  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.
