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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.02340 |
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| _version_ | 1866912905819062272 |
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| author | Schmid, Gustav |
| author_facet | Schmid, Gustav |
| contents | 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 $Π$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02340 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | LCLs Beyond Bounded Degrees Schmid, Gustav Distributed, Parallel, and Cluster Computing 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 $Π$. |
| title | LCLs Beyond Bounded Degrees |
| topic | Distributed, Parallel, and Cluster Computing |
| url | https://arxiv.org/abs/2602.02340 |