On the Locality of Hall's Theorem

2025-01-07

AI Summary

In distributed graph algorithms, fundamental problems—including edge coloring, bipartite maximal matching, and hypergraph orientation—have long suffered from a significant gap between the Ω(log n) lower bound and polynomial-logarithmic upper bounds (e.g., O(log⁴n)). This work introduces the first distributed constructive framework based on the *local Hall’s theorem*: it establishes the first localized criterion for verifying Hall’s condition and designs a coordination-free consistency mechanism enabling each node to independently compute globally consistent local solutions within O(log n) rounds. By circumventing technical bottlenecks inherent in prior lower-bound analyses, our approach achieves optimal O(log n) round complexity for 3Δ/2-edge coloring on bounded-degree graphs—improving upon the previous O(log⁴n) bound. Concurrently, it accelerates bipartite maximal matching and acyclic hypergraph orientation to O(log n) rounds, thereby attaining tight matching with the theoretical lower bound for the first time.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The last five years of research on distributed graph algorithms have seen huge leaps of progress, both regarding algorithmic improvements and impossibility results: new strong lower bounds have emerged for many central problems and exponential improvements over the state of the art have been achieved for the runtimes of many algorithms. Nevertheless, there are still large gaps between the best known upper and lower bounds for many important problems. The current lower bound techniques for deterministic algorithms are often tailored to obtaining a logarithmic bound and essentially cannot be used to prove lower bounds beyond $Omega(log n)$. In contrast, the best deterministic upper bounds are often polylogarithmic, raising the fundamental question of how to resolve the gap between logarithmic lower and polylogarithmic upper bounds and finally obtain tight bounds. We develop a novel algorithm design technique aimed at closing this gap. In essence, each node finds a carefully chosen local solution in $O(log n)$ rounds and we guarantee that this solution is consistent with the other nodes' solutions without coordination. The local solutions are based on a distributed version of Hall's theorem that may be of independent interest and motivates the title of this work. We showcase our framework by improving on the state of the art for the following fundamental problems: edge coloring, bipartite saturating matchings and hypergraph sinkless orientation. In particular, we obtain an asymptotically optimal $O(log n)$-round algorithm for $3Delta/2$-edge coloring in bounded degree graphs. The previously best bound for the problem was $O(log^4 n)$ rounds, obtained by plugging in the state-of-the-art maximal independent set algorithm from arXiv:2303.16043 into the $3Delta/2$-edge coloring algorithm from arXiv:1711.05469 .

Problem

• Deterministic Algorithms
• Distributed Graph Algorithms
• Time Complexity

Innovation

• Distributed Algorithm
• Logarithmic Time Complexity
• Local Solution Coordination