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Towards Optimal Distributed Delta Coloring

Manuel Jakob, Yannic Maus

TL;DR

The paper tackles deterministic and randomized Δ-coloring in the LOCAL model, focusing on dense constant-degree graphs where sparse parts are absent. It introduces slack triads and hyperedge grabbing (HEG) as central mechanisms, organizing the coloring into a multi-phase procedure that colors hard cliques, then uses slack structure to extend to remaining vertices; the deterministic result achieves $O(\min\{\widetilde{O}(\log^{5/3}n), O(\Delta+\log n)\})$ rounds, while for dense graphs the bound collapses to $O(\log n)$, matching the general lower bound. The randomized algorithm leverages shattering to achieve $O(\min\{\widetilde{O}(\log^{5/3}\log n), O(\Delta+\log\log n)\})$ rounds on dense graphs, reflecting the current best-known bounds in this regime. Together, these contributions push toward an optimal distributed Δ-coloring algorithm for general constant-degree graphs and offer a robust framework (ACD, slack triads, HEG) that could extend to sparser settings.

Abstract

The $Δ$-vertex coloring problem has become one of the prototypical problems for understanding the complexity of local distributed graph problems on constant-degree graphs. The major open problem is whether the problem can be solved deterministically in logarithmic time, which would match the lower bound [Chang et al., FOCS'16]. Despite recent progress in the design of efficient $Δ$-coloring algorithms, there is currently a polynomial gap between the upper and lower bounds. In this work we present a $O(\log n)$-round deterministic $Δ$-coloring algorithm for dense constant-degree graphs, matching the lower bound for the problem on general graphs. For general $Δ$ the algorithms' complexity is $\min\{\widetilde{O}(\log^{5/3}n),O(Δ+\log n)\}$. All recent distributed and sublinear graph coloring algorithms (also for coloring with more than $Δ$ colors) decompose the graph into sparse and dense parts. Our algorithm works for the case that this decomposition has no sparse vertices. Ironically, in recent (randomized) $Δ$-coloring algorithms, dealing with sparse parts was relatively easy and these dense parts arguably posed the major hurdle. We present a solution that addresses the dense parts and may have the potential for extension to sparse parts. Our approach is fundamentally different from prior deterministic algorithms and hence hopefully contributes towards designing an optimal algorithm for the general case. Additionally, we leverage our result to also obtain a randomized $\min\{\widetilde{O}(\log^{5/3}\log n), O(Δ+\log\log n)\}$-round algorithm for $Δ$-coloring dense graphs that also matches the lower bound for the problem on general constant-degree graphs [Brandt et al.; STOC'16].

Towards Optimal Distributed Delta Coloring

TL;DR

The paper tackles deterministic and randomized Δ-coloring in the LOCAL model, focusing on dense constant-degree graphs where sparse parts are absent. It introduces slack triads and hyperedge grabbing (HEG) as central mechanisms, organizing the coloring into a multi-phase procedure that colors hard cliques, then uses slack structure to extend to remaining vertices; the deterministic result achieves rounds, while for dense graphs the bound collapses to , matching the general lower bound. The randomized algorithm leverages shattering to achieve rounds on dense graphs, reflecting the current best-known bounds in this regime. Together, these contributions push toward an optimal distributed Δ-coloring algorithm for general constant-degree graphs and offer a robust framework (ACD, slack triads, HEG) that could extend to sparser settings.

Abstract

The -vertex coloring problem has become one of the prototypical problems for understanding the complexity of local distributed graph problems on constant-degree graphs. The major open problem is whether the problem can be solved deterministically in logarithmic time, which would match the lower bound [Chang et al., FOCS'16]. Despite recent progress in the design of efficient -coloring algorithms, there is currently a polynomial gap between the upper and lower bounds. In this work we present a -round deterministic -coloring algorithm for dense constant-degree graphs, matching the lower bound for the problem on general graphs. For general the algorithms' complexity is . All recent distributed and sublinear graph coloring algorithms (also for coloring with more than colors) decompose the graph into sparse and dense parts. Our algorithm works for the case that this decomposition has no sparse vertices. Ironically, in recent (randomized) -coloring algorithms, dealing with sparse parts was relatively easy and these dense parts arguably posed the major hurdle. We present a solution that addresses the dense parts and may have the potential for extension to sparse parts. Our approach is fundamentally different from prior deterministic algorithms and hence hopefully contributes towards designing an optimal algorithm for the general case. Additionally, we leverage our result to also obtain a randomized -round algorithm for -coloring dense graphs that also matches the lower bound for the problem on general constant-degree graphs [Brandt et al.; STOC'16].

Paper Structure

This paper contains 36 sections, 21 theorems, 12 equations, 5 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. The $\mathsf{LOCAL}$ model of distributed computing linial92.
  3. Why should one care about $\Delta$-coloring and its randomized complexity?
  4. What are dense graphs and why are they challenging for the $\Delta$-coloring problem?
  5. Technical Overview & Related Work.
  6. Prior deterministic algorithms.
  7. Finding slack triads in dense graphs.
  8. Why does our approach not (yet) extent to sparse graphs?
  9. Outline
  10. Preliminaries: ACD, Dense Graphs, and Hyperedge Grabbing
  11. Notation.
  12. Hyperedge Grabbing (HEG).
  13. Deterministic Delta-Coloring dense graphs
  14. Definitions: Loopholes and Hard/Easy Almost-Cliques
  15. Coloring hard cliques: High level overview and Pseudocode (Algorithm \ref{['alg:coloring']})
  16. ...and 21 more sections

Key Result

Lemma 2

Let $\varepsilon=1/63$. For any graph $G=(V,E)$, there is a partition (almost-clique decomposition (ACD) of $V$ into sets $V_{sparse}\xspace$ and $C_1, C_2, \ldots, C_t$ such that each node in $V_{sparse}\xspace$ is $\Omega(\epsilon^2\Delta)$-sparse and for every $i\in [t]$, Further, there is a deterministic $O(1)$-round $\mathsf{LOCAL}$ algorithm to compute a valid ACD.

Figures (5)

  • Figure 1: The complexity landscape of LCL problems: Each black dot represents a problem with the depicted asymptotic complexity. For $T_{LLL}$, the randomized lower bound is $\Omega(\log\log n)$, and the best upper bound is $O(\operatorname{\text{\rm poly}}\log\log n)$. The top branch reflects deterministic complexities, the bottom branch randomized ones. An $\mathsf{X}$ marks impossible complexities, e.g., no problem has deterministic complexity $\omega(\log^*n)$ and $o(\log n)$. Infinitely many complexities exist in the shaded areas. Current research focuses on the poorly understood red-shaded area, where $\Delta$-coloring is the only natural problem with its best upper bound inside.
  • Figure 2: For each hard clique, such as $C_1$ and $C_2$, we identify slack triads—for example, $(u, v, w)$ in $C_1$. The slack pair vertices, $v$ and $w$, are then same-colored (see \ref{['fig:slackTriadVirtual']}), providing slack to vertex $u$.
  • Figure 3: For coloring the hard cliques in the graph, we identify slack pair vertices (orange boxes), which provide the slack vertices (checkboard) with one unit of slack. To color the slack pair vertices, we construct a virtual graph where each slack pair (orange box) is treated as a single vertex. Edges between these vertices (orange) exist if there is an underlying edge in the original graph (black).
  • Figure 4: This example illustrates the intuition behind the HEG procedure in \ref{['alg:coloring']}, applied to three cliques with their respective partitions. It demonstrates the process of edge flipping triggered by the HEG. Initially, vertices request to grab an edge, as indicated by the blue arrows based on the function $f$. This request can either target an adjacent matching edge (e.g., vertex $v$) or a matching edge adjacent to a neighbor (e.g., vertex $u$). As the algorithm progresses, the vertices $(u, v, w)$ form the slack triad for the bottom clique assuming $u$ becomes the slack vertex.
  • Figure 5: Example of a loophole (red circle) that is used in the proof of \ref{['lem:hardcliqueprops']}, Part \ref{['lem:hardcliqueprops:notwons']}.

Theorems & Definitions (39)

  • Claim 1: ACK19
  • Lemma 2: ACD computation HSS18AKM22FHM23HM24
  • Definition 4: Dense graph
  • Lemma 5: BMNSU25
  • Theorem 1
  • Definition 6: Loophole
  • Lemma 7: erdos79choosabilityvizing76vertexGHKM21
  • Definition 8: Easy Almost Cliques/Hard Cliques
  • Lemma 9
  • Lemma 10
  • ...and 29 more