Efficient Classification of Locally Checkable Problems in Regular Trees

TL;DR

This paper presents practical algorithms to automatically determine the asymptotic distributed round complexity of locally checkable problems in regular trees, distinguishing between $O( ext{log} n)$ and polynomial complexities.

Contribution

It introduces two algorithms for unrooted and rooted regular trees that classify the complexity of locally checkable problems in the $[ ext{log} n, n]$ range, including exact exponents.

Findings

Algorithms decide $O( ext{log} n)$ solvability.

Algorithms output the exponent $k$ for $ heta(n^{1/k})$ complexity.

Exact classification in rooted trees is achieved; unrooted trees remain open.

Abstract

We give practical, efficient algorithms that automatically determine the asymptotic distributed round complexity of a given locally checkable graph problem in the $[\Theta(\log n), \Theta(n)]$ 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 $\Theta(n^{1/k})$ for some $k = 1, 2, \dotsc$, 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.