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Communication complexity of approximate maximum matching in the message-passing model

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Abstract

We consider the communication complexity of finding an approximate maximum matching in a graph in a multi-party message-passing communication model. The maximum matching problem is one of the most fundamental graph combinatorial problems, with a variety of applications. The input to the problem is a graph G that has n vertices and the set of edges partitioned over k sites, and an approximation ratio parameter \(\alpha \). The output is required to be a matching in G that has to be reported by one of the sites, whose size is at least factor \(\alpha \) of the size of a maximum matching in G. We show that the communication complexity of this problem is \(\varOmega (\alpha ^2 k n)\) information bits. This bound is shown to be tight up to a \(\log n\) factor, by constructing an algorithm, establishing its correctness, and an upper bound on the communication cost. The lower bound also applies to other graph combinatorial problems in the message-passing communication model, including max-flow and graph sparsification.

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Notes

  1. \(\mu _k(b)\) is the marginal distribution of b of the joint distribution \(\mu _k\) (see Sect. 3.2 for the definition)

  2. Since none of the sites can see messages sent by other sites to the coordinator (unless this is communicated by the coordinator), each site needs to communicate with the coordinator at least once to determine the status of the protocol.

  3. The constants used here are slightly different from [18].

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Acknowledgements

The funding was provided by Science and Technology Commission of Shanghai Municipality (Grant No. 17JC1420200), National Natural Science Foundation of China (Grant No. 61802069), National Science Foundation (Grant Nos. CCF-1525024, IIS-1633215) and Shanghai Sailing Program (Grant No. 18YF1401200).

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Authors and Affiliations

  1. School of Data Science, Fudan University, Shanghai, China

    Zengfeng Huang

  2. Microsoft Research, Cambridge, UK

    Bozidar Radunovic

  3. Department of Statistics, London School of Economics (LSE), London, UK

    Milan Vojnovic

  4. Computer Science Department, Indiana University, Bloomington, USA

    Qin Zhang

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  1. Zengfeng Huang
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  2. Bozidar Radunovic
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  3. Milan Vojnovic
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  4. Qin Zhang
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Correspondence to Zengfeng Huang.

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This paper is a revised and extended version of a paper by the same authors that appeared in the Proceedings of the 32nd Symposium on Theoretical Computer Science (STACS), Munich, Germany, March 4–7, 2015.

Q. Zhang: Supported in part by NSF IIS-1633215 and CCF-1844234.

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Huang, Z., Radunovic, B., Vojnovic, M. et al. Communication complexity of approximate maximum matching in the message-passing model. Distrib. Comput. 33, 515–531 (2020). https://doi.org/10.1007/s00446-020-00371-6

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