Completing the Node-Averaged Complexity Landscape of LCLs on Trees
<|reference_start|>Completing the Node-Averaged Complexity Landscape of LCLs on Trees: The node-averaged complexity of a problem captures the number of rounds nodes of a graph have to spend on average to solve the problem in the LOCAL model. A challenging line of research with regards to this new complexity measure is to understand the complexity landscape of locally checkable labelings (LCLs) on families of bounded-degree graphs. Particularly interesting in this context is the family of bounded-degree trees as there, for the worst-case complexity, we know a complete characterization of the possible complexities and structures of LCL problems. A first step for the node-averaged complexity case has been achieved recently [DISC '23], where the authors in particular showed that in bounded-degree trees, there is a large complexity gap: There are no LCL problems with a deterministic node-averaged complexity between $\omega(\log^* n)$ and $n^{o(1)}$. For randomized algorithms, they even showed that the node-averaged complexity is either $O(1)$ or $n^{\Omega(1)}$. In this work we fill in the remaining gaps and give a complete description of the node-averaged complexity landscape of LCLs on bounded-degree trees. Our contributions are threefold.   - On bounded-degree trees, there is no LCL with a node-averaged complexity between $\omega(1)$ and $(\log^*n)^{o(1)}$.   - For any constants $0<r_1 < r_2 \leq 1$ and $\varepsilon>0$, there exists a constant $c$ and an LCL problem with node-averaged complexity between $\Omega((\log^* n)^c)$ and $O((\log^* n)^{c+\varepsilon})$.   - For any constants $0<\alpha\leq 1/2$ and $\varepsilon>0$, there exists an LCL problem with node-averaged complexity $\Theta(n^x)$ for some $x\in [\alpha, \alpha+\varepsilon]$.<|reference_end|>
@article{balliu2024completing,
    title={Completing the Node-Averaged Complexity Landscape of LCLs on Trees},
    author={Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Dennis Olivetti, Gustav
  Schmid},
    journal={arXiv preprint arXiv:2405.01366},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.01366},
    primaryClass={cs.DC}
}