Hardness of minimal symmetry breaking in distributed computing
Abstract
A graph is weakly $2$colored if the nodes are labeled with colors black and white such that each black node is adjacent to at least one white node and vice versa. In this work we study the distributed computational complexity of weak $2$coloring in the standard LOCAL model of distributed computing, and how it is related to the distributed computational complexity of other graph problems. First, we show that weak $2$coloring is a minimal distributed symmetrybreaking problem for regular evendegree trees and highgirth graphs: if there is any nontrivial locally checkable labeling problem that is solvable in $o(\log^* n)$ rounds with a distributed graph algorithm in the middle of a regular evendegree tree, then weak $2$coloring is also solvable in $o(\log^* n)$ rounds there. Second, we prove a tight lower bound of $\Omega(\log^* n)$ for the distributed computational complexity of weak $2$coloring in regular trees; previously only a lower bound of $\Omega(\log \log^* n)$ was known. By minimality, the same lower bound holds for any nontrivial locally checkable problem inside regular evendegree trees.
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 DOI:
 10.48550/arXiv.1811.01643
 arXiv:
 arXiv:1811.01643
 Bibcode:
 2018arXiv181101643B
 Keywords:

 Computer Science  Distributed;
 Parallel;
 and Cluster Computing;
 Computer Science  Computational Complexity