounds for maximal independent set (MIS) and $(Δ+ 1)$-coloring [Ghaffari, Grunau, FOCS'24] and $\tilde{O}(\log^{19/9} n)$ rounds for the more restrictive $Δ$-coloring problem [Ghaffari, Kuhn, FOCS'21; Ghaffari, Grunau, FOCS'24; Bourreau, Brandt, Nolin, STOC'25]. In our work, we show that $Δ$-coloring can be solved deterministically in $\tilde{O}(\log^{5/3} n)$ rounds as well, matching the currently best bound for $(Δ+ 1)$-coloring. We achieve our result by developing a reduction from $Δ$-coloring to MIS that guarantees that the (asymptotic) complexity of $Δ$-coloring is at most the complexity of MIS, unless MIS can be solved in sublogarithmic time, in which case, due to the $Ω(\log n)$-round $Δ$-coloring lower bound from [BFHKLRSU, STOC'16], our reduction implies a tight complexity of $Θ(\log n)$ for $Δ$-coloring. In particular, any improvement on the complexity of the MIS problem will yield the same improvement for the complexity of $Δ$-coloring (up to the true complexity of $Δ$-coloring). Our reduction yields improvements for $Δ$-coloring in the randomized LOCAL model and when complexities are parameterized by both $n$ and $Δ$. We obtain a randomized complexity bound of $\tilde{O}(\log^{5/3} \log n)$ rounds (improving over the state of the art of $\tilde{O}(\log^{8/3} \log n)$ rounds) on general graphs and tight complexities of $Θ(\log n)$ and $Θ(\log \log n)$ for the deterministic, resp.\ randomized, complexity on bounded-degree graphs. In the special case of graphs of constant clique number (which for instance include bipartite graphs), we also give a reduction to the $(Δ+1)$-coloring problem.

Faster Distributed $Δ$-Coloring via a Reduction to MIS

TL;DR

The paper presents a deterministic reduction of Δ-coloring to MIS and hypergraph sinkless orientation (HSO) in the LOCAL model, showing that Δ-coloring’s complexity is at most the MIS complexity (up to poly(Δ) factors) and becomes tight if MIS is sublogarithmic. Building a three-tier cluster framework (DCC, flex, link) and introducing gatherers, the authors reduce the global Δ-coloring task to a sequence of MIS and HSO subproblems, enabling a layer-based coloring scheme. Applying state-of-the-art MIS and HSO bounds yields new deterministic and randomized complexities for Δ-coloring, specifically ⊂ tilde O(log^{5/3} n) deterministic rounds and sublogarithmic randomized bounds, with improvements extending to graphs of bounded clique number. The approach also provides tight results on constant-degree graphs and connects to Δ+1 coloring for special graph classes, highlighting the MIS problem as a central proxy for Δ-coloring improvements and offering potential routes toward resolving the Chang–Pettie conjecture in this setting.

Abstract

Recent improvements on the deterministic complexities of fundamental graph problems in the LOCAL model of distributed computing have yielded state-of-the-art upper bounds of rounds for maximal independent set (MIS) and -coloring [Ghaffari, Grunau, FOCS'24] and rounds for the more restrictive -coloring problem [Ghaffari, Kuhn, FOCS'21; Ghaffari, Grunau, FOCS'24; Bourreau, Brandt, Nolin, STOC'25]. In our work, we show that -coloring can be solved deterministically in rounds as well, matching the currently best bound for -coloring. We achieve our result by developing a reduction from -coloring to MIS that guarantees that the (asymptotic) complexity of -coloring is at most the complexity of MIS, unless MIS can be solved in sublogarithmic time, in which case, due to the -round -coloring lower bound from [BFHKLRSU, STOC'16], our reduction implies a tight complexity of for -coloring. In particular, any improvement on the complexity of the MIS problem will yield the same improvement for the complexity of -coloring (up to the true complexity of -coloring). Our reduction yields improvements for -coloring in the randomized LOCAL model and when complexities are parameterized by both and . We obtain a randomized complexity bound of rounds (improving over the state of the art of rounds) on general graphs and tight complexities of and for the deterministic, resp.\ randomized, complexity on bounded-degree graphs. In the special case of graphs of constant clique number (which for instance include bipartite graphs), we also give a reduction to the -coloring problem.

Paper Structure

This paper contains 28 sections, 26 theorems, 8 equations, 1 figure, 7 algorithms.

Table of Contents

  1. Introduction
  2. The LOCAL Model.
  3. The Big $4$.
  4. Beyond greedy.
  5. $\Delta$-coloring.
  6. Our Contributions
  7. Deterministic complexities.
  8. Randomized complexities.
  9. LOCAL complexities for important graph classes.
  10. Overview of Our Techniques
  11. Preliminaries
  12. Graph and hypergraph notation
  13. Hypergraph sinkless orientation
  14. MIS and coloring
  15. Degree-choosability and DCCs
  16. ...and 13 more sections

Key Result

Theorem 1.1

Deterministic $\Delta$-coloring reduces to solving $O(1)$ instances of deterministic MIS on graphs of maximum degree $O(\mathop{\mathrm{poly}}\nolimits(\Delta))$ and at most $n$ nodes, and solving an instance of deterministic hypergraph sinkless orientation with degree-to-rank ratio $\delta/r \in \O

Figures (1)

  • Figure 1: The $3$ cases for the selection of independent gatherers: a node $v$ of depth ${{\alpha}_{\mathsf{flex}}}-2$ with at least one neighbor of equal depth (left); a node $v$ of depth ${{\alpha}_{\mathsf{flex}}}-2$ with two or more unconnected children (middle); a child $v'$ of a node $v$ of depth ${{\alpha}_{\mathsf{flex}}}-2$, where $v'$ has a neighbor of equal depth as $v$ had in the first case (right). In each case, gray nodes show two unconnected neighbors of the selected gatherer, marked by a cross.

Theorems & Definitions (50)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Corollary 1.2
  • Theorem 1.3
  • Definition 2.1: hypergraph sinkless orientation (HSO)
  • Theorem 2.2: BMNSU_halls_thm_soda25
  • Theorem 2.3: BMNSU_halls_thm_soda25
  • Theorem 2.4
  • Theorem 2.5: Luby1986
  • ...and 40 more