On the inapproximability of minimizing cascading failures under the deterministic threshold model

Given a directed graph G and a threshold L(r) for each node r, the rule of deterministic threshold cascading is that a node r fails if and only if it has at least L(r) failed in-neighbors. The cascading failure minimization problem is to find at most k edges to delete, such that the number of failed nodes is minimized. We prove an n1−ϵ inapproximability result for the general case and a 12nϵ inapproximability result for the special case with the maximum threshold of 1.