Let G be a graph and K be a set of target vertices of G. Assume that all vertices of G, except the vertices in K, may fail with given probabilities. The K-terminal reliability of G is the probability that all vertices in K are mutually connected. This reliability problem is known to be #P-complete for general graphs. This work develops the first polynomial-time algorithm for computing the K-terminal reliability of circular-arc graphs.

Let G be a probabilistic graph, in which the vertices fail independently with known probabilities. Let K represent a specified subset of vertices. The K-terminal reliability of G is defined as the probability that all vertices in K are connected. When |K|=2, the K-terminal reliability is called the 2-terminal reliability, which is the probability that the source vertex is connected to the destination vertex. The problems of computing K-terminal reliability and 2-terminal reliability have been proven to be #P-complete in general. This work demonstrates that on multi-tolerance graphs, the 2-terminal reliability problem can be solved in polynomial-time and the results can be extended to the K-terminal reliability problem on bounded multi-tolerance graphs.