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On the Randomized Locality of Matching Problems in Regular Graphs

Authors Seri Khoury, Manish Purohit , Aaron Schild, Joshua R. Wang

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ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Mathematics of computing → Probabilistic algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • regular graphs
  • maximum matching
  • augmenting paths
  • distributed algorithms
  • Luby’s algorithm
  • martingales

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Abstract

The main goal in distributed symmetry-breaking is to understand the locality of problems: the radius of the neighborhood that a node must explore to determine 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. Our main results are summarized as follows:  
1) 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 Δ ≥ 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. 
2) 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 ≈ 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.

Cite As Get BibTex

Seri Khoury, Manish Purohit, Aaron Schild, and Joshua R. Wang. On the Randomized Locality of Matching Problems in Regular Graphs. In 39th International Symposium on Distributed Computing (DISC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 356, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.DISC.2025.40

Author Details

Seri Khoury
  • INSAIT, Sofia University "St. Kliment Ohridski", Bulgaria
Manish Purohit
  • Google Research, Mountain View, CA, USA
Aaron Schild
  • Google Research, San Francisco, CA, USA
Joshua R. Wang
  • Google Research, San Francisco, CA, USA

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