A rich line of work has been addressing the computational complexity of locally checkable labelings (\LCL{}s), illustrating the landscape of possible complexities. In this paper, we study the landscape of \LCL complexities under bandwidth restrictions. Our main results are twofold. First, we show that on trees, the \CONGEST complexity of an \LCL problem is asymptotically equal to its complexity in the \LOCAL model. An analog statement for general (non-\LCL) problems is known to be false. Second, we show that for general graphs this equivalence does not hold, by providing an \LCL problem for which we show that it can be solved in $O(\log n)$ rounds in the \LOCAL model, but requires $\tilde{\Omega}(n^{1/2})$ rounds in the \CONGEST model.

翻译：处理本地可核对标签的计算复杂性(\ LCL ⁇ s) 是一项内容丰富的工作, 说明了可能的复杂情况。 在本文中, 我们研究了带宽限制下的 LCL 复杂性。 我们的主要结果有双重。 首先, 我们显示在树上,\ LCOL 问题的复杂性与 \ LCOL 模型中的复杂程度无异。 一般( 非 LCL ⁇ s) 问题的类比说明是假的。 第二, 我们通过提供 \ LCL 问题来显示, 对于一般图表来说, 这一等值并不有效, 我们显示, 在 \ LCOL 模型中, 它可以用 $( log n) 来解决, 但是在\ CONEST 模型中, 需要$\ tilde\ Omega} (n\ 1/2} 。