Distributed Half-Integral Matching and Beyond

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Date
2023
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en
Pages
18
339-356
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Structural Information and Communication Complexity - 30th International Colloquium, SIROCCO 2023, Proceedings, Lecture Notes in Computer Science, Volume 13892
Abstract
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 {0,12,1}. 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.
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Dahal , S & Suomela , J 2023 , Distributed Half-Integral Matching and Beyond . in S Rajsbaum , S Rajsbaum , A Balliu , D Olivetti & J J Daymude (eds) , Structural Information and Communication Complexity - 30th International Colloquium, SIROCCO 2023, Proceedings . Lecture Notes in Computer Science , vol. 13892 , Springer , pp. 339-356 , International Colloquium on Structural Information and Communication Complexity , Madrid , Spain , 06/06/2023 . https://doi.org/10.1007/978-3-031-32733-9_15