Towards Optimal Distributed Delta Coloring

2025-04-03

AI Summary

Δ-vertex coloring is a fundamental problem in distributed graph algorithms on constant-degree graphs, where a long-standing polynomial gap existed between the deterministic time complexity lower bound of Ω(log n) (Chang et al., FOCS’16) and the best prior upper bounds. This work presents the first deterministic Δ-coloring algorithm achieving O(log n) rounds on dense constant-degree graphs—matching the lower bound tightly. The key innovation is a novel deterministic coloring framework tailored to dense subgraphs, departing from the conventional sparse-dense decomposition paradigm. It integrates structural graph analysis, deterministic message passing, and hierarchical vertex grouping, and further yields a randomized acceleration variant. As corollaries, we obtain two theoretically optimal randomized algorithms with round complexities Õ(log^{5/3} log n) and O(Δ + log log n), respectively.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Distributed Search

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 $Delta$-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 $Delta$-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 $Delta$-coloring algorithm for dense constant-degree graphs, matching the lower bound for the problem on general graphs. For general $Delta$ the algorithms' complexity is $min{widetilde{O}(log^{5/3}n),O(Delta+log n)}$. All recent distributed and sublinear graph coloring algorithms (also for coloring with more than $Delta$ 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) $Delta$-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 $min{widetilde{O}(log^{5/3}log n), O(Delta+loglog n)}$-round algorithm for $Delta$-coloring dense graphs that also matches the lower bound for the problem on general constant-degree graphs [Brandt et al.; STOC'16].

Problem

• Solve Delta coloring in logarithmic time
• Address complexity gap in distributed algorithms
• Develop deterministic algorithm for dense graphs

Innovation

• Deterministic O(log n)-round Δ-coloring for dense graphs
• Minimizes complexity to min{Õ(log⁵/³n), O(Δ+log n)}
• Novel approach for dense parts, extendable to sparse