Steiner trees in probabilistic networks

Various network reliability problems are #P-complete, however, certain classes of networks such as series-parallel networks, admit polynomial time algorithms. We extend these efficient methods to a superclass of series-parallel networks, the partial 2-trees. In fact, we solve a more general problem: given a probabilistic partial 2-tree and a set T of target nodes, we compute in linear time the probability of obtaining a subgraph connecting all of the target nodes. Equivalently, this is the probability of obtaining a Steiner tree for T. The algorithm exploits a characterization of partial 2-trees as graphs with no subgraph homeomorphic to K₄.