LCLs Beyond Bounded Degrees

TL;DR

This work investigates local labelings on trees with unbounded degrees, focusing on the polynomial-time LOCAL complexity regime. It shows that naive generalizations of LCLs to unbounded degrees eliminate polynomial gaps, but by introducing Locally Finite Labelings (LFLs) which restrict configurations to a finite set while allowing unbounded degrees, polynomial gaps are preserved: any LFL Π on trees yields deterministic LOCAL complexity that is either Θ(n^{1/k}) for some integer k≥1 or O(log n), and the applicable case is computable from Π’s description. The authors develop a robust framework—including a node–edge NEC formalism, the TwigLFL radius-1 reduction, and a pumping-based gap analysis with independent classes and virtual trees—to extend classical gap results from bounded-degree LCLs to unbounded-degree trees. Key technical contributions include a radius-reduction technique that preserves correctness via twig certificates, a finite-types machinery for boundary analysis on trees, and a decomposition-based approach that yields near-optimal algorithms in the polynomial regime. The results delineate the boundary between expressive generalizations that destroy gaps and natural finite-configuration restrictions (LFLs) that restore them, offering a principled path toward understanding distributed complexity on unbounded-degree trees and guiding future work in the lower-time regimes and Δ-dependence. In short, LFLs recover the clean polynomial-gap landscape on unbounded-degree trees, while more expressive generalizations fail to do so; the main theorems provide a decidable dichotomy and concrete algorithmic consequences for distributed graph labeling problems.

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 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, 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 polynomial regime ($Θ(n^{1/k} \mid k \in \mathbb{N})$) and show that whether polynomial gap results persist in the unbounded-degree setting crucially depends on how LCLs are generalized beyond bounded degrees. We first demonstrate that if one allows LCLs to be defined using infinitely many local configurations, then the polynomial gaps disappear entirely: for every real exponent $0 < r \leq 1$, there exists a locally checkable problem on trees with deterministic LOCAL complexity $Θ(n^r)$. Rather than stopping at this negative result, we identify 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 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 $Π$.

Figures (16)

• Figure 1: The deterministic complexity landscape of LCLs on bounded-degree trees. Every such LCL has one of the depicted complexities; no others are possible. In the randomized setting, the class $\Theta(\log\log n)$ also appears.
• Figure 2: These two configurations encode the problem of finding an MIS. Optional edges are illustrated using dashed lines and required edges are solid lines. Any node that outputs 0 must have at least one neighbor that outputs 1 and can have arbitrarily many more. On the other side any node that outputs 1 cannot have another neighbor that outputs 1, but arbitrarily many neighbors that output 0. Therefore this correctly encodes the problem of finding an MIS.
• Figure 3: These three configurations suffice to encode the problem of computing a 3-coloring in unbounded degree graphs. The arrows identify the center nodes and the deges are dashed to identify them as optional edges (there are no required edges for this problem).
• Figure 4: Illustration of the radius-reduction construction. The leftmost panel shows an instance of our LFL, with the set $A$ of nodes at distance $r\!-\!1$ from $v$ highlighted in purple. The node $v$ appears as a twig $u_i$ in each of the four configurations $C_1, C_2, C_3, C_4$. The second panel depicts the resulting configuration $C_X$ of the twigLFL, obtained by combining the four twig configurations $C_{u_1}, C_{u_2}, C_{u_3}, C_{u_4}$. The rightmost panel shows how $C_X$ is combined with the original configuration $C_v$ to obtain a version of the initial LFL augmented with certificates. In the final configuration, each node outputs both its original label and a certificate encoding the twig configurations it can match (a placeholder '*' is put everywhere, except the center node). In particular, the node $v$ outputs the set $X$, certifying that it can match all twig configurations $C_{u_1}, C_{u_2}, C_{u_3}, C_{u_4}$simultaneously, while still correctly solving the original problem.
• Figure 5: By cutting out a piece of the graph at the ends of some path $P$, splits the tree into three parts $T_s, H,T_t$. The parts are connected by the edges $e_s$ and $e_t$.

Theorems & Definitions (107)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 3
• Definition 4: $\alpha$-path
• proof
• Lemma 6
• proof
• Lemma 7
• proof