A Survey of Network Reliability and Domination Theory

We present a brief survey of the current state of the art in network reliability. We survey only exact methods and do not consider Monte Carlo methods. Most network reliability problems are, in the worst case, NP-hard and are, in a sense, more difficult than many standard combinatorial optimization problems. Nevertheless, there are, in fact, linear and polynomial time algorithms for network reliability problems of special structure. We review general methods for network reliability computation and discuss the central role played by domination theory in network reliability computational complexity. We also point out the connection with the more general problem of computing the reliability of coherent structures. The class of coherent structures contains both directed and undirected networks as well as logic (or fault) trees without not gates. This topic is a rich area for further research.