The complexity of the reliable connectivity problem
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.
KeywordsDirected Path Directed Acyclic Graph Undirected Graph Incident Edge Failure Pattern
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