Distributed $(Δ+1)$-Coloring in Graphs of Bounded Neighborhood Independence

TL;DR

The paper tackles deterministic distributed $(\Delta+1)$-coloring in networks with bounded neighborhood independence $\theta$, aiming to close the gap in $\,\Delta$-dependence. It develops a recursive framework around list arbdefective coloring, slack generation, and color-space reduction to color with $\Delta+1$ colors in time $$(\theta\cdot\log\Delta)^{O(\log\log\Delta/\log\log\log\Delta)}+O(\log^* n)$$, which becomes quasipolynomial in $\Delta$ when $\theta$ is polylogarithmic. A core technical contribution is extending list arbdefective coloring to small-$\theta$ settings, enabling efficient reduction of the color space while maintaining feasibility, and coupling this with a lower bound for defective 2-edge coloring in hypergraphs to delineate the limits of certain strategies. The results imply faster deterministic $\Delta+1$-coloring for graphs like line graphs of bounded-rank hypergraphs and related families, while the defeated lower bound shows the limitations of defect-based reductions for hypergraph edge-coloring. Overall, the work advances understanding of the $\Delta$-dependency in distributed coloring and offers a versatile toolkit for near-optimal deterministic coloring in structured graph classes.

Abstract

The distributed coloring problem is arguably one of the key problems studied in the area of distributed graph algorithms. The most standard variant of the problem asks for a proper vertex coloring of a graph with $Δ+1$ colors, where $Δ$ is the maximum degree of the graph. Despite an immense amount of work on distributed coloring problems in the distributed setting, determining the deterministic complexity of $(Δ+1)$-coloring in the standard message passing model remains one of the most important open questions of the area. In this paper, we aim to improve our understanding of the deterministic complexity of $(Δ+1)$-coloring as a function of $Δ$ in a special family of graphs for which significantly faster algorithms are already known. The neighborhood independence $θ$ of a graph is the maximum number of pairwise non-adjacent neighbors of some node of the graph. In general, in graphs of neighborhood independence $θ=O(1)$ (e.g., line graphs), it is known that $(Δ+1)$-coloring can be solved in $2^{O(\sqrt{\logΔ})}+O(\log^* n)$ rounds. In the present paper, we significantly improve this result, and we show that in graphs of neighborhood independence $θ$, a $(Δ+1)$-coloring can be computed in $(θ\cdot\logΔ)^{O(\log\logΔ/ \log\log\logΔ)}+O(\log^* n)$ rounds and thus in quasipolylogarithmic time in $Δ$ as long as $θ$ is at most polylogarithmic in $Δ$. We also show that the known approach that leads to a polylogarithmic in $Δ$ algorithm for $(2Δ-1)$-edge coloring already fails for edge colorings of hypergraphs of rank at least $3$.

Figures (1)

• Figure 1: The recursive structure for solving list arbdefective coloring instances $P_A(2,C)$ with slack $\geq 2$ and colors from a space of size $C$. This observation leads to the statement $T_A(2, C) \leq O(\theta^2) \cdot (\log^3 \Delta \cdot T_A(2, C^\varepsilon) + T_A(2, C^{1-\varepsilon}))$ (cf. \ref{['eq:splitting']}). Taking either $\varepsilon = O(\log \log \log \Delta/ \log \log \Delta)$ or $\varepsilon = 1/2$ (depending on the value of $\theta$) results in the complexity claimed in \ref{['thm:mainNeighborhood']}.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: Kuhn2009KawarabayashiS18
• Definition 2.1: Slack
• Lemma 2.1
• proof
• Lemma 2.1
• proof
• Theorem 2.2
• Lemma 3.1