Understanding the role of randomness when solving locally checkable labeling (LCL) problems in the LOCAL model has been one of the top priorities in the research on distributed graph algorithms in recent years. For LCL problems in bounded-degree graphs, it is known that randomness cannot help more than polynomially, except in one case: if the deterministic complexity of an LCL problem is in $\Omega(\log n)$ and its randomized complexity is in $o(\log n)$, then the randomized complexity is guaranteed to be $poly(\log \log n)$. However, the fundamental question of \emph{which} problems with a deterministic complexity of $\Omega(\log n)$ can be solved exponentially faster using randomization still remains wide open. We make a step towards answering this question by studying a simple, but natural class of LCL problems: so-called degree splitting problems. These problems come in two varieties: coloring problems where the edges of a graph have to be colored with $2$ colors and orientation problems where each edge needs to be oriented. For $\Delta$-regular graphs (where $\Delta=O(1)$), we obtain the following results. - We gave an exact characterization of the randomized complexity of all problems where the edges need to be colored with two colors, say red and blue, and which have a deterministic complexity of $O(\log n)$. - For edge orientation problems, we give a partial characterization of the problems that have a randomized complexity of $\Omega(\log n)$ and the problems that have a randomized complexity of $poly\log\log n$. While our results are cleanest to state for the $\Delta$-regular case, all our algorithms naturally generalize to nodes of any degree $d<\Delta$ in general graphs of maximum degree $\Delta$.

翻译：理解随机性在LOCAL模型中求解局部可检查标记（LCL）问题时所扮演的角色，已成为近年来分布式图算法研究的核心议题之一。对于有界度图中的LCL问题，已知随机性的加速效果不会超过多项式级别，仅存在一种例外情况：若某LCL问题的确定性复杂度为$\Omega(\log n)$而随机复杂度为$o(\log n)$，则可保证其随机复杂度为$poly(\log \log n)$。然而，对于具有$\Omega(\log n)$确定性复杂度的问题中，\emph{哪些}能通过随机化获得指数级加速这一根本问题，至今仍悬而未决。我们通过研究一类简单而自然的LCL问题——度分割问题，向解答该问题迈进一步。这类问题包含两种变体：需要将图边着以$2$种颜色的着色问题，以及需要为每条边确定方向的定向问题。针对$\Delta$-正则图（其中$\Delta=O(1)$），我们获得以下结果：- 对于需要将边着以红蓝两色且确定性复杂度为$O(\log n)$的所有问题，我们给出了其随机复杂度的精确刻画。- 对于边定向问题，我们部分刻画了具有$\Omega(\log n)$随机复杂度的问题与具有$poly\log\log n$随机复杂度的问题。虽然我们的结论在$\Delta$-正则情形下表述最为简洁，但所有算法均可自然推广至最大度为$\Delta$的一般图中任意度$d<\Delta$的节点。