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Improved deterministic distributed matching via rounding

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Abstract

We present improved deterministic distributed algorithms for a number of well-studied matching problems, which are simpler, faster, more accurate, and/or more general than their known counterparts. The common denominator of these results is a deterministic distributed rounding method for certain linear programs, which is the first such rounding method, to our knowledge. A sampling of our end results is as follows:

  • An \(O\mathopen {}\left( \log ^2 \Delta \cdot \log n\right) \mathclose {}\)-round deterministic distributed algorithm for computing a maximal matching, in n-node graphs with maximum degree \(\Delta \). This is the first improvement in about 20 years over the celebrated \(O(\log ^4 n)\)-round algorithm of Hańćkowiak, Karoński, and Panconesi [SODA’98, PODC’99].

  • A deterministic distributed algorithm for computing a \((2+\varepsilon )\)-approximation of maximum matching in \(O\mathopen {}\left( \log ^2 \Delta \cdot \log \frac{1}{\varepsilon } + \log ^ * n\right) \mathclose {}\) rounds. This is exponentially faster than the classic \(O(\Delta +\log ^* n)\)-round 2-approximation of Panconesi and Rizzi [DIST’01]. With some modifications, the algorithm can also find an almost maximal matching which leaves only an \(\varepsilon \)-fraction of the edges on unmatched nodes.

  • An \(O\mathopen {}\left( \log ^2 \Delta \cdot \log \frac{1}{\varepsilon } \cdot \log _{1+\varepsilon } W + \log ^ * n\right) \mathclose {}\)-round deterministic distributed algorithm for computing a \((2+\varepsilon )\)-approximation of a maximum weighted matching, and also for the more general problem of maximum weighted b-matching. Here, W denotes the maximum normalized weight. These improve over the \(O\mathopen {}\left( \log ^4 n \cdot \log _{1+\varepsilon } W\right) \mathclose {}\)-round \((6+\varepsilon )\)-approximation algorithm of Panconesi and Sozio [DIST’10].

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Notes

  1. The accepted standard for efficient is \({\text {{{poly}}}}\log n\) rounds.

  2. For instance, our improvement in the deterministic complexity of maximal matching directly improves the randomized complexity of maximal matching, as we formally state in Corollary 1.3.

  3. Stating this result formally and in full generality requires some definitions. See [8] for the precise statement.

  4. As standard, with high probability indicates a probability at least \(1-1/n^{c}\), for a desirably large constant \(c\ge 2\).

  5. Any fractional maximum matching can be transformed to this format, with at most a 2-factor loss in the total value: simply round down each value to the next power of 2, and then drop edges with values below \(2^{-(\lceil \log {\Delta }\rceil +1)}\).

  6. This simple idea has been used frequently before. For instance, it gives an almost trivial proof of Petersen’s 2-factorization theorem from 1891 [26]. It has also been used by [11, 12, 17].

  7. Our algorithm actually does something slightly different, but describing this ideal procedure is easier.

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Acknowledgements

I want to thank Mohsen Ghaffari for suggesting this topic, for his guidance and his support, as well as for the many valuable and enlightening discussions. I am also thankful to Seth Pettie for several helpful comments.

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  1. ETH Zurich, Zurich, Switzerland

    Manuela Fischer

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  1. Manuela Fischer
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Fischer, M. Improved deterministic distributed matching via rounding. Distrib. Comput. 33, 279–291 (2020). https://doi.org/10.1007/s00446-018-0344-4

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