Year: 2017
Type: preprint
Abstract: LCLs or locally checkable labelling problems (e.g. maximal independent set, maximal matching, and vertex colouring) in the LOCAL model of computation are very well-understood in cycles (toroidal 1-dimensional grids): every problem has a complexity of $O(1)$, $Θ(\log^* n)$, or $Θ(n)$, and the design of optimal algorithms can be fully automated. This work develops the complexity theory of LCL problems for toroidal 2-dimensional grids. The complexity classes are the same as in the 1-dimensional case: $O(1)$, $Θ(\log^* n)$, and $Θ(n)$. However, given an LCL problem it is undecidable whether its complexity is $Θ(\log^* n)$ or $Θ(n)$ in 2-dimensional grids. Nevertheless, if we correctly guess that the complexity of a problem is $Θ(\log^* n)$, we can completely automate the design of optimal algorithms. For any problem we can find an algorithm that is of a normal form $A' \circ S_k$, where $A'$ is a finite function, $S_k$ is an algorithm for finding a maximal independent set in $k$th power of the grid, and $k$ is a constant. Finally, partially with the help of automated design tools, we classify the complexity of several concrete LCL problems related to colourings and orientations.
Source: arXiv (Cornell University)
Authors: Brandt, Sebastian, Hirvonen, Juho, Korhonen, Janne H., Lempiäinen, Tuomo, Östergård, Patric R. J., Purcell, Christopher, Rybicki, Joel, Suomela, Jukka, Uznański, Przemysław
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2950527429