# 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.

## Keywords

Directed Path Directed Acyclic Graph Undirected Graph Incident Edge Failure Pattern## Preview

Unable to display preview. Download preview PDF.

## References

- R. Anderson and E.W. Mayr [1984]
*A P-Complete Problem and Approximations to It*, Technical Report, Computer Science Dept., Stanford University, California.Google Scholar - M.O. Ball [1977],
*Network Reliability and Analysis: Algorithms and Complexity*, Doctoral Thesis, Operations Research Dept., Cornell University, Ithaca, New York.Google Scholar - L.M. Kirousis, M. Serna and P. Spirakis [1989], The parallel complexity of the subgraph connectivity problem,
*Proceedings of the 30th Annual Symposium on Foundations of Computer Science*, (IEEE Computer Society Press), 294–299.Google Scholar - A. Rosenthal [1974],
*Computing Reliability of Complex Systems*, Doctoral Thesis, Dept. of Electrical Engineering and Computer Science, University of California, Berkeley, California.Google Scholar - M. Serna and P. Spirakis [1991], “Tight RNC approximations to max flow,” to appear in:
*Proceedings of the 8th Symposium on Theoretical Aspects of Computer Science*(Lecture Notes in Computer Science, Springer-Verlag).Google Scholar