Distributed half-integral matching and beyond

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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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Theoretical Computer Science, Volume 982
By prior work, it is known that any distributed graph algorithm that finds a maximal matching requires Ω(log⁎⁡n) communication rounds, while it is possible to find a maximal fractional matching in O(1) rounds in bounded-degree graphs. However, all prior O(1)-round algorithms for maximal fractional matching use arbitrarily fine-grained fractional values. In particular, none of them is able to find a half-integral solution, using only values from [Formula presented]. We show that the use of fine-grained fractional values is necessary, and moreover we give a complete characterization on exactly how small values are needed: if we consider maximal fractional matching in graphs of maximum degree Δ=2d, and any distributed graph algorithm with round complexity T(Δ) that only depends on Δ and is independent of n, we show that the algorithm has to use fractional values with a denominator at least 2d. We give a new algorithm that shows that this is also sufficient.
Funding Information: This work was supported in part by the Research Council of Finland , Grant 333837 . We would like to thank the anonymous reviewers for their helpful feedback, and the members of Aalto Distributed Algorithms group for discussions. This is an extended version of a preliminary conference report [8] . Publisher Copyright: © 2023 The Author(s)
Computational complexity, Distributed graph algorithms, Fractional matching, Maximal matching
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Dahal , S & Suomela , J 2024 , ' Distributed half-integral matching and beyond ' , Theoretical Computer Science , vol. 982 , 114278 . https://doi.org/10.1016/j.tcs.2023.114278