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：先前的研究表明，任何寻找最大匹配的分布式图算法都需要$\Omega(\log^* n)$通信轮次，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细粒度的分数值。特别地，它们都无法仅使用$\{0, \frac12, 1\}$中的值找到半整数解。我们证明了使用细粒度分数值的必要性，并进一步给出了所需最小值的完整刻画：若考虑最大度为$\Delta = 2d$的图中的最大分数匹配，且任何轮复杂度$T(\Delta)$仅依赖于$\Delta$而与$n$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，表明这一条件是充分且必要的。