Sameep Dahal · Jukka Suomela

Distributed half-integral matching and beyond

Theoretical Computer Science · to appear · doi:10.1016/j.tcs.2023.114278

authors’ version publisher’s version


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.

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