Locally Checkable Problems in Rooted Trees

Consider any locally checkable labeling problem $Pi$ in rooted regular trees: there is a finite set of labels $Sigma$, and for each label $x in Sigma$ we specify what are permitted label combinations of the children for an internal node of label $x$ (the leaf nodes are unconstrained). This formalism is expressive enough to capture many classic problems studied in distributed computing, including vertex coloring, edge coloring, and maximal independent set. We show that the distributed computational complexity of any such problem $Pi$ falls in one of the following classes: it is $O(1)$, $Theta(log^* n)$, $Theta(log n)$, or $n^{Theta(1)}$ rounds in trees with $n$ nodes (and all of these classes are nonempty). We show that the complexity of any given problem is the same in all four standard models of distributed graph algorithms: deterministic $mathsf{LOCAL}$, randomized $mathsf{LOCAL}$, deterministic $mathsf{CONGEST}$, and randomized $mathsf{CONGEST}$ model. In particular, we show that randomness does not help in this setting, and the complexity class $Theta(log log n)$ does not exist (while it does exist in the broader setting of general trees). We also show how to systematically determine the complexity class of any such problem $Pi$, i.e., whether $Pi$ takes $O(1)$, $Theta(log^* n)$, $Theta(log n)$, or $n^{Theta(1)}$ rounds. While the algorithm may take exponential time in the size of the description of $Pi$, it is nevertheless practical: we provide a freely available implementation of the classifier algorithm, and it is fast enough to classify many problems of interest.