Abstract
We present a complete classification of the deterministic distributed time complexity for a family of graph problems: binary labeling problems in trees. These are locally checkable problems that can be encoded with an alphabet of size two in the edge labeling formalism. Examples of binary labeling problems include sinkless orientation, sinkless and sourceless orientation, 2vertex coloring, perfect matching, and the task of coloring edges red and blue such that all nodes are incident to at least one red and at least one blue edge. More generally, we can encode e.g. any cardinality constraints on indegrees and outdegrees.
We study the deterministic time complexity of solving a given binary labeling problem in trees, in the usual LOCAL model of distributed computing. We show that the complexity of any such problem is in one of the following classes: O(1), Θ(log n), Θ(n), or unsolvable. In particular, a problem that can be represented in the binary labeling formalism cannot have time complexity Θ(log^* n), and hence we know that e.g. any encoding of maximal matchings has to use at least three labels (which is tight).
Furthermore, given the description of any binary labeling problem, we can easily determine in which of the four classes it is and what is an asymptotically optimal algorithm for solving it. Hence the distributed time complexity of binary labeling problems is decidable, not only in principle, but also in practice: there is a simple and efficient algorithm that takes the description of a binary labeling problem and outputs its distributed time complexity.
BibTeX  Entry
@InProceedings{balliu_et_al:LIPIcs:2020:13095,
author = {Alkida Balliu and Sebastian Brandt and Yuval Efron and Juho Hirvonen and Yannic Maus and Dennis Olivetti and Jukka Suomela},
title = {{Classification of Distributed Binary Labeling Problems}},
booktitle = {34th International Symposium on Distributed Computing (DISC 2020)},
pages = {17:117:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771689},
ISSN = {18688969},
year = {2020},
volume = {179},
editor = {Hagit Attiya},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/13095},
URN = {urn:nbn:de:0030drops130957},
doi = {10.4230/LIPIcs.DISC.2020.17},
annote = {Keywords: LOCAL model, graph problems, locally checkable labeling problems, distributed computational complexity}
}
Keywords: 

LOCAL model, graph problems, locally checkable labeling problems, distributed computational complexity 
Collection: 

34th International Symposium on Distributed Computing (DISC 2020) 
Issue Date: 

2020 
Date of publication: 

07.10.2020 