LCL problems on grids

TL;DR

This paper extends the complexity theory of locally checkable labelling problems from 1D to 2D grids, revealing undecidability issues but enabling automated algorithm design for certain complexity classes.

Contribution

It develops the complexity classification for LCL problems on 2D grids, showing similarities to 1D cases and enabling automated algorithm synthesis under certain assumptions.

Findings

Complexity classes in 2D grids mirror 1D: O(1), log* n, and n.

Decidability issues arise in distinguishing log* n and n complexities.

Automated algorithms of a specific normal form can be constructed for problems with log* n complexity.

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 $O(1)$, $\Theta(\log^* n)$, 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. Finally, partially with the help of automated design tools, we classify the complexity of several concrete LCL problems related to colourings and orientations.