Completing the Node-Averaged Complexity Landscape of LCLs on Trees

TL;DR

This work delivers a complete description of the node-averaged complexity landscape for LCLs on bounded-degree trees by introducing weighted and hierarchical problem families, showing infinite density in both polynomial and log^* n regimes, and proving decisive gaps. The authors develop a suite of techniques—weighted colorings, k-hierarchical constructions, and fast decomposition-based solvers—to tightly bound deterministic node-averaged times and to interpolate between worst-case and average performance. They prove that there are no intermediate deterministic node-averaged complexities between constant and sublogarithmic regimes, while demonstrating dense, controllable regions of complexity in polynomial and log^* n regimes, including Θ(n^x) for a wide range of x and Θ((log^* n)^{α}) classes. A central methodological pillar is the decomposition-based framework and the notion of label-sets and classes in the black-white formalism, which enable both upper and lower bound arguments and decidability statements about the existence of constant-time node-averaged algorithms.

Abstract

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 $ω(\log^* n)$ and $n^{o(1)}$. For randomized algorithms, they even showed that the node-averaged complexity is either $O(1)$ or $n^{Ω(1)}$. 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 $ω(1)$ 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 $Ω((\log^* n)^c)$ and $O((\log^* n)^{c+\varepsilon})$. - For any constants $0<α\leq 1/2$ and $\varepsilon>0$, there exists an LCL problem with node-averaged complexity $Θ(n^x)$ for some $x\in [α, α+\varepsilon]$.

Figures (6)

• Figure 1: Everything that was known about the Node-Averaged Complexity Landscape before the results in this paper.
• Figure 2: Everything that is known about the node-averaged complexity landscape after including the results in this paper. The gap between constant and $\log^*(n)^{o(1)}$ is due to \ref{['thm:lowgap']}. The regions of infinite density, represented by red bars, are a result of \ref{['thm:LowDensity', 'thm:UpperDensity']}. The gap between $\omega(\sqrt{n})$ and $o(n)$ is due to \ref{['cor:linearGap']}.
• Figure 3: The figure illustrates the lower bound graph for $2$-hierarchical $3\frac{1}{2}$-coloring. All nodes of level 1 are colored green and all nodes of level 2 are colored red. The nodes in level 1 form paths of length $\sqrt{\log^*n}$ (and $\sqrt{\log^*n}+1$ for the left and rightmost). The level 2 nodes are just one long path of length $\frac{n}{\sqrt{\log^*n}} - 2$
• Figure 4: Showing the construction for $k=3$. The red nodes have level 3, the green nodes level 2 and the purple nodes have level 1 (Some purple paths from the green nodes in the middle are omitted). No trees are attached to level 1 nodes, but every level 2, or 3 node has a tree attached to it. The trees attached to level 3 nodes are larger, as the same number of $\frac{n}{k}$ nodes is evenly distributed among less nodes.
• Figure 5: This figure from fullversion illustrates the way edges are oriented in the Fast Decomposition Algorithm. The two black nodes at the top are nodes that are not yet assigned a layer. The arrows in the tree show how edges are oriented when such a tree is raked away and the bold path in the middle represents a compress path. In such a compress path, only the first and last $\ell$ edges are oriented.

Theorems & Definitions (130)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Definition 8: $k$-hierarchical $2\frac{1}{2}$-coloring
• Definition 9: $k$-hierarchical $3\frac{1}{2}$-coloring
• Corollary 10