LCLs Beyond Bounded Degrees

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 [Alkida Balliu et al., 2021; Balliu et al., 2019; Christoph Grunau et al., 2022; Yi-Jun Chang, 2020; Yi-Jun Chang et al., 2019; Yi-Jun Chang and Seth Pettie, 2019]. A central feature of this complexity landscape is the existence of strong 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 [Yi-Jun Chang, 2020], this has turned out not to be the case [Balliu et al., 2018]. In this work, we investigate a different direction: we remain in the class of trees, but allow arbitrarily large degrees.
We focus on the polynomial regime, i.e. complexities of the form Θ(n^{1/k}) for k ∈ ℕ, and show that whether polynomial gap results persist in the unbounded-degree setting crucially depends on how LCLs are generalized beyond bounded degrees. 
A complex construction in [Bousquet et al., 2025] 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 problem definitions to distinguish infinitely many local cases.
This guides us to a natural class of problems for which polynomial gap results can still be recovered. We introduce Locally Finite Labelings (LFLs), which formalize the intuition that every node must fall into one of finitely many local cases, even in the presence of unbounded degrees.
Our main result shows that this restriction is sufficient to restore the polynomial gaps: for any LFL Π on trees of unbounded degrees, the deterministic LOCAL complexity of Π is either  
- Θ(n^{1/k}) for some integer k ≥ 1, or 
- O(log n).