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.

翻译：根据先前研究，已知任何求解最大匹配的分布式图算法都需要Ω(log* n)轮通信，而在有界度图中可以在O(1)轮内找到最大分数匹配。然而，所有先前的O(1)轮最大分数匹配算法都使用了任意精细的分数值。特别地，没有一个算法能仅使用{0, 1/2, 1}中的值找到半整数解。我们证明了使用精细分数值的必要性，并进一步给出了对所需最小数值的完整刻画：若考虑最大度为Δ = 2d的图中的最大分数匹配，以及任意轮复杂度T(Δ)仅依赖于Δ且与n无关的分布式图算法，我们证明该算法必须使用分母至少为2^d的分数值。我们提出了一种新算法，表明这一下界也是充分的。