Bounding the Number of Tolerable Faults in Majority-Based Systems

Abstract

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 $\lfloor |V|/2 \rfloor$ are given on the minimum sizes of irreversible dynamos depending on whether the majority is strict or simple. Let ε> 0 be any constant. We also show that, unless $\text{NP}\subseteq \text{TIME}(n^{O(\ln \ln n)}),$ no polynomial-time, ((1/2 − ε)ln |V|)-approximation algorithms exist for finding a minimum irreversible dynamo.

The authors are supported in part by the National Science Council of Taiwan under grant 97-2221-E-002-096-MY3 and Excellent Research Projects of National Taiwan University under grant 98R0062-05.