Deterministic Edge Coloring with few Colors in CONGEST

TL;DR

This work presents deterministic edge coloring algorithms in the CONGEST model and improves the complexity of edge coloring with $2\Delta-1$ colors for all ranges of $\Delta$ to $\tilde{O}(\log^{2.5} n+\log^2 \Delta \log n)$.

Abstract

As the main contribution of this work we present deterministic edge coloring algorithms in the CONGEST model. In particular, we present an algorithm that edge colors any $n$-node graph with maximum degree $Δ$ with with $(1+\varepsilon)Δ+O(\sqrt{\log n})$ colors in $\tilde{O}(\log^{2.5} n+\log^2 Δ\log n)$ rounds. This brings the upper bound polynomially close to the lower bound of $Ω(\log n/\log\log n)$ rounds that also holds in the more powerful LOCAL model [Chang, He, Li, Pettie, Uitto; SODA'18]. As long as $Δ\geq c\sqrt{\log n}$ our algorithm uses fewer than $2Δ-1$ colors and to the best of our knowledge is the first polylogarithmic-round CONGEST algorithm achieving this for any range of $Δ$. As a corollary we also improve the complexity of edge coloring with $2Δ-1$ colors for all ranges of $Δ$ to $\tilde{O}(\log^{2.5} n+\log^2 Δ\log n)$. This improves upon the previous $O(\log^8 n)$-round algorithm from [Fischer, Ghaffari, Kuhn; FOCS'17]. Our approach builds on a refined analysis and extension of the online edge-coloring algorithm of Blikstad, Svensson, Vintan, and Wajc [FOCS'25], and more broadly on new connections between online and distributed graph algorithms. We show that their algorithm exhibits very low locality and, if it can additionally have limited local access to future edges (as distributed algorithms can), it can be derandomized for smaller degrees. Under this additional power, we are able to bypass classical online lower bounds and translate the results to efficient distributed algorithms. This leads to our CONGEST algorithm for $(1+\varepsilon)Δ+O(\sqrt{\log n})$-edge coloring. Since the modified online algorithm can be implemented more efficiently in the LOCAL model, we also obtain (marginally) improved complexity bounds in that model.

Figures (6)

• Figure 1: In this graph $d(e,f)=1$ and $d(e,g)=2$.
• Figure 2: At the time of arrival of edge $e = (u,v)$, some other edges have already arrived, while some (dashed) edges will arrive in the future. Edge $e$ can affect the potential $\Phi_{w}$ stored at $w$. In order to compute $\Phi_{w}(t)$, and see how it is affected by different choices of coloring $e$, all information on edges in the 3-hop of $w$ (and thus $5$-hop of $e$) needs to be collected. For example, information stored at edge $f$ is necessary to make the coloring choice at $e$, while information on edge $g$ is not necessary.
• Figure 3: The lower bound construction. Edges in the bottom layer arrive first. The $2$-hop of an edge $e$ in the bottom layer will not see any of the other stars. Thus, the viewpoints of edges $e$ and $f$ at the bottom looks identical, and decisions for coloring, say edge $e$, only depends on colors used on edges already arrived at the star containing $e$.
• Figure 4: A distance-$3$ coloring. Note that $e$ and $i$ can have the same color, while each other edge has a different color. In a distance-$1$ coloring (also edge coloring): two colors would suffice: $g$ could also be red and $f$ and $h$ could both be green.
• Figure 5: The edges $e$ and $f$ are within distance $5$: $d(e,f)=5$. During a color try, they need to check wether they have the same color.

Theorems & Definitions (78)

• Theorem 1
• Corollary 1
• Theorem 2
• Theorem 3
• Definition
• Theorem 4
• Theorem 5
• Theorem 6
• Remark 7
• Corollary 7