LCL problems on grids

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 $O(1)$, $\Theta(\log^* n)$, or $\Theta(n)$, 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: $O(1)$, $\Theta(\log^* n)$, and $\Theta(n)$. However, given an LCL problem it is undecidable whether its complexity is $\Theta(\log^* n)$ or $\Theta(n)$ in 2-dimensional grids. Nevertheless, if we correctly guess that the complexity of a problem is $\Theta(\log^* n)$, 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. With the help of this technique, we study several concrete \lcl{} problems, also in more general settings. For example, for all $d \ge 2$, we prove that: - $d$-dimensional grids can be $k$-vertex coloured in time $O(\log^* n)$ iff $k \ge 4$, - $d$-dimensional grids can be $k$-edge coloured in time $O(\log^* n)$ iff $k \ge 2d+1$. The proof that $3$-colouring of $2$-dimensional grids requires $\Theta(n)$ time introduces a new topological proof technique, which can also be applied to e.g. orientation problems.