On Finding Minimum Deadly Sets for Directed Networks

Given a set S of elements in a directed network that are initially faulty, an element becomes (functionally) faulty if all its in-neighbors or all its outneighbors are (functionally) faulty. A set S of initially faulty elements is called deadly if it causes the entire network to become faulty according to the above rule. We show that finding a minimum deadly set is NP-hard for arbitrary directed networks. For directed acyclic graphs (DAGs), we show that finding a weighted minimum deadly set is no harder than finding a minimum cut. We also study the case where a vertex becomes faulty if at least a certain percentage of its in-neighbors or out-neighbors is faulty. We call a set S of initially faulty elements ε-deadly if it causes the whole network to become faulty using this ε-majority rule. We show that finding a minimum ε-deadly set is NP-hard even for a restricted subclass of directed acyclic graphs.