[论文解读] LCLs Beyond Bounded Degrees

G. H. SCHMID|arXiv (Cornell University)|Feb 2, 2026

Advanced Graph Theory Research被引用 0

一句话总结

论文表明在有界度数树上局部可检标签（LCL）的多项式间隙结果在无界度数时失败，除非将其限制为局部有限标签（LFLs）；随后证明在无界度数树上，LFLs 能恢复多项式间隙现象。

ABSTRACT

The study of Locally Checkable Labelings (LCLs) has led to a remarkably precise characterization of the distributed time complexities that can occur on bounded-degree trees. A central feature of this complexity landscape is the existence of gap results, which rule out large ranges of intermediate complexities. While it was initially hoped that these gaps might extend to more general graph classes, this has turned out not to be the case. In this work, we investigate a different direction: we remain in the class of trees, but allow arbitrarily large degrees. We focus on the \emph{polynomial regime} ($Θ(n^{1/k} \mid k \in \mathbb{N})$) and show that whether polynomial gap results persist in the unbounded-degree setting depends on how LCLs are generalized beyond bounded degrees. There already exists a complex construction that shows that the polynomial gaps also vanish for LCLs on unbounded degree trees. Rather than stopping at this negative result, we give a much simpler set of problems that already contradicts the existence of any polynomial gaps. The insight obtained from this cleaner construction is that for gap results to exist, we cannot allow problems to distinguish infinitely many cases. This guides us to a natural class of problems for which polynomial gap results can still be recovered. We introduce \emph{Locally Finite Labelings} (LFLs), which formalize the intuition that \emph{every node must fall into one of finitely many local cases}. Our main result shows that this restriction is sufficient to restore the polynomial gaps: for any LFL $Π$ on trees with unbounded degrees, the deterministic LOCAL complexity of $Π$ is either - $Θ(n^{1/k})$ for some integer $k \geq 1$, or - $O(\log n)$. Moreover, which case applies, and the corresponding value of $k$, can be determined solely from the description of $Π$.

研究动机与目标

• 推动在有界度数树之外研究 LCLs 的动机，并理解为何在无界度数下间隙结果可能失效。
• 在无界度数设置中识别一个自然的限制条件，以保持多项式时间的间隙。
• 将 LFL 框架发展为一种有限配置描述，适用于无界度数同时保持可控分析。

提出的方法

• 引入使用有限配置集合、边被标记为必需或可选的 Locally Finite Labelings (LFLs)。
• 通过分析将半径降至1，显示 LFLs 与节点-边可检形式之间的等价性。
• 开发一种半径降维技巧，利用 twig 证明来逐步把检查半径从 r 降到 r-1。
• 使用虚拟树来有限表示无界度数树的无限族，并应用泵送论证来约束分析。
• 为树上的 LFLs 证明一个间隙定理：确定性 LOCAL 复杂度要么是 Theta(n^{1/k})，其中 k>=1，或是 O(log n)。
• 提供一个与现有 LCL 间隙结果一致的构造性框架，同时处理无界度数。

Figure 3: These three configurations suffice to encode the problem of computing a 3-colouring in unbounded degree graphs. The arrows identify the center nodes and the edged are dashed to identify them as optional edges (there are no required edges for this problem).

实验结果

研究问题

• RQ1 无界度数树上，已观察到的有界度数树上的 LCL 间隙多项式结果是否可以扩展？
• RQ2 在无界度数树上，哪些对 LCL 的泛化允许或阻止丰富的多项式间隙结构？
• RQ3 Locally Finite Labelings (LFLs) 是否在无界度数设置中保留多项式间隙，并且如何在计算上表征？
• RQ4 如何利用半径降维和节点-边可检形式，将 LCL 技巧迁移到 LFLs？
• RQ5 边界类型和独立类在建立 LFLs 的多项式时间间隙中的作用是什么？

主要发现

• 存在一些可用 {0,1} 标签的 4-可检问题，在无界度数树上复杂度为 Theta(n^{r})，任意 0<r<=1（定理 1）。
• Locally Finite Labelings 将 LCLs 推广到无界度数，同时保持有限性条件，防止退化构造。
• 对于树上的 LFLs，确定性 LOCAL 复杂度为 Theta(n^{1/k})，其中某个整数 k>=1，或为 O(log n)（定理 2）。
• 每个 LFL 都可以在不改变可解性或 LOCAL 复杂度的前提下，被转化为从节点-边可检 LFL，反之亦然（定理 3）。
• 有限集合的虚拟树足以捕捉无界度数树中的边界行为，使得通过泵送/独立类分析得到多项式间隙：
• 研究指出半径降维（降到半径 1）和 twig 证明确为将 LCL 风格的间隙结果扩展到超出有界度数的关键技术工具。

Figure 9: The new configuration $C_{2}^{\prime}$ that $u$ can use to convince $v$ that the output is consistent. The nodes in red are the subgraph that is required in order for $v$ in Figure 8 to be convinced of the correctness of the solution.