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

2025-10-24

AI Summary

This paper studies the time complexity of distributed (Δ+1)-vertex coloring in graphs with bounded neighborhood independence number θ. To overcome the exponential dependence on Δ in prior algorithms, we introduce θ into the complexity analysis and propose a deterministic distributed algorithm based on hierarchical color assignment and local synchronization. In the standard CONGEST model, our algorithm runs in $(θlogDelta)^{O(loglogDelta/logloglogDelta)} + O(log^* n)$ rounds; when θ = polylog(Δ), it achieves quasi-polylogarithmic time—significantly improving upon previous bounds. Our work advances the theoretical understanding of graph coloring for structured graph classes and establishes the first quantitative relationship between neighborhood independence and distributed coloring efficiency. This yields a new paradigm for efficient coloring of structured graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $(θcdotlogΔ)^{O(loglogΔ/ logloglogΔ)}+O(log^* n)$ 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 $3$.

Problem

• Determining deterministic complexity of distributed (Δ+1)-coloring in graphs
• Improving coloring algorithms for graphs with bounded neighborhood independence
• Developing faster deterministic coloring methods for special graph families

Innovation

• Distributed coloring algorithm for bounded neighborhood independence graphs
• Quasipolylogarithmic time complexity in maximum degree
• Improved deterministic complexity for Delta+1 coloring