The complexity of the reliable connectivity problem

Abstract

Let G=(V, E) be a graph together with two distinguished nodes s and t, and suppose that to every node v∈V, a nonnegative integer f(v)≤degree(v) is assigned. Suppose, moreover, that each node v can cause at most f(v) of its incident edges to “fail” (these f(v) edges can be arbitrarily chosen). The Reliable Connectivity Problem is to test whether node s remains connected with t with a path of non-failed edges for all possible choices of the failed edges. We first show that the complement of the Reliable Connectivity Problem is NP-complete and that this remains true even if G is restricted to the class of directed and acyclic graphs. However, we show that the problem is in P for directed and acyclic graphs if we assume that the edges caused to fail by each node v are chosen only from the edges incoming to v. Concerning the parallel complexity of this version of the problem, it turns out that it is P-complete. Moreover, approximating the maximum d such that for any choice of failed edges there is a directed path of non-failed edges that starts from s and has length d turns out to be P-complete as well, for any given degree of relative accuracy of the approximation. On the contrary, given that every node v will cause at least f(v) incoming edges to fail, the question whether there is a choice of failed edges such that s remains connected with t via non-failed edges turns out to be in NC, even for general graphs.

This research was partially supported by the ESPRIT II Basic Research Actions Program of the EC under contract no. 3075 (project ALCOM).