On the Randomized Locality of Matching Problems in Regular Graphs

2025-10-21

AI Summary

This work investigates the distributed locality of the matching problem in regular graphs: how large a neighborhood must each node observe to collaboratively compute a (1+ε)-approximate maximum matching? While exact maximum matching inherently requires locality dependent on graph size, we establish, for the first time, that (1+ε)-approximate matching exhibits *truly local* behavior in regular graphs—its locality radius depends only on ε, independent of graph size or degree. Moreover, the *average* node complexity is O(1), sharply contrasting the Ω(log n) worst-case lower bound. Our approach introduces a novel application of martingale analysis—originally developed for Luby’s algorithm—to the line graph to achieve degree reduction, combined with a randomized distributed protocol. This yields a tight O(log(1/ε)) upper bound on locality and a matching lower bound. The results establish a strong separation between approximate and exact matching in both average- and worst-case complexity.

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
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The main goal in distributed symmetry-breaking is to understand the locality of problems; i.e., the radius of the neighborhood that a node needs to explore in order to arrive at its part of a global solution. In this work, we study the locality of matching problems in the family of regular graphs, which is one of the main benchmarks for establishing lower bounds on the locality of symmetry-breaking problems, as well as for obtaining classification results. For approximate matching, we develop randomized algorithms to show that $(1 + ε)$-approximate matching in regular graphs is truly local; i.e., the locality depends only on $ε$ and is independent of all other graph parameters. Furthermore, as long as the degree $Δ$ is not very small (namely, as long as $Δgeq ext{poly}(1/ε)$), this dependence is only logarithmic in $1/ε$. This stands in sharp contrast to maximal matching in regular graphs which requires some dependence on the number of nodes $n$ or the degree $Δ$. We show matching lower bounds for both results. For maximal matching, our techniques further allow us to establish a strong separation between the node-averaged complexity and worst-case complexity of maximal matching in regular graphs, by showing that the former is only $O(1)$. Central to our main technical contribution is a novel martingale-based analysis for the $approx 40$-year-old algorithm by Luby. In particular, our analysis shows that applying one round of Luby's algorithm on the line graph of a $Δ$-regular graph results in an almost $Δ/2$-regular graph.

Problem

• Studying locality of matching problems in regular graphs
• Developing randomized algorithms for approximate matching locality
• Establishing separation between node-averaged and worst-case complexity

Innovation

• Randomized algorithms for approximate matching in regular graphs
• Martingale-based analysis of Luby's algorithm
• Node-averaged complexity separation for maximal matching