Efficient Classification of Locally Checkable Problems in Regular Trees

We give practical, efficient algorithms that automatically determine the asymptotic distributed round complexity of a given locally checkable graph problem in the [Θ(log n ) , Θ( n )] region, in two settings. We present one algorithm for unrooted regular trees and another algorithm for rooted regular trees. The algorithms take the description of a locally checkable labeling problem as input, and the running time is polynomial in the size of the problem description. The algorithms decide if the problem is solvable in O (log n ) rounds. If not, it is known that the complexity has to be Θ( n 1 /k ) for some k = 1 , 2 , . . . , and in this case the algorithms also output the right value of the exponent k . In rooted trees in the O (log n ) case we can then further determine the exact complexity class by using algorithms from prior work; for unrooted trees the more fine-grained classification in the O (log n ) region remains an open question.