Abstract

By prior work, it is known that any distributed graph algorithm that finds a maximal matching requires $\Omega(\log^* n)$ 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 $\{0, \frac12, 1\}$. 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 $\Delta = 2d$, and any distributed graph algorithm with round complexity $T(\Delta)$ that only depends on $\Delta$ and is independent of $n$, we show that the algorithm has to use fractional values with a denominator at least $2^d$. We give a new algorithm that shows that this is also sufficient.

Publication

Sergio Rajsbaum, Alkida Balliu, Joshua J. Daymude, and Dennis Olivetti (Eds.): Structural Information and Communication Complexity, 30th International Colloquium, SIROCCO 2023, Alcalá de Henares, Spain, June 6–9, 2023, Proceedings, volume 13892 of Lecture Notes in Computer Science, pages 339–356, Springer, Berlin, 2023

ISBN 978-3-031-32733-9

Journal Version

• Sameep Dahal and Jukka Suomela: Distributed half-integral matching and beyond · Theoretical Computer Science, to appear