Completing the Node-Averaged Complexity Landscape of LCLs on Trees
arxiv(2024)
摘要
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 00, 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(n^x) for some x∈ [α,
α+ε].
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