Completing the Node-Averaged Complexity Landscape of LCLs on Trees
Authors: 
Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Dennis Olivetti, Gustav SchmidAuthors Info & Claims
Pages 369 - 379
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 no(1). For randomized algorithms, they even showed that the node-averaged complexity is either O(1) or nΩ(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 < r1 < r2 ≤ 1 and ε > 0, there exists a constant c and an LCL problem with node-averaged complexity between Ω((log* n)c) and O((log* n)c+ε).
• For any constants 0 < α ≤ 1/2 and ε > 0, there exists an LCL problem with node-averaged complexity Θ(nx) for some x ∈ [α, α + ε].
References
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- Completing the Node-Averaged Complexity Landscape of LCLs on Trees
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Published In
June 2024
570 pages
ISBN:9798400706684
DOI:10.1145/3662158
- Chair:
- Ran Gelles,
- Proceedings Chair:
- Dennis Olivetti,
- Program Chair:
- Petr Kuznetsov
This work is licensed under a Creative Commons Attribution-NonCommercial International 4.0 License.
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Association for Computing Machinery
New York, NY, United States
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Published: 17 June 2024
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