Bounding the Number of Tolerable Faults in Majority-Based Systems

Consider the following coloring process in a simple directed graph G(V,E) with positive indegrees. Initially, a set S of vertices are white. Thereafter, a black vertex is colored white whenever the majority of its in-neighbors are white. The coloring process ends when no additional vertices can be colored white. If all vertices end up white, we call S an irreversible dynamic monopoly (or dynamo for short). We derive upper bounds of 0.7732|V | and 0.727|V | on the minimum sizes of irreversible dynamos depending on whether the majority is strict or simple. When G is an undirected connected graph without isolated vertices, upper bounds of |V |/2 and |V |/2 are given on the minimum sizes of irreversible dynamos depending on whether the majority is strict or simple. Let > 0 be any constant. We also show that, unless NP ⊆ TIME(n ), no polynomial-time, ((1/2− ) ln |V |)-approximation algorithms exist for finding a minimum irreversible dynamo.