Abstract
In the deterministic binary majority process we are given a simple graph where each node has one out of two initial opinions. In every round, each node adopts the majority opinion among its neighbors. It is known that this process always converges in O(E) rounds to a twoperiodic state in which every node either keeps its opinion or changes it in every round.
It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the O(E) bound on the convergence time of the deterministic binary majority process is even for dense graphs tight. However, in many graphs such as the complete graph the process converges in just
a constant number of rounds from any initial opinion assignment.
We show that it is NPhard to decide whether there exists an initial opinion assignment for which it takes more than k rounds to converge to the twoperiodic stable state, for a given integer k. We then give a new upper bound on the voting time of the deterministic binary majority process. Our bound can be computed in linear time by carefully exploiting the structure of the potential function by Goles and Olivos. We identify certain modules of a graph G to obtain a new graph G^Delta. This new graph G^Delta has the property that the worstcase convergence time of G^Delta is an upper bound on that of G. Our new bounds asymptotically improve the best known bounds for various graph classes.
BibTeX  Entry
@InProceedings{kaaser_et_al:LIPIcs:2016:6467,
author = {Dominik Kaaser and Frederik MallmannTrenn and Emanuele Natale},
title = {{On the Voting Time of the Deterministic Majority Process}},
booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
pages = {55:155:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770163},
ISSN = {18688969},
year = {2016},
volume = {58},
editor = {Piotr Faliszewski and Anca Muscholl and Rolf Niedermeier},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6467},
URN = {urn:nbn:de:0030drops64675},
doi = {10.4230/LIPIcs.MFCS.2016.55},
annote = {Keywords: distributed voting, majority rule}
}
Keywords: 

distributed voting, majority rule 
Collection: 

41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016) 
Issue Date: 

2016 
Date of publication: 

19.08.2016 