LCLs Beyond Bounded Degrees

TL;DR

This paper investigates the complexity landscape of Locally Checkable Labelings (LCLs) on trees with unbounded degrees, introducing Locally Finite Labelings (LFLs) to recover polynomial gap results previously lost in unbounded degree settings.

Contribution

It defines LFLs as a class of problems where nodes fall into finitely many local cases, restoring polynomial complexity gaps for LCLs on unbounded degree trees.

Findings

Polynomial gaps vanish for general LCLs on unbounded degree trees.

Introducing LFLs allows polynomial gaps to be restored.

Complexity is either Θ(n^{1/k}) or O(log n), with k determined by the problem description.

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} ($\Theta(n^{1/k} \mid k \in \mathbb{N})$) 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 $\Pi$ on trees with unbounded degrees, the deterministic LOCAL complexity of $\Pi$ is either - $\Theta(n^{1/k})$ 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 $\Pi$.