A Framework for Network Reliability Problems on Graphs of Bounded Treewidth

In this paper, we consider problems related to the network reliability problem, restricted to graphs of bounded treewidth. We look at undirected simple graphs with each vertex and edge a number in [0, 1] associated. These graphs model networks in which sites and links can fail, with a given probability, independently of whether other sites or links fail or not. The number in [0, 1] associated to each element is the probability that this element does not fail. In addition, there are distinguished sets of vertices: a set S of servers, and a set L of clients. This paper presents a dynamic programming framework for graphs of bounded treewidth for computing for a large number of different properties Y whether Y holds for the graph formed by the nodes and edges that did not fail. For instance, it is shown that one can compute in linear time the probability that all clients are connected to at least one server, assuming the treewidth of the input graph is bounded. The classical Sterminal reliability problem can be solved in linear time as well using this framework. The method is applicable to a large number of related questions. Depending on the particular problem, the algorithm obtained by the method uses linear, polynomial, or exponential time.