发表机构
Gran Sasso Science Institute; Institut de Recherche en Informatique Fondamentale (IRIF) CNRS and Université Paris Cité; Departamento de Ingeniería Matemática, Universidad de Chile（格兰萨索科学研究所; 基础信息学研究所（IRIF）CNRS 与巴黎西岱大学; 智利大学数学工程系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究构造了适用于大度图的局部可检查问题，填补了LOCAL模型的轮复杂度间隙，扩展了相关结果并提出递增度问题与转换定理。
AI 中文摘要
我们证明，局部可检查标注（LCL）问题的轮复杂度间隙并非源于LCL问题的解必须可局部检查，而仅源于LCL问题仅针对最大度被某个任意常数Δ上界约束的图定义。具体而言，我们证明存在无穷多个局部可检查问题（即解可被局部检查的问题），当这些问题在最大度无界的网络中考虑时，其轮复杂度属于区间[ω(1),o(log log⁎n)]和[ω(log⁎n),o(log n)]。这扩展了Schmid（arXiv，2026）的先前结果，该结果仅适用于多项式范围，以及Bousquet、Feuilloley和Pierron（OPODIS，2025）的结果，该结果仅适用于树。我们的所有上界均使用确定性算法获得，这些算法可在端口编号模型（LOCAL模型的弱变体）下运行，且无需网络中节点数量的任何先验信息。相反，我们的下界适用于随机LOCAL和量子LOCAL，即使节点具有[1,n]范围内的标识符，即使它们知道网络中节点的确切数量，甚至适用于随机在线LOCAL（LOCAL模型的强变体），甚至对树也成立。我们的结果基于两个主要要素：第一个是对名为递增度（Increasing Degree）的新局部可检查问题的分析，该问题由函数f:ℕ→ℕ参数化，通过调整函数f可获得不同的轮复杂度；第二个工具是通用转换定理，该定理可将给定复杂度范围的结果转换为更低复杂度范围的结果。
英文摘要
We show that the complexity gaps in the round complexities of locally checkable labeling (LCL) problems are not due to the fact that solutions to LCL problems must be locally checkable, but solely to the fact that LCL problems are defined only for graphs of maximum degree upper bounded by some arbitrary yet constant value $Δ$. Specifically, we show that there are infinitely many locally checkable problems (i.e., problems whose solutions can be checked locally) whose round complexities belongs to the two intervals $[ω(1),o(\log\log^\star n)]$ and $[ω(\log^\star n),o(\log n)]$ whenever these problems are considered in networks with unbounded maximum degrees. This extends the previous results by Schmid (arXiv, 2026), which hold for the polynomial regime only, and by Bousquet, Feuilloley, and Pierron (OPODIS, 2025), which hold for trees only. All our upper bounds are obtained using deterministic algorithms that can be run under the port-numbering model, which is a weak variant of LOCAL, without any a priori information on the number of nodes in the network. Instead, our lower bounds apply to randomized LOCAL, and quantum LOCAL, even if nodes have identifiers in $[1,n]$, and even if they know the exact number of nodes in the network. They even hold under randomized online LOCAL, a strong variant of the LOCAL model. Finally, our lower bounds hold even for trees. Our results are obtained using two main ingredients. The first one is the analysis of a new locally checkable problem called Increasing Degree, parameterized by a function $f:\mathbb{N}\to\mathbb{N}$. Different round complexities can be obtained by tuning the function $f$ accordingly. Our second tool is a general Translation Theorem that enables to transfer results from a given range of complexities to results for a range of lower complexities.
Comments24 pages, 4 figures
