On irreversible dynamic monopolies in general graphs

Consider the following coloring process in a simple directed graph G(V, E) with positive indegrees. Initially, a set S of vertices are white, whereas all the others are black. Thereafter, a black vertex is colored white whenever more than half 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) under the strict-majority scenario. An irreversible dynamo under the simple-majority scenario is defined similarly except that a black vertex is colored white when at least half of its in-neighbors are white. We derive upper bounds of (2/3) | V | and | V |/2 on the minimum sizes of irreversible dynamos under the strict and the simple-majority scenarios , respectively. For the special case when G is an undirected connected graph, we prove the existence of an irreversible dynamo with size at most ⌈| V |/2⌉ under the strict-majority scenario. Let ǫ > 0 be any constant. We also show that, unless NP ⊆ TIME(n O(ln ln n)), no polynomial-time, ((1/2 − ǫ) ln | V |)-approximation algorithms exist for finding the minimum irreversible dynamo under either the strict or the simple-majority scenario. The inapproximability results hold even for bipartite graphs with diameter at most 8.